Curvature inequalities and rigidity for constant mean curvature and spacetime constant mean curvature surfaces

CUR V A TURE INEQUALITIES AND RIGIDITY F OR CONST ANT MEAN CUR V A TURE AND SP A CETIME CONST ANT MEAN CUR V A TURE SURF A CES ALEJANDR O PEÑUELA DIAZ Abstract. W e establish curv ature inequalities and rigidit y results for surfaces satisfying constan t mean curv ature type conditions in b oth Riemannian and Loren tzian geometry . In the Riemannian setting we study constant mean curv ature (CMC) surfaces in three-dimensional manifolds with scalar curv ature b ounds. Building on the Christo doulou–Y au inequality H 2 ≤ 16 π / | Σ | (with H the mean curv ature and | Σ | the area), w e show that the associated rigidity phenomena p ersist under a weak er notion of stability con trolling only the constant mo de of the second v ariation, combined with an extrinsic curv ature sign condition. This yields Euclidean rigidit y without imp osing in trinsic symmetry or near-roundness assumptions and extends to higher dimensions and to the hyperb olic and spherical settings. In the Lorentzian setting w e consider spacetime constant mean curv ature (STCMC) sur- faces, a natural generalization of CMC surfaces. W e introduce a stabilit y theory for STCMC surfaces and pro ve the sharp inequality |  H | 2 ≤ 16 π / | Σ | under the dominan t energy condition. W e also obtain rigidity for the equality case: under suitable geometric assumptions the sur- face is in trinsically round and the spacetime region it b ounds is ﬂat, with maximal globally h yp erb olic developmen t isometric to a causal diamond in Minko wski spacetime. Finally , we sho w that the canonical asymptotic STCMC foliations kno wn in b oth the spacelik e and null settings hav e leav es that are stable with resp ect to this notion of stability . 1. Intr oduction and Resul ts The study of constan t mean curv ature (CMC) surfaces as critical p oints of the area functional under a volume constraint is a cen tral theme in diﬀerential geometry . Bey ond their role as isop erimetric surfaces, stable CMC surfaces satisfy fundamen tal geometric inequalities that link the curv ature of the ambien t manifold to intrinsic prop erties of the surface. A landmark result in this direction w as established by Christo doulou and Y au [11], who pro v ed that if ( M , g ) is a three-dimensional Riemannian manifold with nonnegativ e scalar curv ature and Σ ⊂ M is a stable CMC surface, then (1) H 2 ≤ 16 π | Σ | , where H denotes the mean curv ature of Σ and | Σ | its area. Equalit y is attained b y round spheres in Euclidean space. This naturally leads to the corresp onding rigidity question: if equalit y holds, m ust the enclosed region b e isometric to a Euclidean ball? Rigidit y phenomena under scalar curv ature low er b ounds hav e b een extensiv ely studied in Riemannian geometry . In man y cases, the theory of stable minimal and constan t mean cur- v ature hypersurfaces plays a central role, as the stability inequalit y relates ambien t scalar curv ature to intrinsic geometric quantities. F or a general o v erview of such rigidity results and their geometric con text, see the survey of Brendle [4]. In the sp eciﬁc setting of inequality (1), rigidity results hav e b een obtained under additional geometric assumptions. In particular, Sun [35] sho wed that if a stable CMC sphere satisﬁes 1 2 equalit y and is suﬃcien tly close to b eing round, then the surface is isometric to a round sphere and the enclosed region is isometric to a Euclidean ball. These results demonstrate that equality in (1) enco des strong geometric information, though existing approac hes rely on in trinsic near-roundness or symmetry assumptions [32, 35]. Note that the inequalit y (1) w as originally motiv ated by general relativit y , as it ensures the nonnegativity of the Ha wking quasi- lo cal energy , while the equality case pro vides a rigidity statement for the Ha wking quasi-lo cal energy within totally geo desic h yp ersurfaces. The ﬁrst goal of this pap er is to reﬁne and generalize these results. W e sho w that Euclidean rigidit y can b e reco v ered under a purely extrinsic curv ature sign condition com bined with a w eak stabilit y hypothesis, namely , a condition controlling only the constan t mo de of the second v ariation, without imp osing intrinsic symmetry or almost-roundness assumptions. Moreov er, w e establish corresp onding rigidit y results in the hyperb olic and spherical settings under ap- propriate scalar curv ature b ounds, as well as higher-dimensional analogues. In eac h case, the mec hanism underlying rigidity is the in teraction b etw een curv ature low er b ounds and a sharp b oundary inequality . The rigidity argumen ts ultimately rely on geometric rigidit y theorems for manifolds with b oundary , in particular the Brown–Y ork mass rigidity theorem of Shi and T am [33] and its extensions. W e then turn to the Loren tzian setting, considering a four-dimensional Lorentzian manifold ( M , g ) satisfying the dominant energy condition. F or a spacelik e tw o-surface Σ ⊂ M , the relev ant curv ature quan tity is the squared norm of the mean curv ature v ector, |  H | 2 = − θ ℓ θ k , where θ ℓ and θ k denote the n ull expansions of Σ . Surfaces with constan t |  H | — kno wn as spacetime constant mean curv ature (STCMC) surfaces — provide a natural analogue of CMC surfaces in Riemannian geometry . While STCMC surfaces ha ve app eared in several con texts in mathematical relativity , a stability theory for such surfaces has not previously b een developed. In this pap er, we in tro duce a natural notion of stability and sho w that it leads to a sharp curv ature inequalit y (2) − θ ℓ θ k = |  H | 2 ≤ 16 π | Σ | . Moreo v er, we establish a rigidit y theorem for the equality case: under a sp eciﬁc curv ature sign condition (or an assumption of even symmetry or intrinsic near-roundness), the surface Σ is in trinsically round. Under the additional assumption that it b ounds a spacelike region with no other b oundary comp onen ts, the enclosed region is ﬂat, and its maximal globally hyperb olic dev elopmen t is isometric to a standard causal diamond in Mink owski spacetime. The pro of of this rigidit y statemen t builds on a theorem of Liu and Y au [19, 20], together with an analysis of the maximal globally h yp erb olic dev elopment of the enclosed region. These results admit a natural interpretation in general relativity in terms of the Hawking quasi-lo cal energy (or quasi-lo cal mass) (3) E (Σ) = s | Σ | 16 π  1 + 1 16 π Z Σ θ ℓ θ k dµ  . Inequalit y (2) is precisely equiv alen t to the nonnegativit y of E (Σ) under the dominan t energy condition, and the equality case corresp onds to rigidity when the Hawking energy v anishes. Bey ond this energetic in terpretation, spacetime constan t mean curv ature surfaces pla y an imp ortan t role in the geometry of asymptotically ﬂat spacetimes. In particular, it has b een 3 sho wn that asymptotically ﬂat initial data sets admit a canonical foliation by STCMC surfaces near inﬁnit y [6], which pro vides a geometric c haracterization of the cen ter of mass of an isolated gra vitational system. These foliations serve as a natural Lorentzian analogue of the classical constan t mean curv ature foliations [16] used to deﬁne the center of mass in the time-symmetric setting. More recen tly , STCMC foliations hav e also been constructed on asymptotically Sch w arzsc hildean null h yp ersurfaces [17]. In the present w ork w e show that the leav es of these known asymptotic STCMC foliations are stable with resp ect to the stabilit y notion in tro duced here. Th us our results connect the p ositivit y and rigidit y prop erties of the Ha wking energy with the geometric structure of asymptotically ﬂat spacetimes, b oth in the spacelik e and n ull settings. These applications also sho w that the stability theory developed here is realized b y the natural STCMC surfaces arising in asymptotically ﬂat geometry . More broadly , they illustrate the geometric meaning of the Lorentzian stability theory and place it in direct analogy with the classical Riemannian theory of stable CMC surfaces. The dominant energy condition plays in the Lorentzian setting a role analogous to that of scalar curv ature low er b ounds in the Riemannian theory . In particular, inequality (2) can b e viewed as a spacetime counterpart of the Christo doulou-Y au inequality for stable CMC surfaces. This rev eals a direct parallel b etw een the tw o settings: curv ature conditions lead to sharp geometric inequalities for surfaces with constant mean curv ature and spacetime constant mean curv ature, and equality rigidly determines the geometry of the region they enclose. 1.1. Main results. W e ﬁrst presen t the results for CMC surfaces with a Euclidean rigidit y result. Theorem 2.3. L et ( M , g ) b e a 3 -dimensional R iemannian manifold with nonne gative sc alar curvatur e. L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either (i) The se c ond variation of ar e a satisﬁes δ 2 ν | Σ | = R Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ − 1 2 H 2 | Σ | , or (ii) Σ is top olo gic al ly a spher e and is variational ly stable. Then (4) H 2 ≤ 16 π | Σ | . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ( ν, ν ) do es not change sign along Σ . If e quality holds in (4), then Ω is isometric to a Euclide an b al l in R 3 . In p articular, Σ is isometric to a r ound spher e. This result under assumption ( i ) is generalized to higher dimensions in Theorem 2.5. The hyperb olic case: An analogous rigidit y statemen t holds when hyperb olic space serves as the reference geometry Theorem 2.6. L et ( M , g ) b e a 3 -dimensional Riemannian manifold with sc alar curvatur e Sc M ≥ 2Λ for a c onstant Λ ≤ 0 . L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either 4 (i) The se c ond variation of ar e a satisﬁes δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ −  1 2 H 2 + 2 3 Λ  | Σ | , or (ii) Σ is top olo gic al ly a spher e and is variational ly stable. Then (5) H 2 ≤ 16 π | Σ | − 4 3 Λ . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ( ν, ν ) − 2 3 Λ do es not change sign along Σ . If e quality holds in (5), then Σ is isometric to a r ound spher e and Ω is isometric to a ge o desic b al l in the hyp erb olic sp ac e of r adius 3 / Λ , H 3 Λ / 3 . The spherical case: W e consider also the case when the sphere serv es as the reference geometry . In contrast to the Euclidean and hyperb olic settings, rigidit y in the spherical case requires a Ricci curv ature lo wer b ound. Theorem 2.9. L et ( M , g ) b e a 3 -dimensional Riemannian manifold with sc alar curvatur e Sc M ≥ 2Λ for a c onstant Λ ≥ 0 . L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either (i) The se c ond variation of ar e a satisﬁes δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ −  1 2 H 2 + 2 3 Λ  | Σ | , or (ii) Σ is top olo gic al ly a spher e and is variational ly stable. Then (6) H 2 ≤ 16 π | Σ | − 4 3 Λ . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ≥ 2 3 Λ g on Ω . If e quality holds in (6), then Σ is total ly ge o desic ( B = 0 ), it is isometric to a r ound spher e and Ω is isometric to the hemispher e S 3 +  q | Σ | / (4 π )  . Just as in the Euclidean setting, this result under assumption ( i ) is generalized to higher dimensions in Theorem 2.10. Results for STCMC surfaces. W e establish the spacetime analogue of the ab o ve rigidity under the dominan t energy condition. Theorem 4.1. L et ( M , g ) b e a 4 -dimensional L or entzian manifold satisfying the dominant ener gy c ondition. L et Σ b e a close d sp ac elike STCMC surfac e with θ ℓ θ k ≤ 0 , such that either (i) Σ is c onstant-mo de stable ( δ 2  H | Σ | ≥ 0 ), or (ii) Σ is top olo gic al ly a spher e and variational ly stable in the sense of Deﬁnition 3.2. 5 Then (7) − θ ℓ θ k = |  H | 2 ≤ 16 π | Σ | . If e quality holds in (7) and Rm M ( k , , , k ) do es not change sign along Σ , then Σ is isometric to a r ound spher e, and any c omp act sp ac elike hyp ersurfac e Ω ⊂ M with b oundary ∂ Ω = Σ emb e ds isometric al ly as a sp ac elike hyp ersurfac e into Minkowski sp ac etime. F urthermor e, the maximal glob al ly hyp erb olic development of the induc e d initial data on Ω is isometric to a standar d c ausal diamond in Minkowski sp ac etime. An alternativ e rigidity criterion under symmetry or near-roundness assumptions is prov ed in Theorem 4.5. The stability hypotheses app earing in Theorem 4.1 are satisﬁed for the canonical STCMC foli- ations kno wn in b oth the spacelike and null settings. On asymptotically Euclidean initial data sets, w e show that the canonical STCMC foliation constructed b y Cederbaum and Sako vic h [6] has leav es that are b oth constant-mode stable and v ariationally stable, provided the am bient Einstein tensor satisﬁes | Ein ( ν σ , ν σ ) | = O ( r − 2 ) along the foliation; see Theorem 5.3. W e also pro v e that the asymptotic STCMC foliation of asymptotically Sc hw arzsc hildean lightcones constructed by Krönc ke and W olﬀ [17] has leav es that are b oth constant-mode stable and v ariationally stable; see Theorem 5.4. This pap er is organized as follo ws. Section 2 in tro duces a weak stability condition con trolling the constan t mo de of the second v ariation and uses it to reco ver Euclidean, hyperb olic, and spherical rigidit y under appropriate curv ature bounds. Section 3 dev elops the Lorentzian framew ork for spacelik e surfaces, in tro duces STCMC surfaces, and derives the associated stabilit y op erator. In Section 4 we pro v e the sharp curv ature inequalit y and rigidity theorem for stable STCMC surfaces under the dominan t energy condition. Finally , Section 5 presents examples and applications of the Lorentzian stability theory , including the stabilit y of the canonical STCMC foliation in asymptotically Euclidean initial data sets and of the asymptotic STCMC foliation on asymptotically Sc h warzsc hildean n ull hypersurfaces. 2. Const ant mean cur v a ture surf aces in Riemannian manif olds with scalar cur v a ture bounds W e recall imp ortan t prop erties of stable constan t mean curv ature (CMC) surfaces in a Rie- mannian manifold with nonnegative scalar curv ature, which can also b e viewed as a totally geo desic hypersurface in an ambien t Loren tzian manifold satisfying the dominant energy con- dition. These results also provide imp ortan t motiv ation for the STCMC theory developed here. A CMC surface is a critical point of the area functional under volume-preserving v ariations, that is, under v ariations whose normal sp eed α satisﬁes (8) Z Σ α dµ = 0 . In this setting, stability means that the second v ariation of area in the normal direction ν , δ 2 αν | Σ | satisﬁes δ 2 αν | Σ | = Z Σ  |∇ α | 2 − ( | B | 2 + Ric M ( ν, ν )) α 2  dµ ≥ 0 for all smo oth functions α satisfying (8). 6 Stable CMC surfaces ha v e several remarkable geometric prop erties. In particular, Christo doulou and Y au pro v ed that, in manifolds with nonnegativ e scalar curv ature, such surfaces satisfy a sharp curv ature inequalit y . Theorem 2.1 (Christo doulou-Y au [11]) . L et ( M , g ) b e a 3 -dimensional R iemannian manifold with nonne gative sc alar curvatur e. If Σ ⊂ M is a stable c onstant me an curvatur e surfac e, then H 2 ≤ 16 π | Σ | . Her e, stability is understo o d in the usual volume-pr eserving sense r e c al le d ab ove. In the time-symmetric setting of an initial data set in general relativity , the Hawking energy of a surface Σ reduces to E (Σ) = s | Σ | 16 π  1 − 1 16 π Z Σ H 2 dµ  . Th us, the Christo doulou-Y au inequality implies that the Hawking energy is nonnegative for stable CMC surfaces. Related p ositivity results for the Ha wking energy of CMC surfaces under h yp otheses diﬀeren t from stabilit y w ere obtained in [27]. 2.1. The Euclidean case: The inequalit y in Theorem 2.1 identiﬁes Euclidean space as the mo del geometry in the regime of nonnegativ e scalar curv ature. Indeed, equality in the Hawking energy b ound is achiev ed by round spheres in R 3 . It is therefore natural to ask under what additional conditions equality forces the enclosed region to b e isometric to a Euclidean ball. W e b egin by recalling the follo wing rigidity result. Theorem 2.2 ([32, 35, Theorem 2, Theorem 1]) . L et ( M , g ) b e a 3 -dimensional Riemannian manifold with nonne gative sc alar curvatur e, and let Ω ⊂ M b e a r elatively c omp act domain with smo oth b oundary Σ = ∂ Ω . A ssume Σ is a stable c onstant me an curvatur e spher e satisfying H 2 = 16 π | Σ | . If either (i) Σ has even symmetry, i.e. ther e exist an isometry ρ : Σ → Σ with ρ 2 = id and ρ ( x )  = x for x ∈ Σ , or (ii) its Gauss curvatur e K Σ is C 0 -close to 4 π | Σ | , i.e. | K Σ − 4 π | Σ | | C 0 < δ 0 for some δ 0 ≪ 1 . Then Ω is isometric to a Euclide an b al l in R 3 . In p articular, Σ is isometric to the r ound spher e in R 3 . The rigidity theorem just recalled relies on auxiliary assumptions b ey ond the equality , namely either the existence of a ﬁxed-p oin t-free isometry (ev en symmetry) or the requiremen t that the Gauss curv ature b e suﬃciently close to a constant. These are essentially in trinsic condi- tions on the geometry of Σ . By con trast, the follo wing theorem sho ws that Euclidean rigidity also follo ws from a purely extrinsic curv ature sign condition, provided the surface satisﬁes one of t w o v ariational criteria. The ﬁrst of these, condition ( i ) , is a "w eak stability" assumption that prescrib es a lo w er b ound for the second v ariation of area sp eciﬁcally for constan t normal v ariations. This condition is notably less restrictiv e than full v ariational stability , as it only con trols the "constan t mo de" rather than the entire sp ectrum of the Jacobi op erator. Specif- ically , w e denote b y δ 2 αν | Σ | the second v ariation of area in the direction of the normal v ector ﬁeld with sp eed α ; condition ( i ) concerns the case where α ≡ 1 is a constan t. 7 Theorem 2.3. L et ( M , g ) b e a 3 -dimensional Riemannian manifold with nonne gative sc alar curvatur e. L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either (i) The second v ariation of area satisﬁes δ 2 ν | Σ | = R Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ − 1 2 H 2 | Σ | , or (ii) Σ is top ologically a sphere and is v ariationally stable. Then (9) H 2 ≤ 16 π | Σ | . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ( ν, ν ) do es not change sign along Σ . If e quality holds in (9), then Ω is isometric to a Euclide an b al l in R 3 . In p articular, Σ is isometric to a r ound spher e. Pr o of. W e ﬁrst show that, under either assumption, the inequality (9) holds, and that equalit y implies Sc Σ = 1 2 H 2 . ( i ) Note that the condition on the second v ariation reduces to (10) Z Σ | ˚ B | 2 + Ric M ( ν, ν ) dµ ≤ 0 Recall the Gauss equation (11) Sc Σ = Sc M − 2Ric M ( ν, ν ) + 1 2 H 2 − | ˚ B | 2 , where ˚ B is the tracefree second fundamental form. Then in tegrating the equation and using (10) (12) 1 2 Z Σ H 2 dµ = Z Σ Sc Σ dµ − Z Σ Sc M dµ + 2 Z Σ Ric M ( ν, ν ) + | ˚ B | 2 dµ − Z Σ | ˚ B | 2 dµ ≤ 8 π where we also used Sc M ≥ 0 , and that R Σ Sc Σ dµ ≤ 8 π b y Gauss Bonnet theorem. No w if 1 2 R Σ H 2 dµ = 8 π . Then R Σ 1 2 Sc Σ dµ = 4 π , Σ is a top ological sphere and | ˚ B | 2 = Sc M = 0 . Now b y (10) Z Σ Ric M ( ν, ν ) dµ = 0 and since Ric M ( ν, ν ) do es not change sign we obtain Ric M ( ν, ν ) = 0 on Σ . This implies that Sc Σ = 1 2 H 2 . ( ii ) First note that b ecause of Theorem 2.1, H 2 ≤ 16 π | Σ | . Now for the rigidit y results, note that b y the uniformization theorem, Σ is conformally equiv alen t to the round sphere. Moreov er, b y Hersch’s lemma [15] (see also [18]), there exists a conformal map ϕ : Σ → S 2 ⊂ R 3 suc h that Z Σ ϕ dµ = 0 . 8 In particular, each co ordinate function ϕ i has zero mean and is an admissible test function in the stability inequalit y for CMC surfaces. (13) Z Σ |∇ ϕ i | 2 dµ ≥ Z Σ  | B | 2 + Ric M ( ν, ν )  ϕ 2 i dµ. F or a surface conformal to S 2 ⊂ R 3 , we ha v e (14) 3 X i =1 Z Σ |∇ ϕ i | 2 dµ = 3 X i =1 Z S 2 |∇ x i | 2 dµ S 2 = 3 X i =1 − Z S 2 x i ∆ x i dµ S 2 = 3 X i =1 2 Z S 2 x 2 i dµ S 2 = 8 π . Then since eac h ϕ i is in the sphere they satisfy P 3 i =1 ϕ 2 i = 1 then adding we obtain (15) 8 π ≥ Z Σ  | B | 2 + Ric( ν, ν )  dµ = 1 2 Z Σ H 2 dµ + Z Σ  | ˚ B | 2 + Ric M ( ν, ν )  dµ. Then we obtain the inequality (10) obtained under the assumption of ( i ) (16) Z Σ | ˚ B | 2 + Ric M ( ν, ν ) dµ ≤ 0 then just as b efore, w e obtain: Sc Σ = 1 2 H 2 and since H is constan t Sc Σ is also a p ositiv e constan t with Sc Σ = 1 2 H 2 , in particular Sc Σ = 2 r 2 where r is the area radius of Σ . Now with this, w e can apply the rigidity result of Theorem A.1 (Brown-Y ork mass). Since the Gauss curv ature is constant and p ositive, Σ is isometric to the round sphere of radius r , hence its isometric em b edding into R 3 is the standard sphere and H 0 = 2 r . Then since also H = 2 r b y Theorem A.1 we ha ve our result. □ Remark 2.4. The same conclusion holds if ∂ Ω has ﬁnitely man y connected comp onents and one comp onent Σ satisﬁes the h yp otheses of Theorem 2.3. Assume that all other b oundary comp onen ts hav e p ositive Gaussian curv ature and p ositive mean curv ature with resp ect to the outw ard normal. Then Theorem A.1 implies that equalit y in (87) can o ccur only if ∂ Ω is connected. Consequently , Ω is isometric to a Euclidean ball. W e can generalize this result to higher dimensions under assumption ( i ) . Theorem 2.5. L et ( M , g ) b e a R iemannian manifold of dimension n ≥ 3 with nonne gative sc alar curvatur e. L et Σ ⊂ M b e a close d hyp ersurfac e of c onstant me an curvatur e H ≥ 0 , such that its se c ond variation of ar e a satisﬁes (17) δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ − H 2 n − 1 | Σ | . Then (18) H 2 ≤ ( n − 1) R Σ Sc Σ dµ ( n − 2) | Σ | . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M with outwar d unit normal ν , and that Ric M ( ν, ν ) do es not change sign along Σ . If Σ admits an isometric emb e dding into R n as a c onvex hyp ersurfac e and e quality holds in (18) , then Ω is isometric to a Euclide an b al l in R n . In p articular, Σ is isometric to a r ound spher e. 9 Pr o of. Note that the condition on the second v ariation reduces to (19) Z Σ  | ˚ B | 2 + Ric M ( ν, ν )  dµ ≤ 0 . In our case, the Gauss equation is (20) Sc Σ = Sc M − 2Ric M ( ν, ν ) + H 2 − | B | 2 = Sc M − 2Ric M ( ν, ν ) + n − 2 n − 1 H 2 − | ˚ B | 2 . Isolating the mean curv ature term, integrating o v er Σ , and using (19) yields: (21) n − 2 n − 1 Z Σ H 2 dµ = Z Σ Sc Σ dµ − Z Σ Sc M dµ + 2 Z Σ Ric M ( ν, ν ) + | ˚ B | 2 dµ − Z Σ | ˚ B | 2 dµ ≤ Z Σ Sc Σ dµ where we used Sc M ≥ 0 , the nonpositivity of − R Σ | ˚ B | 2 dµ , and the w eak stability condition (19). No w if n − 2 n − 1 R Σ H 2 dµ = R Σ Sc Σ dµ , the inequalities must b e equalities. Then | ˚ B | 2 = Sc M = 0 , and by (19), Z Σ Ric M ( ν, ν ) dµ = 0 . Since Ric M ( ν, ν ) do es not change sign, we obtain Ric M ( ν, ν ) = 0 p oint wise on Σ . Substituting these v anishing terms back in to the Gauss equation implies that Sc Σ = n − 2 n − 1 H 2 . Since H is constan t, Sc Σ is a p ositiv e constant. By Ros’s Constant-Scalar-Curv ature Rigidity Theorem A.5, the only closed, embedded h yp ersurfaces in Euclidean space with constan t scalar curv ature are round spheres. Hence, the conv ex isometric em b edding of Σ into R n is a round sphere. Because a round sphere in R n is totally um bilic ( | ˚ B 0 | 2 = 0 ), its Gauss equation dictates n − 2 n − 1 H 2 0 = Sc Σ = n − 2 n − 1 H 2 , where H 0 is the mean curv ature of the isometric embedding. W e may therefore apply the rigidity result of Theorem A.3 to conclude that Ω is isometric to a Euclidean ball in R n . □ 2.2. The hyperb olic case: The preceding rigidity results identify Euclidean space as the mo del geometry in the case of nonnegative scalar curv ature. An analogous rigidity statement holds when hyperb olic space serves as the reference geometry under the scalar curv ature b ound Sc ≥ 2Λ with Λ < 0 . In that setting, the Brown-Y ork mass rigidit y theorem of Shi-T am (see Theorem A.6 in App endix) implies that equalit y in the corresp onding b oundary inequality forces the region to b e isometric to a domain in h yp erb olic space. Using this rigidit y input, results analogous to Theorem 2.2 were obtained in [32, 35, Theorem 3, Theorem 2]. Similarly , one obtains a hyperb olic analogue of Theorem 2.3 under an appropriate extrinsic curv ature sign condition. Theorem 2.6. L et ( M , g ) b e a 3 -dimensional R iemannian manifold with sc alar curvatur e Sc M ≥ 2Λ for a c onstant Λ ≤ 0 . L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either 10 (i) The se c ond variation of ar e a satisﬁes δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ −  1 2 H 2 + 2 3 Λ  | Σ | , or (ii) Σ is top olo gic al ly a spher e and is variational ly stable. Then (22) H 2 ≤ 16 π | Σ | − 4 3 Λ . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ( ν, ν ) − 2 3 Λ do es not change sign along Σ . If e quality holds in (22), then Σ is isometric to a r ound spher e and Ω is isometric to a ge o desic b al l in the hyp erb olic sp ac e of r adius 3 / Λ , H 3 Λ / 3 . Pr o of. W e will see ﬁrst that under the t wo assumptions, w e obtain that Sc Σ is a p ositive constan t, then the rest of the pro of follo ws the same path for b oth assumptions. ( i ) The condition on the second v ariation reduces to (23) Z Σ | ˚ B | 2 + Ric M ( ν, ν ) − 2 3 Λ dµ ≤ 0 By integrating the Gauss equation (24) Sc Σ = Sc M − 2Ric M ( ν, ν ) + 1 2 H 2 − | ˚ B | 2 , and using (23) (25) 1 2 Z Σ H 2 dµ = Z Σ Sc Σ dµ − Z Σ Sc M dµ + 2 Z Σ Ric M ( ν, ν ) + | ˚ B | 2 dµ − Z Σ | ˚ B | 2 dµ ≤ 8 π + ( 4 3 Λ − 2Λ) | Σ | = 8 π − 2 3 Λ | Σ | where we also used Sc M ≥ 2Λ , and that R Σ Sc Σ dµ ≤ 8 π by Gauss Bonnet theorem. Now if 1 2 R Σ H 2 dµ = 8 π − 2 3 Λ | Σ | . Then R Σ 1 2 Sc Σ dµ = 4 π , Σ is a top ological sphere, | ˚ B | 2 = Sc M = 0 and R Σ Ric M ( ν, ν ) − 2 3 Λ dµ = 0 . Since Ric M ( ν, ν ) − 2 3 Λ do es not change sign on Σ , it follows that Ric M ( ν, ν ) = 2 3 Λ p oin t wise on Σ . Substituting | ˚ B | = 0 , Sc M = 2Λ , and Ric M ( ν, ν ) = 2 3 Λ in to the Gauss equation yields that Sc Σ is a p ositiv e constan t. Hence (Σ , h ) is intrinsically round. ( ii ) By uniformization and b y Hersch’s lemma [15] (see also [18]), there exists a conformal map ϕ : Σ → S 2 whose co ordinate functions hav e zero mean. Using these as test functions in the volume-preserving stabilit y inequality yields (cf. (15)) (26) 8 π ≥ Z Σ  | B | 2 + Ric M ( ν, ν )  dµ = 1 2 Z Σ H 2 dµ + Z Σ  | ˚ B | 2 + Ric M ( ν, ν )  dµ. Using the Gauss equation Sc Σ = Sc M − 2Ric M ( ν, ν ) + 1 2 H 2 − | ˚ B | 2 , together with Gauss-Bonnet R Σ 1 2 Sc Σ dµ = 4 π and the lo wer b ound Sc M ≥ 2Λ , inequality (26) implies 8 π ≥ − 4 π + Z Σ  3 4 H 2 + | ˚ B | 2 + Λ  dµ, 11 and therefore Z Σ H 2 dµ ≤ 16 π − 4 3 Λ | Σ | − Z Σ | ˚ B | 2 dµ ≤ 16 π − 4 3 Λ | Σ | . Assume no w that equality holds: R Σ H 2 dµ = 16 π − 4 3 Λ | Σ | . Then the previous inequality forces | ˚ B | ≡ 0 on Σ and Sc M = 2Λ along Σ . In tegrating the Gauss equation giv es Z Σ  2Ric M ( ν, ν ) − 4 3 Λ  dµ = Z Σ  1 2 H 2 + 2 3 Λ − Sc Σ  dµ = 0 . Since Ric M ( ν, ν ) − 2 3 Λ do es not c hange sign on Σ , it follows that Ric M ( ν, ν ) = 2 3 Λ p oin t wise on Σ . As b efore we obtain that Sc Σ is a p ositiv e constan t and (Σ , h ) is intrinsically round. Let r := q | Σ | / (4 π ) b e its area radius, so Sc Σ = 2 /r 2 . By Theorem A.6, Σ admits a con vex isometric em b edding in to H 3 Λ / 3 with mean curv ature H 0 . Since the induced metric is round, the image is a geo desic sphere in H 3 Λ / 3 , and its mean curv ature satisﬁes H 2 0 = − 4 3 Λ + 16 π | Σ | . Using the equalit y assumption, the right-hand side equals H 2 ; since H , H 0 > 0 we obtain H 0 = H . Therefore R Σ ( H 0 − H ) dµ = 0 , and Shi-T am rigidit y (Theorem A.6) implies that Ω is isometric to a domain in H 3 Λ / 3 . Because Σ is round and H is constant, this domain is a geo desic ball. □ Remark 2.7. In related w ork [29], the author obtained analogous rigidity results for ar e a- c onstr aine d Wil lmor e surfac es , that is, surfaces satisfying the equation (27) 0 = λH + ∆ H + H | ˚ B | 2 + H Ric M ( ν, ν ) , for some constan t λ . In that setting one obtains the geometric inequalit y Z Σ H 2 dµ ≤ 16 π , whose equality case leads to the same rigidity conclusions in the Euclidean and h yp erb olic settings as those obtained ab o ve. How ev er, when one considers a spherical bac kground, the rigidit y case forces the surface to b e minimal, and the ab ov e equation then degenerates since H = 0 . Consequently , the Willmore framew ork does not readily yield the spherical rigidit y result, whereas the CMC approac h used here remains applicable, as we will see next. 2.3. The spherical case: Finally , one ma y also consider the p ositively curv ed mo del geom- etry . Let S n ( r ) denote the round n –sphere of radius r , and S n + ( r ) := { x ∈ R n : | x | = r , x n ≥ 0 } its upp er hemisphere. In contrast to the Euclidean and hyperb olic settings, rigidit y in the spherical case requires a Ricci curv ature low er b ound. In this setting, Melo established the follo wing rigidity result for stable constant mean curv ature spheres. Theorem 2.8 ([24, Theorem 1.1, Theorem 1.2]) . L et ( M , g ) b e a 3 -dimensional R iemannian manifold with sc alar curvatur e Sc M ≥ 6 , and let Ω ⊂ M b e a r elatively c omp act domain with smo oth b oundary Σ = ∂ Ω . A ssume Σ is a stable c onstant me an curvatur e spher e satisfying Z Σ ( H 2 + 4) dµ = 16 π . 12 If either (i) Σ admits an even symmetry, or (ii) the Gauss curvatur e K Σ is suﬃciently C 0 –close to 4 π | Σ | , then Σ is isometric to a r ound spher e of r adius | Σ | / (4 π ) . Mor e over, if Ric M ≥ 2 g , then Ω is isometric to the hemispher e S 3 +  q | Σ | / (4 π )  . Finally , w e will see that the condition Ric M ≥ 2 g is enough to prov e rigidity and that in this case the almost roundness or symmetry conditions are not necessary . Theorem 2.9. L et ( M , g ) b e a 3 -dimensional R iemannian manifold with sc alar curvatur e Sc M ≥ 2Λ for a c onstant Λ ≥ 0 . L et Σ ⊂ M b e a close d surfac e of c onstant me an curvatur e H ≥ 0 , satisfying either (i) The se c ond variation of ar e a satisﬁes δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ − ( 1 2 H 2 + 2 3 Λ) | Σ | , or (ii) Σ is top olo gic al ly a spher e and is variational ly stable. Then (28) H 2 ≤ 16 π | Σ | − 4 3 Λ . Supp ose additional ly that Σ is the b oundary of a r elatively c omp act domain Ω ⊂ M , with outwar d unit normal ν , and that Ric M ≥ 2 3 Λ g on Ω . If e quality holds in (28), then Σ is total ly ge o desic ( B = 0 ), it is isometric to a r ound spher e and Ω is isometric to the hemispher e S 3 +  q | Σ | / (4 π )  . Pr o of. As in the pro of of Theorem 2.6, the sign of Λ plays no role in the deriv ation of the inequalit y . Under either assumption ( i ) or ( ii ) , the same argument yields Z Σ H 2 dµ ≤ 16 π − 4 3 Λ | Σ | − Z Σ | ˚ B | 2 dµ, and hence H 2 ≤ 16 π | Σ | − 4 3 Λ . Assume now that equality holds and that Ric M ≥ 2 3 Λ g on Ω . Then necessarily | ˚ B | 2 = 0 on Σ , Sc M = 2Λ along Σ . In tegrating the Gauss equation Sc Σ = Sc M − 2Ric M ( ν, ν ) + 1 2 H 2 − | ˚ B | 2 , and using equalit y in the previous estimate gives 0 = Z Σ  Ric M ( ν, ν ) − 2 3 Λ  dµ. 13 Since Ric M ( ν, ν ) ≥ 2 3 Λ on Σ , it follo ws that Ric M ( ν, ν ) = 2 3 Λ p oin t wise on Σ . Substituting | ˚ B | = 0 , Sc M = 2Λ , and Ric M ( ν, ν ) = 2 3 Λ into the Gauss equation yields Sc Σ = 2 3 Λ + 1 2 H 2 = 8 π | Σ | = 2 r 2 , r := s | Σ | 4 π . Th us Σ has constant p ositive scalar curv ature and is isometric to a round sphere (b y W eyl- Niren b erg-Pogorelo v theorem A.2). Since the second fundamen tal form of Σ is nonnegative, w e may therefore apply Theorem A.7 ( after rescaling the metric by ˜ g := R − 2 g where R = q 6 2Λ so that Ric ˜ g ≥ 2 ˜ g ). It follo ws that (Ω , g ) is isometric to the hemisphere S 3 +  q | Σ | / (4 π )  and Σ is a round sphere of radius r = q | Σ | / (4 π ) . This implies that Sc M = 2Λ = 6 /r 2 , then Λ = 12 π | Σ | = 3 r 2 and H 2 = 16 π | Σ | − 4 3 Λ = 0 . □ Just as in the Euclidean setting, this result generalizes to higher dimensions under the weak stabilit y condition of assumption ( i ) . Theorem 2.10. L et ( M , g ) b e a Riemannian manifold of dimension n ≥ 3 with sc alar curva- tur e Sc M ≥ 2Λ for a c onstant Λ ≥ 0 . L et Σ ⊂ M b e a close d hyp ersurfac e of c onstant me an curvatur e H ≥ 0 , such that its se c ond variation of ar e a satisﬁes (29) δ 2 ν | Σ | = Z Σ − ( | B | 2 + Ric M ( ν, ν )) dµ ≥ −  H 2 n − 1 + 2 n Λ  | Σ | . Then (30) H 2 ≤ ( n − 1) R Σ Sc Σ dµ ( n − 2) | Σ | − 2( n − 1) n Λ . Supp ose additional ly that Σ admits an isometric emb e dding into R n as a c onvex hyp ersurfac e, and it is the b oundary of a r elatively c omp act domain Ω ⊂ M which satisﬁes that Ric M ≥ 2 n Λ g on Ω . If e quality holds in (30), then Σ is total ly ge o desic ( B = 0 ), it is isometric to a r ound spher e and Ω is isometric to the hemispher e S n +  q | Σ | /ω n − 1  , wher e ω n − 1 is the volume of the n − 1 -dimensional r ound spher e. Pr o of. The condition on the second v ariation reduces to (31) Z Σ  | ˚ B | 2 + Ric M ( ν, ν ) − 2 n Λ  dµ ≤ 0 . The Gauss equation is (32) Sc Σ = Sc M − 2Ric M ( ν, ν ) + n − 2 n − 1 H 2 − | ˚ B | 2 . Isolating the mean curv ature term, integrating o v er Σ , and using (31) yields: (33) n − 2 n − 1 Z Σ H 2 dµ = Z Σ Sc Σ dµ − Z Σ Sc M dµ + 2 Z Σ Ric M ( ν, ν ) − 2 n Λ + | ˚ B | 2 dµ − Z Σ | ˚ B | 2 dµ + 4 n Λ | Σ | ≤ Z Σ Sc Σ dµ − 2Λ | Σ | + 4 n Λ | Σ | = Z Σ Sc Σ dµ − 2( n − 2) n Λ | Σ | 14 where we used Sc M ≥ 2Λ and (31). No w, if the equality holds in (33). Then | ˚ B | 2 = 0 . Sc M = 2Λ , and b y (31), Z Σ Ric M ( ν, ν ) − 2 n Λ dµ = 0 . Since Ric M ( ν, ν ) ≥ 2 n Λ on Σ , it follo ws that Ric M ( ν, ν ) = 2 n Λ p oin t wise on Σ . Substituting these v anishing terms back in to the Gauss equation implies that Sc Σ = 2Λ − 4 n Λ + n − 2 n − 1 H 2 = 1 | Σ | Z Σ Sc Σ dµ. Then Sc Σ is a constan t. By Ros’s Constan t-Scalar-Curv ature Rigidit y Theorem A.5, the only closed, embedded hypersurfaces in Euclidean space with constant scalar curv ature are round spheres. Hence, the conv ex isometric em b edding of Σ in to R n is a round sphere. Let r =  | Σ | /ω n − 1  1 n − 1 b e its area radius, so Sc Σ = ( n − 1)( n − 2) r 2 . Since H ≥ 0 and ˚ B = 0 , Σ has nonegativ e second fundamental form, we may therefore apply Theorem A.7 (after rescaling the metric b y ˜ g := R − 2 g where R = q n ( n − 1) 2Λ so that Ric ˜ g ≥ ( n − 1) ˜ g ), which implies that (Ω , g ) is isometric to the hemisphere S n + ( r ) and Σ is a round sphere of radius r . Because (Ω , g ) is strictly isometric to the n -dimensional hemisphere of radius r , its am bient scalar curv ature is precisely Sc M = n ( n − 1) r 2 . F rom our earlier equalit y conditions, w e also established Sc M = 2Λ . Equating these giv es: 2Λ = n ( n − 1) r 2 = ⇒ Λ = n ( n − 1) 2 r 2 . Finally , we substitute this v alue of Λ and the scalar curv ature Sc Σ = ( n − 1)( n − 2) r 2 bac k in to our expression for H 2 w e obtain H = 0 . Consequen tly , the second fundamen tal form v anishes ( Σ is totally geo desic), which fully completes the pro of. □ In the hyperb olic and spherical cases, the normalization Sc M ≥ 2Λ is c hosen to match the dom- inan t energy condition for a spacetime with cosmological constan t Λ ; in the time-symmetric case, this is precisely the normalization for which the resulting inequalities and rigidit y state- men ts coincide with the p ositivity and rigidit y of the Hawking energy with cosmological con- stan t. W e now turn to STCMC surfaces: in the next section we introduce the corresp onding stabilit y framework in the general spacetime setting, and in the subsequent sections w e estab- lish the analogous p ositivity and rigidity results. 3. Geometr y of sp acelike surf aces and STCMC surf a ces In this section w e in tro duce the geometric framew ork for spacelike surfaces in Lorentzian manifolds and deﬁne the class of sp ac etime c onstant me an curvatur e (STCMC) surfaces that will be studied throughout the pap er. W e ﬁrst recall the n ull geometry of spacelik e surfaces in a four-dimensional spacetime and introduce the asso ciated null expansions. W e then deﬁne STCMC surfaces and sho w that they admit a natural v ariational characterization in terms of the area functional. Finally , w e derive the stability op erator asso ciated with v ariations in the spacelik e mean curv ature direction. 15 Our presen tation follows the standard framew ork used for studying surfaces in Lorentzian manifolds; see for instance [23]. 3.1. Spacelik e surfaces and n ull geometry. Let ( M , g ) b e a 4 -dimensional spacetime and let Σ ⊂ M b e a smo oth closed spacelik e surface. W e denote b y Φ Σ : Σ  → M the embedding of Σ in to M , and we will often identify Σ with its image. The induced Riemannian metric on Σ will b e denoted by h . At each p oin t p ∈ Σ the tangent space of the spacetime decomp oses orthogonally as T p M = T p Σ ⊕ N p Σ . Given tangent vector ﬁelds X , Y ∈ T Σ , the second fundamen tal form of Σ in ( M , g ) is deﬁned b y χ ( X , Y ) := − ( ∇ X Y ) ⊥ , where ∇ denotes the Levi-Civita connection of ( M , g ) . Its trace with resp ect to h deﬁnes the mean curv ature v ector  H := tr h χ ∈ N Σ . F or a normal vector ﬁeld n ∈ N Σ we deﬁne the second fundamen tal form in the direction n b y χ n ( X , Y ) = ⟨ χ ( X , Y ) , n ⟩ , and its trace θ n = tr h χ n is called the exp ansion along n . When n is n ull, θ n is referred to as the nul l exp ansion . The normal bundle N Σ admits a frame consisting of t wo future-directed null vector ﬁelds { , k } , which w e normalize by ⟨ , k ⟩ = − 2 . This normalization leav es the usual b o ost freedom  ′ = f  , k ′ = f − 1 k , for any p ositive function f on Σ . The corresp onding null expansions are deﬁned by θ ℓ := tr h χ ℓ , θ k := tr h χ k . Asso ciated with the choice of null frame is the normal connection one-form s ℓ ( X ) = − 1 2 ⟨ k , ∇ X  ⟩ , X ∈ T Σ . With resp ect to the n ull frame { , k } the mean curv ature v ector can b e written as  H = 1 2 ( θ k  + θ ℓ k ) , and its squared norm satisﬁes ⟨  H ,  H ⟩ = − θ ℓ θ k . If w e restrict to the regime θ ℓ θ k < 0 , then the mean curv ature vector is spacelike and w e can deﬁne the v ector ﬁeld (34) U := 1 2   s − θ k θ ℓ  + s − θ ℓ θ k k   . Since θ ℓ θ k < 0 , the quan tities under the square ro ots are p ositive and U is well deﬁned. A direct computation sho ws that ⟨ U, U ⟩ = − 1 , ⟨ U,  H ⟩ = 0 . Th us U is a unit timelike v ector ﬁeld orthogonal to the spacetime mean curv ature v ector. 16 Geometrically , the tangen t space T Σ together with the spacelik e v ector  H span a three- dimensional spacelik e subspace of T M , and U is the unique future-directed unit timelik e normal v ector orthogonal to this subspace. Moreov er, U is in v ariant under the b o ost trans- formation, so it deﬁnes a canonical observ er ﬁeld asso ciated with Σ . 3.2. Spacetime constan t mean curv ature surfaces. Deﬁnition 3.1 (STCMC surfaces) . A spacelik e surface Σ is called a surfac e of c onstant sp ac etime me an curvatur e if θ ℓ θ k = const . These surfaces are referred to as STCMC surfac es (spacetime constant mean curv ature sur- faces) since they provide a natural Loren tzian analogue of constant mean curv ature (CMC) surfaces in Riemannian geometry . In the time-symmetric case where the spacetime splits as M × R and the slice M is totally geo desic, one has θ ℓ = θ k = H , and the STCMC condition reduces to the classical constan t mean curv ature condition. The STCMC condition admits a natural geometric interpretation in terms of the mean cur- v ature v ector. Since ⟨  H ,  H ⟩ = − θ ℓ θ k , it is equiv alen t to requiring that the length of the mean curv ature v ector b e constant along Σ . In particular, when θ ℓ θ k ≤ 0 the mean curv ature v ector is spacelike and has constant norm. These surfaces arise naturally in mathematical relativity and hav e b een studied in several recen t w orks. See, for example, [6, 17, 25, 37, 38, 39]. Sev eral geometric constructions and foliations by suc h surfaces ha ve b een studied in these w orks. STCMC surfaces admit a simple v ariational in terpretation. Consider normal v ariations of Σ of the form v = α  H , with α ∈ C ∞ (Σ) . The ﬁrst v ariation of the area functional in this direction is δ α  H | Σ | = − Z Σ ⟨  H , α  H ⟩ dµ = Z Σ α θ ℓ θ k dµ. Imp osing the constrain t Z Σ α dµ = 0 , w e see that Σ is a critical p oint of the area functional under such v ariations if and only if θ ℓ θ k is constant. Th us STCMC surfaces are precisely the critical p oints of the area functional under inﬁnitesimally v olume-preserving v ariations in the spacelik e mean curv ature direction, whic h is the Lorentzian analogue of the classical v ariational c haracterization of CMC surfaces. 3.3. Second v ariation of area. T o study the stabilit y of STCMC surfaces w e need to com- pute the second v ariation of the area functional under such v ariations. F or this purp ose we recall the v ariation form ulas for the n ull expansions. Let ξ b e a normal v ariation vector ﬁeld along Σ and let ϕ τ denote the asso ciated ﬂo w, with deformed surfaces Σ τ = ϕ τ (Σ) . Cho osing a corresp onding family of null normals  τ along Σ τ , w e deﬁne the ﬁrst v ariation of the null expansion θ ℓ b y δ ξ θ ℓ = d dτ ϕ ∗ τ ( θ ℓ τ )    τ =0 . W e decomp ose the v ariation ﬁeld along the null frame as ξ | Σ = α  − ψ 2 k , where α, ψ ∈ C ∞ (Σ) . 17 The ﬁrst v ariation of θ ℓ is given b y (35) δ ξ θ ℓ = − ∆ h ψ + 2 s ℓ · ∇ h ψ + ψ  div h s ℓ − ∥ s ℓ ∥ 2 h + 1 2 θ ℓ θ k + 1 2 Sc Σ − 1 2 Ein( , k )  − α  Ein( ,  ) + ∥ χ ℓ ∥ 2 h  + κ ξ θ ℓ , where κ ξ := − 1 2 ⟨ k , ∇ ξ  ( ξ ) ⟩ . Int erchanging the roles of  and k , the v ariation of θ k along ξ = α k − ψ 2  is (36) δ ξ θ k = − ∆ h ψ − 2 s ℓ · ∇ h ψ + ψ  − div h s ℓ − ∥ s ℓ ∥ 2 h + 1 2 θ ℓ θ k + 1 2 Sc Σ − 1 2 Ein( , k )  − α  Ein( k , k ) + ∥ χ k ∥ 2 h  − κ ξ θ k . Using the v ariation formulas ab o ve one obtains the second v ariation of the area functional in the spacelike mean curv ature direction. F or v ariations of the form v = α  H a direct computa- tion yields (37) δ 2 α  H | Σ | = Z Σ α  θ k ∆ h ( αθ ℓ ) + θ ℓ ∆ h ( αθ k ) − 2 θ k s ℓ ( ∇ h ( αθ ℓ )) + 2 θ ℓ s ℓ ( ∇ h ( αθ k )) − αθ 2 k 2  Ein( ,  ) + ∥ χ ℓ ∥ 2 h  − αθ 2 ℓ 2  Ein( k , k ) + ∥ χ k ∥ 2 h  + 2 αθ ℓ θ k  ∥ s ℓ ∥ 2 h − 1 2 Sc Σ + 1 2 Ein( , k )  dµ = Z Σ α L α dµ, By integration b y parts the second v ariation form ula can b e written in the form (38) δ 2 α  H | Σ | = Z Σ  − 2 θ ℓ θ k |∇ h α | 2 + W α 2  dµ, where (39) W = θ k ∆ h θ ℓ + θ ℓ ∆ h θ k − 2 θ k s ℓ ( ∇ h θ ℓ ) + 2 θ ℓ s ℓ ( ∇ h θ k ) − θ 2 k 2  Ein( ,  ) + ∥ χ ℓ ∥ 2  − θ 2 ℓ 2  Ein( k , k ) + ∥ χ k ∥ 2  + 2 θ ℓ θ k  ∥ s ℓ ∥ 2 − 1 2 Sc Σ + 1 2 Ein( , k )  . On an STCMC surface with Σ with θ ℓ θ k  = 0 the p oten tial admits a more geometric represen- tation. Since θ ℓ θ k is constant, taking the gradient of log | θ ℓ θ k | yields (40) ∇ h θ k θ k = ∇ h log | θ k | = −∇ h log | θ ℓ | = − ∇ h θ ℓ θ ℓ . Moreo v er, (41) ∆ h θ ℓ θ ℓ = ∆ h log | θ ℓ | + ∥∇ h log | θ ℓ |∥ 2 h , ∆ h θ k θ k = ∆ h log | θ k | + ∥∇ h log | θ k |∥ 2 h . F actoring out the constan t pro duct θ ℓ θ k and using (40) and (41), w e obtain (42) W = θ ℓ θ k " 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h − Sc Σ + 2 Ein( U, U ) − θ k 2 θ ℓ ∥ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ χ k ∥ 2 h # . 18 Here w e used that the Einstein tensor terms can b e expressed in terms of the canonical timelik e v ector ﬁeld U deﬁned in (34), since (43) 2 Ein ( U, U ) = Ein( , k ) − θ k 2 θ ℓ Ein( ,  ) − θ ℓ 2 θ k Ein( k , k ) . Consequen tly , the stability op erator takes the form (44) L = θ ℓ θ k (2∆ h + V ) . where V = 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h − Sc Σ + 2 Ein( U, U ) − θ k 2 θ ℓ ∥ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ χ k ∥ 2 h . In particular, the second v ariation of the area functional can b e written as δ 2 α  H | Σ | = Z Σ α L α dµ, for a surface Σ with θ ℓ θ k  = 0 . Since L is a second-order elliptic op erator and Σ is compact, its sp ectrum is discrete. This v ariational structure naturally leads to a notion of stabilit y for STCMC surfaces. 3.4. Stabilit y of STCMC surfaces. Motiv ated b y the quadratic form asso ciated with the op erator L , we introduce the follo wing notions of stabilit y . As in the classical theory of constan t mean curv ature surfaces, stability is deﬁned b y requiring the second v ariation of area to b e nonnegativ e under appropriate constrain ts on the v ariation function α . Deﬁnition 3.2 (Stability of STCMC surfaces) . Let Σ b e a spacelik e STCMC surface. ( i ) W e say that Σ is variational ly stable if for ev ery α ∈ C ∞ (Σ) satisfying Z Σ α dµ = 0 , the second v ariation of area in the direction α  H satisﬁes δ 2 α  H | Σ | ≥ 16 π | Σ | |  H | 2 Z Σ α 2 dµ. ( ii ) W e say that Σ is c onstant-mo de stable if the second v ariation in the direction of the mean curv ature v ector satisﬁes δ 2  H | Σ | ≥ 0 . In the case θ ℓ θ k  = 0 , the sp ectrum of L on L 2 (Σ) is discrete and v ariational stabilit y can b e in terpreted as a sp ectral b ound on the quadratic form restricted to the mean-zero subspace H 0 =  α ∈ C ∞ (Σ) : Z Σ α dµ = 0  . More precisely , inf α ∈H 0 \{ 0 } R Σ α L α dµ R Σ α 2 dµ ≥ 16 π | Σ | |  H | 2 . V ariational stability and constant-mode stabilit y control diﬀerent parts of the sp ectrum of L : v ariational stabilit y provides a low er b ound on the quadratic form on the mean-zero subspace H 0 , while constan t-mo de stabilit y tests the constan t mo de α ≡ 1 . In general, these t wo conditions are indep enden t. Ho w ever, in the spherical case an imp ortant diﬀerence emerges b et ween the CMC and STCMC settings. F or CMC surfaces, v ariational stability do es not in general imply con trol of the 19 constan t mo de. By contrast, for STCMC surfaces that are top ologically a sphere, the sharp form of the v ariational stabilit y inequality do es imply constant-mode stability . Lemma 3.3. L et Σ b e a smo oth STCMC surfac e in a 4 -dimensional L or entzian manifold, and assume that Σ is top olo gic al ly a spher e. If Σ is variational ly stable, then it is c onstant-mo de stable. Pr o of. If θ ℓ θ k = 0 , then the v ariational stabilit y inequalit y reduces immediately to constan t- mo de stability . W e may therefore assume that θ ℓ θ k < 0 . Since Σ is top ologically a sphere, by the uniformization theorem it is conformally equiv alen t to S 2 . Moreov er, by Hersc h’s lemma [15] (see also [18]), there exists a conformal map ϕ = ( ϕ 1 , ϕ 2 , ϕ 3 ) : Σ → S 2 ⊂ R 3 suc h that Z Σ ϕ i dµ = 0 for i = 1 , 2 , 3 . Th us each ϕ i is an admissible test function in the stability inequalit y . F or each i , Z Σ ϕ i L ϕ i dµ ≥ 16 π | Σ | |  H | 2 Z Σ ϕ 2 i dµ. W riting L = θ ℓ θ k (2∆ h + V ) , dividing b y − θ ℓ θ k = |  H | 2 > 0 yields Z Σ  − 2 ϕ i ∆ h ϕ i − V ϕ 2 i  dµ ≥ 16 π | Σ | Z Σ ϕ 2 i dµ. Summing ov er i = 1 , 2 , 3 and using 3 X i =1 ϕ 2 i = 1 , 3 X i =1 Z Σ |∇ ϕ i | 2 dµ = 8 π , w e obtain 16 π − Z Σ V dµ ≥ 16 π . Th us R Σ V dµ ≤ 0 . Since δ 2  H | Σ | = Z Σ 1 · L (1) dµ = θ ℓ θ k Z Σ V dµ, it follows that δ 2  H | Σ | ≥ 0 . Hence Σ is constant-mode stable. □ Remark 3.4 (Mo del op erator on round spheres) . Let Σ ⊂ R 3 b e a round sphere. Then H and Sc Σ are constant and the op erator L app earing in (44) reduces to L = − 2 H 2 ∆ h + H 2 Sc Σ − 1 2 H 4 = H 2  − 2∆ h + Sc Σ − 1 2 H 2  . Since for a round sphere Sc Σ = 1 2 H 2 , we obtain the further simpliﬁcation L = − 2 H 2 ∆ h . Since the ﬁrst nonzero eigenv alue of the Laplace-Beltrami op erator on a round sphere of radius r is λ 1 ( − ∆ h ) = 2 r 2 , and H = 2 r , we obtain λ 1 ( L ) = 2 H 2 λ 1 ( − ∆ h ) = 2  4 r 2  2 r 2 = 16 r 4 . 20 Recalling that | Σ | = 4 π r 2 and |  H | 2 = H 2 = 4 r 2 , we compute 16 π | Σ | |  H | 2 = 16 π 4 π r 2 · 4 r 2 = 16 r 4 . Therefore equality holds in Deﬁnition 3.2 for round spheres. In particular, the constan t 16 π is sharp. F or comparison, the classical Jacobi op erator go v erning volume-preserving stability of CMC surfaces in R 3 is L Jac = − ∆ h − | B | 2 , and on a round sphere | B | 2 = 1 2 H 2 . Therefore, on the round sphere, − 2∆ h = 2 L Jac + H 2 , and hence L = H 2 (2 L Jac + H 2 ) . In particular, the STCMC stability operator is a constant scaled translation of the classical CMC Jacobi op erator (up to the prefactor H 2 ). The preceding eigen v alue computation sho ws that the stabilit y inequalit y in Deﬁnition 3.2 is sharp, and that the constant 16 π | Σ | |  H | 2 is precisely calibrated by the round sphere. Finally , w e observe that round spheres are also constan t-mo de stable. Indeed, since L = − 2 H 2 ∆ h and the Laplace-Beltrami op erator annihilates constants, w e hav e δ 2  H | Σ | = Z Σ 1 L (1) dµ = 0 . Th us the constant-mode stabilit y inequality is also saturated by round spheres. In particular, b oth stability inequalities in Deﬁnition 3.2 are sharp. Remark 3.5 (The case θ ℓ θ k = 0 ) . In this case, the second v ariation formula simpliﬁes sub- stan tially: the gradien t terms v anish and the second v ariation reduces to δ 2 α  H | Σ | = R Σ W α 2 dµ , where W = − θ 2 k 2  Ein( ,  ) + ∥ χ ℓ ∥ 2 h  − θ 2 ℓ 2  Ein( k , k ) + ∥ χ k ∥ 2 h  . If the ambien t spacetime satisﬁes the null energy condition, then Ein ( ,  ) ≥ 0 , Ein( k , k ) ≥ 0 , and therefore W ≤ 0 on Σ . Consequen tly , if such a surface is constant-mode or v ariationally stable, then necessarily W ≡ 0 . In particular, • If θ ℓ = 0 ( Σ is a MOTS), then χ ℓ = 0 and Ein( ,  ) = 0 , so Σ is an outgoing nonex- panding horizon section. • If θ k = 0 , then χ k = 0 and Ein ( k , k ) = 0 , Σ is an incoming nonexpanding horizon section. Finally , other notions of stabilit y for spacelik e surfaces ha v e b een prop osed in [1]. Inspired b y the stabilit y theory of marginally outer trapp ed surfaces (MOTS), the authors in tro duce a stability condition based on the ﬁrst v ariation of |  H | 2 = − θ ℓ θ k along suitable directions. Under this assumption they obtain b ounds for the in tegral R Σ θ ℓ θ k dµ. The stability notion in tro duced here is instead formulated in terms of the second v ariation of the area functional in the spacetime mean curv ature direction, and can b e viewed as the natural Loren tzian analogue of the classical CMC stabilit y condition in Riemannian geometry . 21 4. Cur v a ture inequalities and rigidity f or st able STCMC surf aces In this section we establish a sharp curv ature inequality for stable spacetime constan t mean curv ature surfaces and analyze the rigidity case. The inequality can b e viewed as a Lorentzian analogue of the classical Christo doulou-Y au estimate for stable CMC surfaces in Riemannian geometry . The pro of combines the stability inequality derived in the previous section with the dominant energy condition. In the rigidit y case w e exploit the p ositivit y and rigidity prop erties of the Kijo wski-Liu-Y au quasi-lo cal energy together with structural results for maximal globally h yp erb olic dev elopments. Throughout this section w e assume that the spacetime ( M , g ) satisﬁes the dominant ener gy c ondition (DEC), namely Ein( X , Y ) ≥ 0 for all future-directed causal v ectors X , Y . The rigidity part of our argument also uses the rigidit y prop erties of the Kijowski-Liu-Y au quasi-lo cal en ergy . F or a surface Σ with positive Gaussian curv ature and spacelik e mean curv ature v ector, this energy is deﬁned b y E K LY (Σ) = 1 8 π Z Σ  H 0 − q − θ ℓ θ k  dµ, where H 0 is the mean curv ature of the isometric em b edding of Σ in to R 3 . W e will use the corresp onding rigidity theorem of Liu-Y au, Theorem A.8, together with the stronger rigidit y statemen t in Mink owski spacetime, Theorem A.9; b oth are recalled in the app endix. In the time-symmetric case K = 0 , the Kijowski-Liu-Y au energy reduces to the Bro wn-Y ork mass (see Theorem A.1). T o pass from rigidit y of the induced initial data to rigidity of the am bient spacetime region, w e will also use maximal globally hyperb olic dev elopmen ts. 4.1. Maximal globally h yp erb olic dev elopmen ts. Let ( M , g ) b e a spacetime and let Ω ⊂ M b e a spacelike hypersurface. The induced Riemannian metric g | Ω together with the second fundamental form K of Ω in M constitute an initial data set (Ω , g | Ω , K ) . More generally , given an initial data set (Ω , g , K ) , a development of this data consists of a spacetime ( M , g ) together with an em b edding ι : Ω  → M suc h that ι (Ω) is a spacelike h yp ersurface in ( M , g ) whose induced metric and second funda- men tal form coincide with g and K , resp ectively . A dev elopment is called glob al ly hyp erb olic if ι (Ω) is a Cauc h y h yp ersurface for ( M , g ) . The Cho quet-Bruhat-Gero ch theorem guaran tees that for every v acuum initial data set (Ω , g , K ) satisfying the Einstein constraint equations there exists a unique (up to isometry) maximal glob al ly hyp erb olic development . This spacetime is maximal among all globally hyperb olic de- v elopmen ts of the given initial data and is completely determined by the data ( g , K ) on Ω (see, e.g., [9]). In particular, if Ω is con tained in a globally h yp erb olic region of the spacetime ( M , g ) , then the maximal globally hyperb olic dev elopmen t coincides with the domain of dep endence D (Ω) ⊂ M . 22 With these preliminaries in place we can state the main curv ature inequalit y for stable STCMC surfaces together with the asso ciated rigidity statemen t. W e establish a sharp curv ature inequality for STCMC surfaces and analyze the equalit y case. The ﬁrst rigidity theorem assumes an additional sign condition on the n ull sectional curv ature Rm M ( k , , , k ) along Σ . Theorem 4.1. L et ( M , g ) b e a 4 -dimensional L or entzian manifold satisfying the dominant ener gy c ondition. L et Σ b e a close d sp ac elike STCMC surfac e with θ ℓ θ k ≤ 0 , such that either (i) Σ is c onstant-mo de stable ( δ 2  H | Σ | ≥ 0 ), or (ii) Σ is top olo gic al ly a spher e and variational ly stable in the sense of Deﬁnition 3.2. Then (45) − θ ℓ θ k ≤ 16 π | Σ | . If e quality holds in (45) and Rm M ( k , , , k ) do es not change sign along Σ , then Σ is isometric to a r ound spher e, and any c omp act sp ac elike hyp ersurfac e Ω ⊂ M with b oundary ∂ Ω = Σ emb e ds isometric al ly as a sp ac elike hyp ersurfac e in Minkowski sp ac etime. F urthermor e, the maximal glob al ly hyp erb olic development of the induc e d initial data on Ω is isometric to a standar d c ausal diamond in Minkowski sp ac etime. Pr o of. First note that if θ ℓ θ k = 0 , then (45) is immediate. W e may therefore assume that θ ℓ θ k < 0 . Moreov er, by Lemma 3.3, if Σ is top ologically a sphere and v ariationally stable, then Σ is constant-mode stable. Thus it suﬃces to prov e the theorem under assumption ( i ) . W e use the geometric formulation of the stability op erator L = θ ℓ θ k (2∆ h + V ) , where V is giv en by (46) V = 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h − Sc Σ + 2 Ein( U, U ) − θ k 2 θ ℓ ∥ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ χ k ∥ 2 h . Substituting the orthogonal decomp osition of the squared second fundamen tal forms, ∥ χ ∥ 2 h = 1 2 θ 2 + ∥ ˚ χ ∥ 2 h , we can extract the STCMC constant: (47) − θ k 2 θ ℓ ∥ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ χ k ∥ 2 h = − 1 2 θ ℓ θ k − θ k 2 θ ℓ ∥ ˚ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ ˚ χ k ∥ 2 h . Th us, (48) V = 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h − Sc Σ + 2 Ein( U, U ) − 1 2 θ ℓ θ k − θ k 2 θ ℓ ∥ ˚ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ ˚ χ k ∥ 2 h . Assume that Σ is constan t-mo de stable. Then 0 ≤ δ 2  H | Σ | = Z Σ 1 · L (1) dµ = θ ℓ θ k Z Σ V dµ, Dividing by θ ℓ θ k , substituting (48) and rearranging giv es − 1 2 Z Σ θ ℓ θ k dµ ≤ Z Σ Sc Σ dµ − Z Σ 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h + 2Ein( U, U ) − θ k 2 θ ℓ ∥ ˚ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ ˚ χ k ∥ 2 h ! dµ. Because θ ℓ and θ k ha v e opp osite signs, the ratios − θ k /θ ℓ and − θ ℓ /θ k are strictly p ositiv e. F urthermore, since U is a timelik e v ector, the dominant energy condition ensures Ein ( U, U ) ≥ 0 . Th us, ev ery term in the subtracted integral is nonnegativ e. Applying the Gauss-Bonnet b ound R Σ Sc Σ dµ ≤ 8 π , w e obtain − 1 2 R Σ θ ℓ θ k dµ ≤ 8 π , whic h yields (45). 23 If − θ ℓ θ k = 16 π | Σ | , then equality must hold in eac h step of the argument ab o ve. In particular, R Σ V dµ = 0 , this implies Z Σ Sc Σ dµ = 8 π , then Σ is top ologically a sphere, and (49) Ein( U, U ) = 0 , ∥ ˚ χ ℓ ∥ 2 h = 0 , ∥ ˚ χ k ∥ 2 h = 0 , ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h = 0 along Σ . Because U is a strictly timelike vector, the v anishing of Ein( U, U ) together with the dominan t energy condition implies that Ein = 0 everywhere along Σ (a pro of of this can b e found in [10, Lemma B1]). T aking the trace of the Einstein tensor leads to Sc M = 0 and Ric M = 0 on Σ . Since for STCMC surfaces ∇ h log | θ ℓ | = −∇ h log | θ k | then we also hav e s ℓ = ∇ h log | θ ℓ | = −∇ h log | θ k | . Note that the pro duct θ ℓ θ k is b o ost inv arian t, so b y c ho osing a b o ost q := − log | θ ℓ | we obtain  ′ = e q , k ′ = e − q k , s ℓ ′ = s ℓ + ∇ h q = 0 , θ ℓ ′ = e q θ ℓ = ± 1 . In this b o osted frame, we hav e s ℓ ′ = 0 and θ k ′ is also constan t. Recall the Gauss equation for surfaces in a 4 -dimensional Loren tzian manifold (this form ula can b e found in the literature, for example in [2, equation (9)]): (50) Sc M + 2Ric M ( k ,  ) − 1 2 Rm M ( k , , , k ) = Sc Σ + 1 2 θ ℓ θ k + ∥ ˚ χ ℓ ∥ 2 h + ∥ ˚ χ k ∥ 2 h . Applying our v anishing constrain ts, this reduces to (51) − 1 2 Rm M ( k , , , k ) = Sc Σ + 1 2 θ ℓ θ k . In tegrating this iden tit y o ver Σ , and recalling that R Σ Sc Σ dµ = 8 π (since Σ is a sphere) and − 1 2 R Σ θ ℓ θ k dµ = 8 π , we obtain (52) − Z Σ 1 2 Rm M ( k , , , k ) dµ = Z Σ Sc Σ dµ + 1 2 Z Σ θ ℓ θ k dµ = 8 π − 8 π = 0 . Since Rm M ( k , , , k ) do es not change sign along Σ , it follows that Rm M ( k , , , k ) = 0 p oin t wise on Σ . Consequen tly , the Gauss equation further reduces to Sc Σ = − 1 2 θ ℓ θ k , which means Sc Σ is a p ositiv e constan t. Since Σ is a top ological sphere with constant p ositiv e scalar curv ature, it has constant Gaussian curv ature and is therefore isometric to a round sphere of radius r in Euclidean space, in particular, Sc Σ = 2 r 2 . F or the isometric embedding into R 3 , the mean curv ature is H 0 = 2 r . Moreov er, since − θ ℓ θ k = |  H | 2 = 4 r 2 , we obtain H 0 = q − θ ℓ θ k . Th us, the Kijo wski-Liu-Y au energy of Σ v anishes, E K LY (Σ) = 0 . Let Ω ⊂ M b e an y compact spacelik e h yp ersurface region with b oundary ∂ Ω = Σ , and let (Ω , g Ω , K Ω ) denote the induced initial data. Since ( M , g ) satisﬁes the dominant energy condition, so do es (Ω , g Ω , K Ω ) . By Theorem A.8, the v anishing of E K LY (Σ) implies that the spacetime is ﬂat along the ﬁlling region Ω . F urthermore, the initial data (Ω , g Ω , K Ω ) embeds isometrically into Mink owski spacetime R 3 , 1 as a compact spacelike graph ˜ Ω ov er a domain in R 3 , with b oundary ˜ Σ . The remainder of the argumen t is carried out in this Minko wski am bien t space. Since ˜ Σ spans the compact spacelik e hypersurface ˜ Ω in R 3 , 1 , p ossesses p ositiv e Gaussian 24 curv ature, and has a spacelik e mean curv ature v ector, we can apply Theorem A.9 [26]. Because E K LY ( ˜ Σ) = 0 , this theorem dictates that ˜ Σ lies en tirely on a ﬂat spatial hyperplane in R 3 , 1 . Because ˜ Σ is intrinsically a round sphere lying in a hyperplane, it b ounds a totally geo desic ﬂat Euclidean ball B = B r within that same hyperplane. Crucially , since ˜ Ω is a spacelike graph t = f ( x ) ov er the Euclidean ball B = B r with f | ∂ B = 0 , we hav e |∇ f | < 1 , and hence f is 1 -Lipschitz. Since the domain of dep endence of a Ball in Minko wski spacetime is given by D ( B ) = { ( t, x ) ∈ R 3 , 1 : | t | + | x | ≤ r } , the estimate | f ( x ) | ≤ dist( x, ∂ B ) = r − | x | guaran tees that ( f ( x ) , x ) ∈ D ( B ) for ev ery x ∈ B , and therefore ˜ Ω ⊂ D ( B ) . Moreo v er, we can track an y future-directed causal curve γ ( s ) = ( t ( s ) , x ( s )) in D ( B ) b y deﬁning the function Φ( s ) = t ( s ) − f ( x ( s )) . Because γ is causal ( t ′ ( s ) ≥ | x ′ ( s ) | ) and ˜ Ω is strictly space- lik e ( |∇ f | < 1 ), diﬀerentiating Φ yields Φ ′ ( s ) = t ′ ( s ) − ∇ f ( x ( s )) · x ′ ( s ) ≥ t ′ ( s ) − |∇ f || x ′ ( s ) | > 0 . Th us, Φ( s ) is strictly increasing. Any inextendible causal curve in D ( B ) m ust originate on the past b oundary (where Φ ≤ 0 ) and terminate on the future b oundary (where Φ ≥ 0 ). By con tin uity and strict monotonicity , Φ( s ) must equal zero at exactly one p oin t, meaning ev ery such curve crosses ˜ Ω exactly once. Hence, their domains of dep endence strictly coincide: D ( ˜ Ω) = D ( B ) . Finally , b ecause the initial data on Ω is isometric to the data on ˜ Ω , the Cho quet-Bruhat- Gero c h theorem for the uniqueness of the maximal globally h yp erb olic Cauc hy dev elopment implies that the domain of dependence of Ω in M m ust b e isometric to D ( ˜ Ω) in R 3 , 1 . Since D ( ˜ Ω) = D ( B ) , the maximal globally h yp erb olic dev elopment of our original ﬁlling Ω is pre- cisely isometric to the domain of dep endence of the ﬂat ball B , whic h is a standard causal diamond in Mink o wski spacetime. □ Figure 1. Sc hematic picture of the causal diamond D (Ω) in Mink o wski space- time. The surface Σ is the edge of the diamond, and the shaded region is a spacelik e ﬁlling. Remark 4.2. The rigidity conclusion in Theorem 4.1 is formulated in terms of the maximal globally h yp erb olic developmen t of the initial data induced on Ω , rather than directly in terms of the ambien t spacetime M . This av oids p ossible obstructions in the causal future or past of Ω within M , such as holes or other causal pathologies, which may prev en t the corresp onding Mink o wski causal diamond from b eing realized inside M . If, ho wev er, Ω is contained in a 25 globally hyperb olic region of M , then the maximal globally hyperb olic developmen t agrees with the domain of dep endence D (Ω) ⊂ M . In that case, the standard Mink owski causal diamond is realized directly inside the am bien t spacetime. Remark 4.3 (Interpretation of the deﬁcit) . F or an STCMC surface Σ satisfying the hypothe- ses of Theorem 4.1, deﬁne the scale-in v arian t deﬁcit D (Σ) := 16 π | Σ | − |  H | 2 ≥ 0 . The pro of of Theorem 4.1 yields D (Σ) ≥ 4 | Σ | Z Σ Ein( U, U ) dµ + 2 | Σ | Z Σ " − θ k 2 θ ℓ ∥ ˚ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ ˚ χ k ∥ 2 h # dµ + 4 | Σ | Z Σ ∥∇ h log | θ ℓ |− s ℓ ∥ 2 h dµ. If Σ is top ologically a sphere, then this is an exact b ound. This decomp osition exhibits three distinct contributions to the deﬁcit: (1) Energy density term. The quantit y Ein ( U, U ) is nonnegativ e under the dominant energy condition. Via the Einstein equations, it ma y b e interpreted as the lo cal matter- energy density measured by the canonical observer U . (2) F ailure of umbilicit y (shear). The trace-free null second fundamental forms ˚ χ ℓ and ˚ χ k measure the failure of the surface to be um bilic. Their nonv anishing measures the deviation of the surface from a round sphere. (3) Normal bundle twist and optimal framing. The one-form s ℓ enco des the twist of the chosen null frame. The term ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h is gauge inv ariant and measures the failure of the frame to b e optimally adapted to the STCMC condition. When it v anishes, one can b o ost to a frame with s ℓ ′ = 0 and constan t n ull expansions. This decomp osition clariﬁes the rigidity mechanism in Theorem 4.1: v anishing of the deﬁcit forces the matter term and all geometric defect terms to v anish, thereb y forcing the surface to b e totally umbilic with an optimally aligned normal frame. Remark 4.4. Note that the proof of the inequalit y − θ ℓ θ k ≤ 16 π / | Σ | , under assumption ( i ) , also holds with only minor mo diﬁcation in the higher-dimensional case. Let M b e an n -dimensional Lorentzian manifold satisfying the dominan t energy condition, and Σ a n − 2 - dimensional closed submanifold, assume that Σ is a STCMC surface satisfying δ 2  H | Σ | ≥ 0 then (53) − θ ℓ θ k ≤ n − 2 n − 3 R Σ Sc Σ dµ | Σ | This, in particular, implies that the higher-dimensional Hawking energy (54) E n (Σ) = 1 2( n − 2)( n − 3) ω n − 2 | Σ | ω n − 2 ! 1 n − 2 Z Σ  Sc Σ + n − 3 n − 2 θ ℓ θ k  dµ is nonnegativ e. Here ω n − 2 denotes v olume of the n − 2 -dimensional round sphere in Euclidean space. W e also hav e an alternative rigidit y result. Theorem 4.5. L et ( M , g ) b e a 4 -dimensional L or entzian manifold satisfying the dominant ener gy c ondition. L et Σ b e a variational ly stable sp ac elike STCMC spher e with θ ℓ θ k ≤ 0 , and assume that one of the fol lowing c onditions holds: 26 (i) Σ has even symmetry, that is, ther e exists a ﬁxe d-p oint fr e e isometry ρ : Σ → Σ with ρ 2 = id . (ii) The Gauss curvatur e K Σ is suﬃciently C 0 -close to 4 π | Σ | , that is, ther e exists a c onstant 0 < δ 0 ≪ 1 such that ∥ K Σ − 4 π | Σ | ∥ C 0 < δ 0 . If − θ ℓ θ k = 16 π | Σ | , then Σ is isometric to a r ound spher e, and any c omp act sp ac elike hyp ersurfac e Ω ⊂ M with b oundary ∂ Ω = Σ emb e ds isometric al ly into Minkowski sp ac etime. F urthermor e, the maximal glob al ly hyp erb olic development of the induc e d initial data on Ω is isometric to a standar d c ausal diamond in Minkowski sp ac etime. Pr o of. W e follow the argumen t of Sun [35], based on the El Souﬁ-Ilias Theorem A.10. By the equalit y case analysis in Theorem 4.1, w e ha v e Ein = Ric M = 0 , ∥ ˚ χ ℓ ∥ 2 h = ∥ ˚ χ k ∥ 2 h = 0 , and s ℓ = 0 on Σ . In this setting, the stabilit y op erator reduces to (55) L = 2 θ ℓ θ k ∆ h − θ ℓ θ k Sc Σ − 1 2 θ 2 ℓ θ 2 k . Deﬁne the scaled op erator (56) ˆ L := − 1 2 θ ℓ θ k L = − ∆ h + Sc Σ 2 + 1 4 θ ℓ θ k . This op erator has the form − ∆ h + q with q = Sc Σ 2 + 1 4 θ ℓ θ k . Then w e can use El Souﬁ–Ilias Theorem A.10 to estimate the second eigen v alue, obtaining that (57) λ 2 ( ˆ L ) | Σ | ≤ 8 π + Z Σ q dµ. with equalit y only if Σ admits a conformal map in to the standard S 2 whose co ordinate functions are second eigenfunctions of ˆ L . Then applying Gauss-Bonnet theorem, the upper b ound b ecomes (58) λ 2 ( ˆ L ) | Σ | ≤ 8 π + Z Σ Sc Σ 2 + 1 4 θ ℓ θ k ! dµ = 8 π . Con v ersely , since Σ is v ariationally stable, the original op erator satisﬁes λ 2 ( L ) ≥ 16 π | Σ | |  H | 2 = − 16 π | Σ | θ ℓ θ k . Scaling this b ound, we obtain λ 2 ( ˆ L ) = − 1 2 θ ℓ θ k λ 2 ( L ) ≥ 8 π | Σ | . T ogether with the upp er b ound ab ov e, this yields the exact equalit y λ 2 ( ˆ L ) = 8 π | Σ | = 2 r 2 , 27 where r is the area radius of Σ . Since w e hav e equalit y in (57) there exists a conformal map ϕ : Σ → S 2 ⊂ R 3 suc h that its comp onen ts ϕ i ( i = 1 , 2 , 3 ) satisfy R Σ ϕ i dµ = 0 , P 3 i =1 ϕ 2 i = 1 , and (59) − ∆ h ϕ i + Sc Σ 2 ϕ i + 1 4 θ ℓ θ k ϕ i − 2 r 2 ϕ i = 0 . Since | ϕ | 2 = P 3 i =1 ϕ 2 i = 1 , w e hav e 0 = ∆ h | ϕ | 2 = P 3 i =1 2 ϕ i ∆ h ϕ i + 2 |∇ h ϕ | 2 , where |∇ h ϕ | 2 = P 3 i =1 |∇ h ϕ i | 2 . Substituting ∆ h ϕ i from (59) yields (60) |∇ h ϕ | 2 = 2 r 2 − 1 4 θ ℓ θ k − Sc Σ 2 = 3 r 2 − Sc Σ 2 , where we used the constraint − θ ℓ θ k = 16 π | Σ | = 4 r 2 . No w, identifying Σ with S 2 via the diﬀeomorphism ϕ , we can write the metric on Σ as h = e u g S 2 , where g S 2 is the standard round metric. Because ϕ is a conformal map, the trace of the pullbac k metric dictates that (61) e − u = 1 2 |∇ h ϕ | 2 . F urthermore, b y the standard conformal transformation law in tw o dimensions, the scalar curv ature of Σ satisﬁes (62) Sc Σ 2 = e − u  1 − 1 2 ∆ g S 2 u  . Com bining (60) with (61) giv es Sc Σ 2 = 3 r 2 − 2 e − u . Inserting this in to (62) and rearranging, we ﬁnd that u must satisfy the partial diﬀerential equation (63) ∆ g S 2 u = 6 − 6 r 2 e u . A dditionally , the co ordinate center-of-mass condition implies (64) Z S 2 x i e u dµ S 2 = Z Σ ϕ i e u e − u dµ = Z Σ ϕ i dµ = 0 . Without loss of generality , we rescale the metric so that r = 1 . Equation (63) then b ecomes ∆ g S 2 u = 6 − 6 e u , sub ject to the normalization constraint (64). This is precisely the mean-ﬁeld equation consid- ered in [32, 35]. In [32, Proposition 1 and Lemma 10], it was sho wn that if Σ satisﬁes the ev en symmetry condition ( i ) , then the only solution to (63) under the constrain t (64) is the trivial solution u ≡ 0 . Similarly , [35, Lemma 4] pro ves that if the Gauss curv ature is suﬃcien tly C 0 -close to constan t as required by ( ii ) , then (63) again admits only u ≡ 0 . Therefore, in either case, we conclude that u ≡ 0 , meaning h = g S 2 , and hence Σ is isometric to the standard round sphere. In particular, Sc Σ = 2 r 2 = − 1 2 θ ℓ θ k , and the remainder of the pro of follows exactly as in Theorem 4.1. □ Remark 4.6 (Comparison with PMC rigidit y results) . Classical rigidity results for subman- ifolds in higher codimension often assume that the mean curv ature vector is parallel in the normal bundle, ∇ ⊥  H = 0 , 28 the so-called p ar al lel me an curvatur e (PMC) condition. Results of Chen and Y au [8, 40] and subsequen t extensions to Loren tzian settings [7] sho w that this strong p oint wise assumption sev erely restricts the geometry of compact submanifolds. In ﬂat ambien t spaces it forces the submanifold to lie in a totally geo desic hypersurface, reducing the problem to co dimension one and leading to spherical rigidity . The PMC condition implies the STCMC condition since d |  H | 2 = 2 ⟨∇ ⊥  H ,  H ⟩ = 0 , but the conv erse is false: STCMC only ﬁxes the length of the mean curv ature vector and lea v es its direction in the normal bundle unconstrained. The rigidit y results of Theorems 4.1 and 4.5 show that no PMC hypothesis is required in our setting. Instead, v ariational stability together with the dominant energy condition pro vides suﬃcien t con trol of the normal connection, forcing the relev an t square terms in the stabilit y iden tit y to v anish in the equalit y case and yielding the same rigidity conclusion. Remark 4.7 (Ha wking energy in terpretation) . Theorems 4.1 and 4.5 sho w that STCMC surfaces are particularly well suited to the Ha wking quasi-lo cal energy (see (3)). The Hawking energy [14, 36] is one of the simplest prop osals for measuring the energy contained in a b ounded spacetime region and satisﬁes sev eral desirable prop erties, including the ADM limit, the small- sphere limit, and monotonicit y along in v erse mean curv ature ﬂow. How ev er, it could be negativ e on arbitrary surfaces: even in Euclidean space, ev ery nonround sphere has strictly negativ e Hawking energy . This highlights the need to identify geometrically distinguished surfaces on whic h p ositivit y can b e recov ered and the quasi-lo cal energy is w ell b eha ved. The results ab ov e sho w that STCMC surfaces pro vide suc h a natural geometric setting. On these surfaces the Ha wking energy is nonnegativ e under the dominant energy condition, and the rigidit y statemen ts sho w that v anishing energy o ccurs only in the ﬂat case. Ph ysically , this means that the Hawking energy measured on STCMC surfaces distinguishes b etw een ﬂat and curved spacetimes. 5. Examples and st ability on STCMC folia tions In this section, we illustrate the stability notions introduced ab o ve through sev eral geometric examples and applications. W e ﬁrst discuss basic classes of stable STCMC surfaces and then turn to the stabilit y of asymptotic foliations by STCMC surfaces in b oth the spacelike and n ull settings. 5.1. Examples of stable STCMC surfaces. Round spheres in Mink o wski spacetime. As sho wn abov e, a round sphere lying in a totally geo desic slice of Minko wski spacetime is a v ariational and constan t-mode stable STCMC surface. MOTS in stationary spacetimes. Marginally outer trapp ed surfaces (MOTS) arising in stationary v acuum spacetimes pro vide natural examples. A t ypical case is the ev ent horizon of a Kerr blac k hole. On such surfaces the outgoing n ull expansion v anishes, θ ℓ ≡ 0 , and therefore |  H | 2 = − θ ℓ θ k = 0 . Stationarity implies that the horizon is shear-free in the outgoing direction, while the v acuum Einstein equations give Ein = 0 . Substituting these relations into the stabilit y op erator (37) shows that L ≡ 0 . Since the righ t-hand side of Deﬁnition 3.2 also v anishes when |  H | 2 = 0 , such MOTS satisfy the STCMC stabilit y inequalities with equality . 29 Spherically symmetric spacetimes. In spherically symmetric spacetimes such as Sch w arzsc hild or Reissner-Nordström, the round co ordinate spheres are STCMC surfaces. The null expan- sions θ ℓ and θ k are constant on eac h sphere and the trace-free parts of the n ull second funda- men tal forms v anish by symmetry , so all tangential deriv atives in the stability op erator (37) disapp ear. If the spacetime is v acuum ( Ein = 0 ), the remaining terms reduce to a purely algebraic expression inv olving the expansions and the in trinsic curv ature of the sphere. In particular, in the Sc hw arzsc hild spacetime the op erator ev aluated on a constan t is prop ortional to − θ ℓ θ k m , where m is the mass parameter, showing that the symmetry spheres satisfy the constan t-mo de stabilit y condition δ 2  H | Σ | ≥ 0 outside the horizon. STCMC surfaces arising from asymptotic foliations. The stability notions introduced here apply to the kno wn asymptotic foliations b y STCMC surfaces, b oth in asymptotically Euclidean initial data sets [6] and on asymptotically Sch w arzsc hildean null hypersurfaces [17]. The precise statemen ts are pro ved in Theorem 5.3 and Theorem 5.4. 5.2. STCMC surfaces on spacelike h yp ersurfaces. Let ( N 3+1 , g ) b e a Lorentzian manifold and let ( M , g , K ) b e a spacelik e h yp ersurface with induced Riemannian metric g and second fundamen tal form K . Let Σ ⊂ M b e a smo oth closed surface with induced metric h . Denote by n the future-p ointing unit normal to M and b y ν the outer unit normal to Σ in M . The asso ciated n ull normals are deﬁned by (65)  = n + ν, k = n − ν , so that ⟨ , k ⟩ = − 2 . I n this setting, the n ull expansions are (66) θ ℓ = H + P , θ k = − H + P , where P := tr h K = tr K − K ( ν, ν ) denotes the trace of K along Σ . The spacetime mean curv ature satisﬁes (67) − θ ℓ θ k = H 2 − P 2 . Th us STCMC surfaces generalize constant mean curv ature surfaces, reducing to the classical CMC condition in the time-symmetric case K = 0 . W e now express the stability op erator (37) in terms of initial data. F or a v ector ﬁeld X tangent to Σ , the normal connection one-form asso ciated to  is giv en b y s ℓ ( X ) = − 1 2 ⟨ k , ∇ X  ⟩ = ⟨ ν , ∇ X n ⟩ = K ( X , ν ) . Hence, ∥ s ℓ ∥ 2 h = ∥ K ( · , ν ) ∥ 2 h . The n ull second fundamental forms satisfy χ ℓ = K ⊤ + B , χ k = K ⊤ − B , where B is the second fundamen tal form of Σ ⊂ M , and K ⊤ is the restriction of K to T Σ . The Einstein constrain t equations on ( M , g , K ) tak e the form Sc M − | K | 2 + (tr g K ) 2 = 2 µ, div M ( K − (tr g K ) g ) = J, where the energy density µ and momentum densit y J are deﬁned along M by µ := Ein ( n, n ) , J ( X ) := − Ein( n, X ) for all X ∈ T M . 30 Here Ein denotes the Einstein tensor of the ambien t spacetime. The canonical timelik e normal v ector ﬁeld U introduced in (34) takes the form U = H √ H 2 − P 2 n − P √ H 2 − P 2 ν. Consequen tly , 2 Ein ( U, U ) = 2 H 2 − P 2  H 2 µ + 2 H P J ( ν ) + P 2 Ein( ν, ν )  , Substituting these iden tities into the geometric expression L = θ ℓ θ k (2∆ h + V ) yields δ 2 α  H | Σ | = Z Σ α L α dµ = ( H 2 − P 2 ) Z Σ  − 2 α ∆ h α − V α 2  dµ, where (68) V =2 ∥∇ h log | H + P | − K ( · , ν ) ∥ 2 h − Sc Σ + H − P 2( H + P ) ∥ K ⊤ + B ∥ 2 h + H + P 2( H − P ) ∥ K ⊤ − B ∥ 2 h + 2 H 2 − P 2  H 2 µ + 2 H P J ( ν ) + P 2 Ein( ν, ν )  . Remark 5.1. Unlik e the Riemannian stabilit y condition for CMC surfaces, the stabilit y op er- ator for STCMC surfaces in volv es the spacetime curv ature comp onent Ein( ν , ν ) , which cannot b e expressed solely in terms of the initial data ( M , g , K ) . Consequently , STCMC stability is intrinsically a Loren tzian notion, dep ending on the embedding of the initial data set into spacetime. If w e supp ose that we are in a totally geo desic hypersurface, that is K = 0 , then our stability op erator is (69) Z Σ α L αdµ = H 2 Z Σ − 2 α ∆ h α + α 2 (Sc Σ − 1 2 H 2 − ∥ ˚ B ∥ 2 h − Sc M ) dµ = 2 H 2 Z Σ − α ∆ h α − α 2 (Ric M ( ν, ν ) + ∥ B ∥ 2 h − 1 2 H 2 ) dµ where we used the Gauss equation Sc Σ = Sc M − 2Ric M ( ν, ν ) + 1 2 H 2 − | ˚ B | 2 , Then w e ha ve that in this case the STCMC stability op erator reduces to a scaled and shifted v ersion of the classical CMC Jacobi op erator, L = 2 H 2 L Jac + H 4 . 5.3. Stabilit y of the canonical spacelik e STCMC foliation. Spacetime constan t mean curv ature surfaces pla y an imp ortant role in the geometry of asymptotically ﬂat spacetimes. In particular, Cederbaum and Sako vic h pro ved that asymptotically Euclidean initial data sets admit a canonical foliation b y STCMC surfaces near inﬁnity [6]. This foliation provides a geometric characterization of the center of mass of an isolated gravitational system and serves as a natural Loren tzian analogue of the classical constant mean curv ature foliation constructed b y Huisken and Y au in the time-symmetric case [16]. Lo cal foliations by STCMC surfaces ha v e also b een constructed inside spacelik e h yp ersurfaces [25]. In the works cited ab ov e, a notion of stability arises from the linearization in normal direction of the spacetime mean curv ature θ ℓ θ k within the initial data h yp ersurface. This op erator 31 is generally not self-adjoint and is used to establish nondegeneracy and solv e the asso ciated elliptic problem. By con trast, the stabilit y notions considered here are those introduced in Section 3.4. As an application of our theory , w e show that the lea ves of the canonical STCMC foliation in an asymptotically Euclidean initial data set are stable with resp ect to b oth of them. As an application of our theory , we sho w that the leav es of the canonical STCMC foliation in an asymptotically Euclidean initial data set are stable with resp ect to b oth notions of stability in tro duced in this pap er. W e ﬁrst recall the Big- O notation. F or functions f , g , we write f ( x ) = O ( g ( x )) ( x → ∞ ) if there exist constants C, ˆ δ > 0 suc h that | f ( x ) | ≤ C | g ( x ) | for all x ≥ ˆ δ . Deﬁnition 5.2 (Asymptotically Euclidean initial data set) . Let ε ∈ (0 , 1 2 ] and let ( M 3 , g , K, µ, J ) b e a smo oth initial data set for the Einstein equations. W e say that ( M , g , K, µ, J ) is C 2 1 / 2+ ε - asymptotic al ly Euclide an if there exist a compact set B ⊂ M and a smo oth co ordinate c hart x : M \ B − → R 3 \ B R (0) suc h that, in the co ordinates x = ( x 1 , x 2 , x 3 ) with r = | x | , the following estimates hold: | g ij − δ ij | + r | ∂ k g ij | + r 2 | ∂ k ∂ l g ij | = O ( r − 1 / 2 − ε ) , (70) | K ij | + r | ∂ k K ij | = O ( r − 3 / 2 − ε ) , (71) | µ | + | J i | = O ( r − 3 − ε ) , (72) for all r suﬃcien tly large and for all indices i, j, k , l ∈ { 1 , 2 , 3 } . Here δ ij denotes the Euclidean metric in the co ordinates x . Theorem 5.3 (Stabilit y of the canonical STCMC foliation) . L et ( M 3 , g , K, µ, J ) b e a C 2 1 / 2+ ε - asymptotic al ly Euclide an initial data set, for some ε ∈ (0 , 1 2 ] , with strictly p ositive ADM ener gy E ADM > 0 . L et { Σ σ } σ >σ 0 denote the c anonic al foliation by surfac es of c onstant sp ac etime me an curvatur e c onstructe d in [6] , wher e e ach le af satisﬁes H 2 − P 2 = 4 σ 2 . L et ν σ denote the outwar d unit normal to Σ σ in ( M , g ) , let r denote the area radius deﬁne d by | Σ σ | = 4 π r 2 , and let Ein denote the Einstein tensor of the ambient sp ac etime in which ( M , g, K ) is emb e dde d. If | Ein ( ν σ , ν σ ) | = O ( r − 2 ) , then for suﬃciently lar ge σ the le aves Σ σ ar e b oth strict c onstant- mo de stable and variational ly stable. Pr o of. First, we recall the decay of certain geometric quantities on each leaf Σ σ (see [6]): (73) ∥ ˚ B ∥ 2 = O ( r − 3 − 2 ε ) , Sc M = O ( r − 3 − ε ) W e also recall the expression of the Ha wking energy in this setting E (Σ σ ) = s | Σ σ | 16 π  1 − 1 16 π Z Σ σ H 2 − P 2 dµ  = 1 8 π s | Σ σ | 16 π  Z Σ σ Sc Σ σ − 1 2 ( H 2 − P 2 ) dµ  32 A ccording to [6, Prop osition 5.6], the ADM energy is related to the Hawking energy of eac h leaf by: (74) | E ADM − E (Σ σ ) | = O ( r − ε ) Since H 2 − P 2 = 4 /σ 2 is constan t and | Σ σ | = 4 π r 2 , the Hawking energy in tegral ev aluates exactly to: (75) E (Σ σ ) = r 2 1 − r 2 σ 2 ! . Since E ADM > 0 , for suﬃcien tly large surfaces w e ha v e E (Σ σ ) > 0 , whic h implies r < σ . Rearranging (75) yields r 2 σ 2 = 1 − 2 E (Σ σ ) r . F or large surfaces where r ≈ σ , we in v ert this relation to obtain the explicit expansion of the area radius in terms of σ : (76) 1 r 2 = 1 σ 2 + 2 E (Σ σ ) σ 3 + O ( σ − 4 ) Consequen tly , an y error term of order O ( r − k ) translates directly to O ( σ − k ) . W e note that the term V in the stabilit y op erator given b y (68) reduces under our decays to (77) V = ( H 2 − P 2 )  Sc Σ σ − 1 2 ( H 2 − P 2 )  + O ( r − 5 − ε ) . W e ﬁrst see that the lea ves are constan t-mo de stable. (78) δ 2  H | Σ σ | = Z Σ σ V dµ = ( H 2 − P 2 ) Z Σ σ Sc Σ σ − 1 2 ( H 2 − P 2 ) dµ + O ( r − 3 − 2 ε ) = 32 π σ 2 s 16 π | Σ σ | E (Σ σ ) + O ( r − 3 − 2 ε ) = 64 π E ADM σ 2 r + O ( r − 3 − 2 ε ) Since E ADM > 0 , the leading O ( r − 3 ) term strictly dominates the error, ensuring δ 2  H | Σ σ | > 0 . No w w e show that the leav es are v ariationally stable. Consider the eigenv alues λ i of − ∆ Σ σ . By [6, Lemma 5.3], λ 0 = 0 , | λ i − 2 r 2 | = O ( σ − 5 / 2 − ε ) for i = 1 , 2 , 3 , and λ i > 5 σ 2 for i ≥ 4 . Letting f i denote the corresp onding eigenfunctions, [6, Lemma 5.4] provides the estimates: (79)      λ i − 2 σ 2 + 6 E (Σ σ ) σ 3 + Z Σ σ Ric M ( ν σ , ν σ ) f 2 i dµ !      = O ( σ − 3 − ε ) , i = 1 , 2 , 3 ,     Z Σ σ Ric M ( ν σ , ν σ ) f i f j dµ     = O ( σ − 3 − ε ) , i  = j, i, j = 1 , 2 , 3 . Consider an arbitrary function α with R Σ σ αdµ = 0 and R Σ σ α 2 dµ = 1 . W e decomp ose α = α t + α d , where α t := P 3 i =1 c i f i = P 3 i =1 R Σ αf i dµf i is the translational part and α d := α − α t = P i ≥ 4 c i f i is the deformational part. Note that P 3 i =1 c 2 i = ∥ α t ∥ 2 L 2 (Σ σ ) and P i =4 c 2 i = ∥ α d ∥ 2 L 2 (Σ σ ) . Ev aluating the stability op erator yields: (80) Z Σ σ α L α dµ = Z Σ σ α t L α t dµ + 2 Z Σ σ α t L α d dµ + Z Σ σ α d L α d dµ 33 F or the deformational part, since ( H 2 − P 2 )  Sc Σ σ − 1 2 ( H 2 − P 2 )  = O ( r − 9 2 − ε ) , we ha v e: Z Σ σ α d L α d dµ = Z Σ σ 2( H 2 − P 2 ) X i =4 c 2 i λ i f 2 i dµ + Z Σ σ ( α d ) 2 V dµ ≥ 2( H 2 − P 2 ) λ 4 X i =4 c 2 i Z Σ σ f 2 i dµ + Z Σ σ ( α d ) 2 ( H 2 − P 2 )  Sc Σ σ − 1 2 ( H 2 − P 2 )  dµ + O ( ∥ α d ∥ 2 L 2 (Σ σ ) r − 5 − ε ) ≥ 40 σ 4 ∥ α d ∥ 2 L 2 (Σ σ ) − O ( ∥ α d ∥ 2 L 2 (Σ σ ) r − 9 2 − ε ) Because α t and α d are orthogonal, the cross-term is purely go v erned b y V = O ( r − 9 2 − ε ) . (81) Z Σ σ α t L α d dµ = Z Σ σ α t α d V dµ = O  ( ∥ α t ∥ 2 L 2 (Σ σ ) + ∥ α d ∥ 2 L 2 (Σ σ ) ) r − 9 2 − ε  No w using the estimate (79) for λ i Z Σ σ α t L α t dµ =2( H 2 − P 2 ) 3 X i =1 c 2 i λ i Z Σ σ f 2 i dµ + Z Σ σ ( α t ) 2 V dµ =2( H 2 − P 2 )  2 σ 2 + 6 E (Σ σ ) σ 3  3 X i =1 c 2 i + 2( H 2 − P 2 ) Z Σ σ Ric M ( ν σ , ν σ ) 3 X i =1 c 2 i f 2 i dµ + Z Σ σ ( α t ) 2 V dµ + O ( ∥ α t ∥ 2 L 2 (Σ σ ) r − 5 − ε ) No w using (79) (82) Z Σ σ Ric M ( ν σ , ν σ ) 3 X i =1 c 2 i f 2 i dµ = Z Σ σ Ric M ( ν σ , ν σ )( α t ) 2 dµ − Z Σ σ Ric M ( ν σ , ν σ ) X i  = j c i f i c j f j dµ = Z Σ σ Ric M ( ν σ , ν σ )( α t ) 2 dµ + O ( σ − 3 − ε ) Note that R Σ σ ( α t ) 2 V dµ = R Σ σ ( α t ) 2 ( H 2 − P 2 )  Sc Σ σ − 1 2 H 2  dµ + O ( ∥ α t ∥ 2 L 2 (Σ σ ) r − 5 − ε ) furthermore, b y the Gauss equation Sc Σ σ + 2Ric M ( ν σ , ν σ ) − 1 2 H 2 = Sc M − ∥ ˚ B ∥ 2 = O ( r − 3 − ε ) obtaining Z Σ σ α t L α t dµ =2( H 2 − P 2 )  2 σ 2 + 6 E (Σ σ ) σ 3  ∥ α t ∥ 2 L 2 (Σ σ ) + Z Σ σ ( α t ) 2 ( H 2 − P 2 )  Sc Σ σ + 2Ric M ( ν σ , ν σ ) − 1 2 H 2  dµ + O ( ∥ α t ∥ 2 L 2 (Σ σ ) r − 5 − ε ) =  16 σ 4 + 48 E (Σ σ ) σ 5  ∥ α t ∥ 2 L 2 (Σ σ ) + O ( ∥ α t ∥ 2 L 2 (Σ σ ) r − 5 − ε ) Summing the con tributions, the stabilit y op erator is b ounded b elo w by: Z Σ σ α L αdµ ≥  16 σ 4 + 48 E (Σ σ ) σ 5  ∥ α t ∥ 2 L 2 (Σ σ ) + 40 σ 4 ∥ α d ∥ 2 L 2 (Σ σ ) + O  ( ∥ α t ∥ 2 L 2 (Σ σ ) + ∥ α d ∥ 2 L 2 (Σ σ ) ) r − 9 2 − ε  F or v ariational stabilit y , we require this integral to be strictly greater than the threshold constan t H 2 − P 2 | Σ σ | 16 π = 16 r 2 σ 2 . Note that by using (74) we can replace the Hawking energy b y the ADM energy E ADM and by using (76) to replace σ b y r w e ha v e R Σ σ α L αdµ > 16 r 2 σ 2 for E ADM > 0 . Th us, the surfaces are strictly v ariationally stable for suﬃcien tly large σ . □ 34 5.4. STCMC surfaces on null h yp ersurfaces. In [17], Krönc k e and W olﬀ construct an asymptotic foliation by STCMC surfaces on asymptotically Sc hw arzsc hildean ligh tcones. This pro vides a genuinely Loren tzian family of spacelik e STCMC surfaces, since the lea v es foliate a null hypersurface rather than a spacelik e initial data set. Moreov er, the foliation is unique within a suitable a-priori class. W e now show that the leav es of this foliation also satisfy our notions of constan t-mo de and v ariational stabilit y . Theorem 5.4 (Stability in the null STCMC foliation) . L et N b e an asymptotic al ly Schwarzschilde an lightc one of p ositive mass m > 0 , and let { Σ σ } σ ≥ σ 0 b e the asymptotic foliation by STCMC sur- fac es c onstructe d in [17] . Then, for σ suﬃciently lar ge, the le aves Σ σ ar e strictly c onstant-mo de stable and strictly variational ly stable. Pr o of. W e com bine the geometric stability op erator (44) with the asymptotic estimates for the null STCMC foliation obtained in [17]. W e ﬁrst pro ve constant-mode stability . Since L = θ ℓ θ k (2∆ h + V ) and − θ ℓ θ k = |  H | 2 > 0 , ha ving constan t-mo de stability is equiv alen t to R Σ σ V dµ ≤ 0 . Note that the p oten tial in (44) is (83) V = 2 ∥∇ h log | θ ℓ | − s ℓ ∥ 2 h − Sc Σ σ + 2 Ein( U, U ) − 1 2 |  H | 2 − θ k 2 θ ℓ ∥ ˚ χ ℓ ∥ 2 h − θ ℓ 2 θ k ∥ ˚ χ k ∥ 2 h = 1 2 |  H | 2 − Sc Σ σ + O ( σ − 4 ) where we used the asymptotics of the foliation from [17, Lemma 2.14]. Moreov er, the leav es satisfy (84) |  H | 2 = 4 σ 2 − 8 m σ 3 + O ( σ − 4 ) , Sc Σ σ = 2 σ 2 + O ( σ − 4 ) , | Σ σ | = 4 π σ 2 . Therefore, using that Σ σ is a sphere, (83) and (84): − Z Σ σ V dµ = 8 π − 1 2  4 σ 2 − 8 m σ 3 + O ( σ − 4 )  (4 π σ 2 ) + O ( σ − 2 ) = 16 π m σ + O ( σ − 2 ) . Since m > 0 , the righ t-hand side is positive for σ suﬃcien tly large. Hence R Σ σ V dµ < 0 , and this prov es strict constant-mode stabilit y . W e no w turn to v ariational stability . Let α ∈ C ∞ (Σ σ ) satisfy Z Σ σ α dµ = 0 . T o pro v e v ariational stabilit y , it suﬃces to sho w that (85) Z Σ σ  2 |∇ h α | 2 − V α 2  dµ ≥ 16 π | Σ σ | Z Σ σ α 2 dµ. T o estimate the gradient term, w e use the Jacobi op erator J considered in [17]. Although this op erator giv es a diﬀerent notion of stabilit y , its prop erties will b e useful. The pro of of [17, Prop osition 3.10] yields, after in tegration b y parts, Z Σ σ J ( α ) α dµ = Z Σ σ  2 |∇ h α | 2 − 2Sc Σ σ α 2 − ( |  H | 2 − 2Sc Σ σ ) α 2 − E σ α 2  dµ, where ∥ E σ ∥ L ∞ (Σ σ ) = O ( σ − 4 ) . After cancellation of the scalar curv ature terms, this b ecomes Z Σ σ 2 |∇ h α | 2 dµ = Z Σ σ J ( α ) α dµ + Z Σ σ |  H | 2 α 2 dµ + Z Σ σ E σ α 2 dµ. 35 By [17, Prop osition 3.10], Z Σ σ J ( α ) α dµ ≥ 6 m σ 3 Z Σ σ α 2 dµ, and therefore (86) Z Σ σ 2 |∇ h α | 2 dµ ≥ Z Σ σ  |  H | 2 + 6 m σ 3 − O ( σ − 4 )  α 2 dµ. Com bining (86) with the p oten tial expansion (83), w e obtain Z Σ σ  2 |∇ h α | 2 − V α 2  dµ ≥ Z Σ σ  1 2 |  H | 2 + Sc Σ σ + 6 m σ 3 − O ( σ − 4 )  α 2 dµ. Using (84), 1 2 |  H | 2 + Sc Σ σ = 4 σ 2 − 4 m σ 3 + O ( σ − 4 ) , and hence Z Σ σ  2 |∇ h α | 2 − V α 2  dµ ≥ Z Σ σ  4 σ 2 + 2 m σ 3 − O ( σ − 4 )  α 2 dµ. Since 16 π | Σ σ | = 4 σ 2 , the p ositive term 2 m σ 3 dominates the O ( σ − 4 ) error for σ suﬃcien tly large. Therefore Z Σ σ  2 |∇ h α | 2 − V α 2  dµ > 16 π | Σ σ | Z Σ σ α 2 dµ. the leav es are strictly v ariationally stable for all suﬃciently large σ . □ Appendix A. Auxiliar y Rigidity Theorems In this app endix w e collect rigidity results used in the equalit y cases throughout the pap er. These results are classical and are stated here for the reader’s conv enience. Riemannian mo del geometries. Euclidean mo del: Bro wn-Y ork rigidit y. In the time-symmetric case K = 0 , the Kijo wski- Liu-Y au energy reduces to the Bro wn-Y ork mass, and rigidit y follo ws from the fundamental result of Shi-T am. Theorem A.1 (Shi-T am [33, Theorem 1]) . L et (Ω , g ) b e a c omp act manifold of dimension thr e e with a smo oth b oundary and with nonne gative sc alar curvatur e. Supp ose ∂ Ω has ﬁnitely many c omp onents Σ i such that e ach c omp onent has p ositive Gaussian curvatur e and p ositive me an curvatur e H i with r esp e ct to the unit outwar d normal. Then for e ach c omp onent, (87) Z Σ i H i dµ ≤ Z Σ i H i 0 dµ, wher e H i 0 denotes the me an curvatur e of the unique c onvex isometric emb e dding of Σ i into R 3 . Mor e over, if e quality holds for some Σ i , then ∂ Ω is c onne cte d and Ω is isometric to a domain in R 3 . The existence and uniqueness (up to rigid motions) of the conv ex isometric em b edding in to R 3 follo w from the classical W eyl-Nirenberg-Pogorelo v theorem. Theorem A.2 (W eyl-Nirenberg-Pogorelo v) . L et ( S 2 , g ) b e a C k,α ( k ≥ 3 , α ∈ (0 , 1) ) Rie- mannian 2 -spher e with Gaussian curvatur e K g > 0 . Then ther e exists a strictly c onvex iso- metric emb e dding X : ( S 2 , g )  → ( R 3 , g Eucl ) , unique up to orientation-pr eserving rigid motions of R 3 . In [33] Shi and T am also prov ed a higher dimensional version of Theorem A.1: 36 Theorem A.3 ([33, Theorem 4.1]) . L et (Ω , g ) b e a c omp act Riemannian manifold of dimen- sion n ≥ 3 , with smo oth b oundary ∂ Ω and nonne gative sc alar curvatur e. A ssume 3 ≤ n ≤ 7 or Ω is spin. Supp ose the b oundary has ﬁnitely many c onne cte d c omp onents Σ i such that e ach c omp onent has p ositive me an curvatur e H i with r esp e ct to the unit outwar d normal and c an b e isometric al ly emb e dde d in R n as a c onvex hyp ersurfac e. Then for e ach c omp onent Z Σ i H i dµ ≤ Z Σ i H i 0 dµ, wher e H i 0 is the me an curvatur e of the isometric emb e dding of Σ i in the Euclide an sp ac e. Mor e over, if e quality holds for some Σ i , then ∂ Ω is c onne cte d (i.e., ∂ Ω = Σ i ) and Ω is isometric to a domain in R n . Remark A.4. The dimensional and spin restrictions in Theorem A.3 arise from the use of the p ositiv e mass theorem in the original pro of. Extensions of the p ositiv e mass theorem to higher dimensions without the spin assumption ha ve been announced by Lohkamp [21, 22] and by Sc ho en-Y au [31]. More recen tly , a preprin t [3] claims a pro of of the p ositiv e mass theorem up to dimension 19 . Whenev er Theorem A.3 is applied in dimensions n ≥ 8 in this pap er, the corresp onding state- men ts should therefore b e understo o d under the assumption that the p ositiv e mass theorem holds in that dimension. T o identify the geometry of these higher-dimensional embeddings when the in trinsic scalar curv ature is constant, w e rely on the follo wing classical rigidity theorem by Ros. Theorem A.5 (Ros [30, Theorem 1]) . L et Σ n − 1 ⊂ R n b e a close d, c onne cte d, emb e dde d hyp ersurfac e with c onstant intrinsic sc alar curvatur e. Then Σ is a r ound spher e. Hyp erb olic mo del. Shi and T am also established the corresp onding rigidit y result when h yp erb olic space serves as the reference geometry . Theorem A.6 (Shi-T am [34, Theorem 3.8]) . L et (Ω , g ) b e a c omp act Riemannian manifold with smo oth b oundary Σ . A ssume: (i) Sc Ω ≥ 2Λ for some Λ < 0 , (ii) Σ is a top olo gic al spher e with Gaussian curvatur e K Σ > Λ 3 and p ositive me an curvatur e H . Then Σ admits a c onvex isometric emb e dding into hyp erb olic sp ac e H 3 Λ / 3 with me an curvatur e H 0 , and Z Σ ( H 0 − H ) dµ ≥ 0 . Equality holds if and only if (Ω , g ) is isometric to a domain in H 3 Λ / 3 . Spherical mo del. In the p ositive curv ature setting, rigidity requires stronger curv ature as- sumptions. The follo wing theorem of Hang-W ang pro vides the appropriate mo del rigidity . Theorem A.7 (Hang-W ang [13, Theorem 2]) . L et ( M , g ) b e a c omp act n -dimensional Rie- mannian manifold ( n ≥ 2 ) with nonempty b oundary Σ . A ssume Ric M ≥ ( n − 1) g , (Σ , g Σ ) is isometric to a r ound spher e, and the se c ond fundamental form of Σ is nonne gative. Then ( M , g ) is isometric to the hemispher e S n + . 37 This result can be viewed as a Ricci-strengthened version of Min-Oo’s conjecture [28]. The original scalar-curv ature formulation of the conjecture w as later dispro ved by Brendle, Mar- ques, and Nev es [5]. Spacetime quasi-lo cal rigidit y. F or the Lorentzian rigidit y arguments in Section 4, w e also use the follo wing p ositivit y and rigidit y results for the Kijowski-Liu-Y au quasi-lo cal energy . Theorem A.8 (Liu-Y au [19, 20, Theorem 1]) . L et (Ω , g , K ) b e a c omp act initial data set satisfying the dominant ener gy c ondition. Supp ose ∂ Ω has ﬁnitely many c omp onents Σ i , e ach with p ositive Gaussian curvatur e and sp ac elike me an curvatur e ve ctor. Then E K LY (Σ α ) ≥ 0 . Mor e over, if e quality holds for some c omp onent, then ∂ Ω is c onne cte d and Ω is isometric to a sp ac elike hyp ersurfac e in Minkowski sp ac etime. Sp e ciﬁc al ly, Ω c an b e isometric al ly emb e dde d in R 3 , 1 as a sp ac elike gr aph ( x, f ( x )) over a sp atial domain Ω 0 ⊂ R 3 , wher e f is a smo oth function on Ω 0 that vanishes on ∂ Ω 0 . The rigidity of the Kijowski-Liu-Y au energy in Minko wski spacetime is even stronger, this w as ﬁrst observed by Ó Murchadha and Szabados in [41] and w as later fully characterized by Miao, Shi, and T am in the following result. Theorem A.9 ([26, Theorem 4.1]) . L et Σ b e a close d, c onne cte d, smo oth, sp ac elike 2 -surfac e in Minkowski sp ac etime R 3 , 1 . Supp ose Σ sp ans a c omp act sp ac elike hyp ersurfac e in R 3 , 1 . If Σ has p ositive Gaussian curvatur e and a sp ac elike me an curvatur e ve ctor, then E K LY (Σ) ≥ 0 . Mor e over, E K LY (Σ) = 0 if and only if Σ lies on a hyp erplane in R 3 , 1 . Sp ectral estimates on the sphere. F or the alternative rigidit y argumen t in Section 4, we also use the following estimate of El Souﬁ and Ilias for the second eigen v alue of a Sc hrö dinger op erator on a genus-zero surface. Theorem A.10 (El Souﬁ-Ilias [12]) . L et (Σ , h ) b e a close d surfac e of genus zer o, let q ∈ C 0 (Σ) , and deﬁne the se c ond eigenvalue λ 2 ( − ∆ h + q ) := inf α ∈ C ∞ (Σ) R Σ α dµ =0 Z Σ  |∇ h α | 2 + q α 2  dµ Z Σ α 2 dµ . Then λ 2 ( − ∆ h + q ) | Σ | ≤ 8 π + Z Σ q dµ. Mor e over, e quality holds if and only if Σ admits a c onformal map into the standar d S 2 whose c o or dinate functions ar e se c ond eigenfunctions of − ∆ h + q . References 1. Aghil Alaee, Martin Lesourd, and Shing-T ung Y au, Stable surfac es and fr e e b oundary mar ginal ly outer tr app e d surfac es , Calculus of V ariations and Partial Diﬀerential Equations 60 (2021), no. 5, 186. 2. 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