A proof by calibration of an isoperimetric inequality in the Heisenberg group H^n

A PR OOF BY CALIBRA TION OF AN ISOPERIMETRIC INEQUALITY IN THE HEISENBE R G GR OUP H n MANUEL RITOR ´ E Abstract. Let D b e a closed disk cente red at t he origin in the hori zon tal hyper- plane { t = 0 } of the sub-Riemannian Heisenberg group H n , and C the vertical cylinder o v er D . W e pr o ve that any finite p erim eter set E such that D ⊂ E ⊂ C has p erimeter l ar ger than or equal to the one of the rotationally symmetric sphere with constan t mean curv ature of the same vo lume, and that equality holds only for these spheres using a recen t r esult by M on ti and Vi ttone [ 12]. 1. Introduction It was conjectured by P . Pansu in 1 983 [14] that the isop erimetric regions, minimiz- ing per imeter under a volume c o nstraint, in the sub-Riemannian Heisenberg gr o up H 1 are the top olog ic al balls enclosed by the one-para meter family { S λ } λ> 0 of r otationally symmetric spher es of co nstant mean curv a tur e λ des crib ed in [14], se e also [7], [16]. The existence of isop erimetric regions in Carno t gr oups was prov en b y G.-P . Leo nar- di a nd S. Rigot [6]. Every C a rnot g roup G is equipp ed with an one- pa rameter family of dilations whic h has a well-known effect on the p erimeter and the Haar meas ure, the volume of the group. Hence the is op erimetric profile of G , the function assig ning to each volume v > 0 the infimum of the p erimeter o f the sets of volume v , is given by I G ( v ) = C G v ( Q − 1) /Q , where Q > 0 is the homogeneous dimension o f G , and C G > 0 is a constant. The isop erimetric profile must be seen as a n optimal iso p er imetric inequa lit y in G . F or any finite p erimeter s et E ⊂ G , (*) P G ( E ) > C G | E | ( Q − 1) /Q , where P G is the sub-Riemannian p erimeter and | E | is the volume of E . Pansu’s conjecture then states that, for G = H 1 , which has ho mo geneous dimension Q = 4, equality is attained in (*) prec is ely when E is the to p o logical ball enclosed by s ome sphere S λ . This conjecture can be extended to the higher dimensional Heisenberg groups H n , n > 2. Several attempts to so lve this conjecture have b een made. R. Monti [10] and G.- P . Leo nardi and S. Masnou [7] have shown that there is no dir ect counterpart in H 1 to the Brunn-Minko wski inequa lit y in Euclidean spa ce. In fact, such a Br unn-Minko wsk i t yp e inequality would imply that the metric ba lls for the Car no t-Carath´ edory distance Date : August 27, 2018. 2000 Mathematics Subje c t Classific ation. 53C17, 53C42, 49Q20. Key wor ds and phr ases. Sub-Ri emannian geometry , Heisenberg group, ar ea-stationary surface, constan t mean curv ature surface, isop erimetric problem, is operi m etric region, calibration. Researc h supported by MCyT-F eder gran t MTM 2010-21206-C02-01 and Jun ta de Andaluc ´ ıa gran ts FQM-325 and P09-FQM-5088. 2 M. RITOR ´ E would b e iso p e rimetric r egions, which is known to b e false [9]. T he author a nd C. Ros- ales show ed in [16] that the only compa ct ro tationally symmetric C 2 hypersurfaces in H n with co nstant mean curv a ture are the spher e s S λ , see als o [13]. The sa me a uthors prov ed in [17] tha t the sphere s S λ are the o nly compact C 2 surfaces in H 1 which are ar ea-statio na ry under a volume c o nstraint, thus solving the isop erimetric problem assuming C 2 regular ity of the solutions. R. Monti and M. Rickly [1 1] prov ed that the spheres S λ are isop erimetric in H 1 under the additional ass umption of Euclidean con- vexit y . In [3], D. Da nielli et al. pr ovided a pro of of the isop er imetric prop er ty of the spheres S λ ⊂ H n in the class of se ts that are the union of the gr aphs of a no n-negative and a non- po sitive function a nd a negative gr aph o f class C 2 ov er a E uclidean disk centered at the orig in in the hor izontal hyperplane { t = 0 } in H n , and enclo sing the same volume ab ov e and below such hyperplane. F or a des c ription of these results, and some other appr oaches, the reader may consult Cha pter 8 of the mono graph by L. Ca po gna et al. [2]. In this pa per we extend the main result in [3]. W e prov e in Theor e m 3 .1 that, if C is the vertical cy linder in H n ov er a closed disk D centered a t the or igin in the hor i- zontal hyper plane { t = 0 } , a nd E ⊂ H n is a finite p erimeter s e t so that D ⊂ E ⊂ C , then the p erimeter o f E is lar ger tha n o r equa l to the one o f the ball B λ enclosed by the spher e S λ with | B λ | = | E | . Equalit y characterize s the spher es S λ by a rec ent result of R. Monti and D. Vittone [12], who prov ed that a set in H n of lo ca lly finite per imeter with contin uous hor izontal unit norma l has H - regular b o unda ry . Theo- rem 3.1 can b e applied to a set E ⊂ H n rotationally s ymmetric with resp ect to a vertical axis pas s ing through the origin and sa tisfying D ⊂ E . Assumption D ⊂ E has been recently removed by R. Monti [8], who has prov en, us ing Theorem 3.1 and a symmetrizatio n ar gument, that the spheres S λ ⊂ H n are isop erimetric in the class of rota tio nally sy mmetr ic s ets of finite per imeter. Theorem 3.1 and the result o f R. Monti and D. Vittone [12] to gether provide the only known character ization result for solutio ns of the isop erimetric problem in H n in the class of finite p erimeter s ets. One of the cla ssical pro o fs of the isop er imetric inequality in R 3 was given by H. A. Sch warz [22], and extended to higher dimensional Euclidean space s, spheres and hyperb olic space s in a series of pap ers by E. Schmidt [18], [19], [20], [21]. Sch warz’s pro of had tw o main ingredients: wha t it is k nown now adays a s Sch warz’s s ymmetriza- tion and a v ariatio nal a rgument to prove that in the c lass of ro tationally symmetric sets spheres hav e the smallest p erimeter under a volume co nstraint. A unified ar gu- men t, using ca libr ations, ca n b e used to g ive a pro o f of the second par t of Sch warz’s argument in a wide class of homogeneous Riemannian manifolds [15, § 1.3.1 ], and in sub-Riemannia n manifolds. A s y mmetrization in the sub-Riemannia n Heisenberg group H n is difficult to pro duce due to the lack of re flections with resp ect to hyper- planes, on which a ll classica l symmetrizations are bas ed. In the pro of o f Theor e m 3.1 we cons ider the r ight cy linder C ⊂ H n ov er a c lo sed Euclidean disk D in the horizo nt al hyper plane { t = 0 } . O n C we constr uct tw o foliations by vertically translating the upp er hemisphere and the low er one of the only s pher e S λ int ersecting { t = 0 } at ∂ D . Using these foliations we prove that the sphere S λ minimize the functional ar ea − n λ volume in the class of finite p erimeter sets E ⊂ H n satisfying D ⊂ E ⊂ C . Then we minimize ov er the spheres S µ the functiona l area − nµ (volume − | E | ) to g e t the desired result. The reader sho uld co mpare our pro of with the one given by E. Schmidt [1 9] of the isop er imetr ic pr op erty of balls in the n -dimens io nal sphere S n in the class of rotatio nally symmetric sets. AN ISOPERIME TRIC INEQUALITY IN THE HEISENBERG GR OUP H n 3 W e hav e o rganized this pap er into tw o sections. In the following one we state some mater ial needed in the pro of o f Theo rem 3.1. Pro o fs of the results which are essential but cannot b e found in the literature, in particula r of Lemmae 2.2, 2.3, 2.4 are outlined. In section 3 w e give the pr o of of o ur main result Theo rem 3.1. The autho r is extremely grateful to Rob er to Monti and Davide Vittone for sending him a copy o f their manuscript [12], and to the referee for his v a luable sugg estions. 2. Preliminaries 2.1. The Heise n b erg group. The Heisenberg gro up H n is the Lie group ( R 2 n +1 , · ), where we c o nsider in R 2 n +1 ≡ C n × R its usual differ ent iable structure and the pr o duct ( z , t ) · ( w , s ) = ( z + w, t + s + n X i =1 Im( z i ¯ w i )) . A basis of left-inv a riant vector fields is given by { X 1 , ..., X n , Y 1 , ..., Y n , T } , wher e X i = ∂ ∂ x i + y i ∂ ∂ t , Y i = ∂ ∂ y i − x i ∂ ∂ t , i = 1 , . . . , n ; T = ∂ ∂ t . The only non-trivia l br ack et relations are [ X i , Y i ] = − 2 T , i = 1 , . . . , n . The horizontal distribution at a p oint p ∈ H n is defined by H p := s pa n { ( X i ) p , ( Y i ) p : i = 1 , . . . , n } . W e shall consider on H n the left-in v ariant Riema nnian metric g =  · , ·  so that the basis { X 1 , ..., X n , Y 1 , ..., Y n , T } is ortho no rmal. The horizontal pr oje ction of a vector field U in H n , denoted by U H , is the or thogonal pro jection of U ov er H . The L evi-Civita connection o n ( H n , g ) is denoted b y D . F ro m Kos zul formula and the Lie brack et relations we get D X i X j = D Y i Y j = D T T = 0 , D X i Y j = − δ ij T , D X i T = Y i , D Y i T = − X i , (2.1) D Y i X j = δ ij T , D T X i = Y i , D T Y i = − X i . F or any vector field U on H n we define J ( U ) := D U T . It follows from (2 .1 ) that J ( X i ) = Y i , J ( Y i ) = − X i , and J ( T ) = 0, s o that J defines a linear isometry when restricted to the ho r izontal distribution. 2.2. V olum e and sub-Riem annian p e rimeter. The volume | E | of a Bor el set E ⊂ H n is the Riemannia n volume of E with resp ect to the metric g . The ar e a A (Σ) of a C 1 hypersurface Σ ⊂ H n is defined as A (Σ) := ˆ Σ | N H | d Σ , where d Σ is the area element induced on Σ by the Riemannian metric g , and N is a lo cally defined unit vector norma l to Σ. F or a C 2 hypersurface enclos ing a bo unded region E , the a rea coincides with the sub-Riemannian per imeter | ∂ E | , defined as | ∂ E | (Ω) := sup  ˆ Ω div U dv : U horizontal of class C 1 , | U | 6 1 , supp( U ) ⊂ Ω  , where Ω ⊂ H n is a n o p en s et, div U is the Riemannia n divergence of the vector field U , and dv is the volume element ass o ciated to g . A set E ⊂ H n is o f lo c al ly fin ite p erimeter if | ∂ E | (Ω) < + ∞ for all b ounded op en sets Ω ⊂ H n . It is of finite p erimeter if | ∂ E | := | ∂ E | ( H n ) < + ∞ . W e normalize 4 M. RITOR ´ E any finite p erimeter s et E ⊂ H n to include its density one p oints and to exclude its density zero p oints [5 , P rop. 3.1]. The r e duc e d b oundary ∂ ∗ E of a set E ⊂ H n of lo ca lly finite p erimeter, as defined in [4, Def. 2.1 7], is co mpo sed of those p oints p ∈ H n such that (1) | ∂ E | ( U ( p, r )) > 0 for all r > 0, (2) there exists lim r → 0 ffl U ( p,r ) ν H d | ∂ E | , (3)   lim r → 0 ffl U ( p,r ) ν H d | ∂ E |   = 1 . Here U ( p, r ) is the o p e n metric ball with r esp ect to the distance induced by the homogeneous no r m (2.2) || p || ∞ := ma x {| z | , | t | 1 / 2 } , p = ( z , t ) , which is globally equiv alent to the Carno t-Carath´ eo dory distance, [4, Prop.2 .7]. The per imeter measure o f a set of lo cally finite p erimeter is supp orted on the reduce d bo undary a s shown in [4, Thm. 7 .1 ]. 2.3. Hyp ersurfaces in H n and v ariational formulae. F or a C 1 hypersurface Σ ⊂ H n , the singular set Σ 0 ⊂ Σ consists of the p oints where the tange nt hyper - plane coincides with the hor izontal distribution. The s et Σ 0 is clo sed and has e mpty int erior in Σ, and so the r e gu lar set Σ \ Σ 0 of Σ is op en and dense in Σ. F or any p ∈ Σ \ Σ 0 , the tangent h yp erplane meets transversally the ho rizontal distr ibution, and so T p Σ ∩ H p is (2 n − 1)-dimensiona l. W e say that Σ is two-side d if there is a globally defined unit vector field normal to Σ. Every C 1 hypersurface is lo cally t wo-sided. Let Σ b e a C 2 hypersurface in H n , a nd N a unit vector nor mal to Σ. The singular set Σ 0 ⊂ Σ can b e describ ed as Σ 0 = { p ∈ Σ : N H ( p ) = 0 } . In the r egular part Σ \ Σ 0 , we c a n define the horizontal unit normal ve ctor ν H by ν H := N H / | N H | . Co nsider the unit vector field Z on Σ \ Σ 0 given by Z := J ( ν H ). As Z is hor iz o ntal and or thogonal to ν H , it follows that Z is tangent to Σ. The integral curves o f Z in Σ \ Σ 0 will be called char acteristic curves . Character istic curves foliate the reg ular part o f Σ. Consider a C 1 vector field U with compact supp or t on H n , a nd deno te by { ϕ t } t ∈ R the asso c iated gr oup o f diffeomo rphisms. Let E b e a b o unded reg ion enclo sed by a hypersurface Σ. The families { E t } , { Σ t } , for t small, are the v ariations of E and Σ induced by U . Let V ( t ) = | E t | and A ( t ) = A (Σ t ). W e say that the v aria tion is volume-pr eserving if V ′ (0) = 0. W e s ay that Σ is ar e a-stationary if A ′ (0) = 0 for any v a riation, a nd volume-pr eserving ar e a-stationary if A ′ (0) = 0 for any volume preserving v a riation. If Σ is a C 1 hypersurface enclo sing a bo unded re g ion E , it is well-kno wn that (2.3) V ′ (0) = ˆ E div U dv = − ˆ Σ u d Σ , where u =  U, N  and N is the unit vector norma l to Σ p ointing into E . If Σ is C 2 , and N is a unit vector field nor mal to Σ, the me an curvatu r e of Σ \ Σ 0 is given by − nH := div Σ ν H , where div Σ U ( p ) := P 2 n i =1  D e i U, e i  for any orthonor mal basis { e i } of T p Σ. W e s ay that Σ ha s constant mean cur v ature if H is c o nstant on Σ \ Σ 0 . By combining [16, Lemma 3.2] and [17, Lemma 4.3 (4.7 )] we hav e Lemma 2.1 . L et Σ ⊂ H n b e a C 2 hyp ersurfac e enclosing a b ounde d r e gion E , with inner unit normal ve ctor N . Consider a variation induc e d by a ve ctor field U , and let AN ISOPERIME TRIC INEQUALITY IN THE HEISENBERG GR OUP H n 5 u =  U, N  . Assume that Σ is volume-pr eserving ar e a-stationary, and let H b e the me an cu rvatur e of Σ . Then we have A ′ (0) = − ˆ Σ nH u d Σ . W e remark that the v aria tion ass o ciated to U in the sta temen t o f Le mma 2.1 is not assumed to b e volume-preserving. F rom Lemma 2.1 a nd (2.3) we eas ily obtain Lemma 2.2. L et Σ ⊂ H n b e a C 2 hyp ersurfac e enclosing a b ounde d r e gion E . As- sume that Σ is volume-pr eserving ar e a-stationary, and let H b e the ( c onstant ) me an curvatur e of Σ . Then Σ is a critic al p oint of the funct ional A − n H V for any variation. Let Σ b e a tw o -sided C 2 hypersurface without singular p oints. W e can translate it vertically to get a foliation of the vertical cylinder C ov er Σ . Denote by N the unit normal to the foliation, a nd by ν H the hor izontal unit normal obtained from N . F or any p ∈ C , let { e i } b e an ortho normal basis of the leaf pas sing through p . Then div ν H ( p ) = 2 n X i =1  D e i ν H , e i  +  D N p ν H , N p  = − nH ( p ) +  D N p ν H , N p  . Since N = | N H | ν H +  N , T  T , and  D U ν H , ν H  ,  D T ν H , T  , a nd  D ν H T , ν H  =  J ( ν H ) , ν H  v anish, we conclude that  D N p ν H , N p  = 0 . Hence (2.4) − nH = div ν H . In [16, Lemma 3.1 (3.3)] it is pr ov en that (2.5) D u ν H = | N H | − 1 2 n − 1 X i =1  D u N , z i  −  N , T   J ( u ) , z i  z i +  z , u  T , where u ∈ T p Σ for a given p oint p ∈ Σ, and { z 1 , . . . , z 2 n − 1 } is an orthono r mal basis of T p Σ ∩ H p , with z 1 = z = J (( ν H ) p ). Completing { z i } to a n orthono rmal ba sis of T p Σ by adding a v ector v , we o btain from (2.5) that  D v ν H , v  = 0 . Hence we conclude − nH ( p ) = 2 n − 1 X i =1  D z i ν H , z i  , where { z i } is an orthonor mal ba sis of T p Σ ∩ H p . F rom (2 .5), it follows that the en- domorphism v 7→ − D v ν H − | N H | − 1  N , T  J ( v ) ⊤ , defined in the subspace T p Σ ∩ H p , p ∈ Σ \ Σ 0 , is s elfadjoint. Th us there exists an orthonor mal bas is { v 1 , . . . , v 2 n − 1 } of T p Σ ∩ H p comp osed o f eig env ector s with eigenv alues κ 1 , . . . , κ 2 n − 1 . By ana logy with the Riemannian ca se they will b e named princip al curvatu r es , and we hav e − nH = κ 1 + . . . + κ 2 n − 1 . 2.4. Geo desics in H n . W e refer the reader to [16, § 3] for detailed ar g ument s. Geo- desics in H n are horizontal cur ves γ : I → H n which ar e critica l p oints of the Rie- mannian length L ( γ ) := ´ I | ˙ γ | for any v aria tion by horiz o ntal curves γ ε . A vector field U a long γ induces a v ar iation by hor izontal curves if a nd only if (2.6) ˙ γ  U, T   + 2  ˙ γ , J ( U )  = 0 . The der iv ative of length fo r such a v ar iation is given by (2.7) d dε     ε =0 L ( γ ε ) = − ˆ I  D ˙ γ ˙ γ , U  . 6 M. RITOR ´ E Observe that D ˙ γ ˙ γ is o rthogona l to b oth ˙ γ and T . Along γ consider the o rthonorma l basis of T H n given by T , ˙ γ , J ( ˙ γ ), Z 1 , . . . , Z 2 n − 2 . In the same wa y as for the cas e of H 1 , see [17, § 3], we take any smo o th f : I → R v anishing a t the endp oints of I so that ´ I f = 0. The vector field U along γ so that U H = f J ( ˙ γ ), and  U, T  = 2 ´ I f , satisfies (2.6). Hence (2.7) a llows us to conclude that  D ˙ γ ˙ γ , J ( ˙ γ )  is cons tant. Now let f : I → R b e a n y smo o th function v anis hing at the endpo int s of I . Then the vector field U = f Z i , for any i = 1 , . . . , 2 n − 2, satisfies (2.6), and hence D ˙ γ ˙ γ is orthogo nal to Z i for all i = 1 , . . . , 2 n − 2. So we obtain that the hor izontal geo desic γ : I → H n satisfies the equation (2.8) D ˙ γ ˙ γ + 2 λ J ( ˙ γ ) = 0 , for some consta nt λ ∈ R . F or λ ∈ R , p ∈ H n , and v ∈ T p H n , | v | = 1 , the geo desic γ : I → H n of cur v ature λ with initia l conditions γ (0 ) = p , ˙ γ (0) = v , will b e denoted by γ λ p,v . The equations of a g eo desic ca n b e co mputed in co ordina tes in the following way: let γ ( s ) = ( x 1 ( s ) , y 1 ( s ) , . . . , x n ( s ) , y n ( s ) , t ( s )) be a horizontal geo desic. Then ˙ γ ( s ) = n X i =1 ˙ x i ( s ) ( X i ) γ ( s ) + ˙ y i ( s ) ( Y i ) γ ( s ) , and ˙ t ( s ) = n X i =1 ( ˙ x i y i − x i ˙ y i )( s ) . So equation (2.8) is transfor med in ¨ x i = 2 λ ˙ y i , ¨ y i = − 2 λ ˙ x i , with initial conditions x i (0) = ( x 0 ) i , y i (0) = ( y 0 ) i , and ˙ x i (0) = A i , ˙ y i (0) = B i , with P n i =1 ( A 2 i + B 2 i ) = 1. Int egrating these equations, fo r λ = 0, we o btain x i ( s ) = ( x 0 ) i + A i s, y i ( s ) = ( y 0 ) i + B i s, t ( s ) = t 0 + n X i =1 ( A i ( y 0 ) i − B i ( x 0 ) i ) s, which are horizontal E uclidean stra ight lines in H n . Int egrating, for λ 6 = 0, we obtain x i ( s ) = ( x 0 ) i + A i  sin(2 λs ) 2 λ  + B i  1 − cos(2 λs ) 2 λ  , y i ( s ) = ( y 0 ) i − A i  1 − cos(2 λs ) 2 λ  + B i  sin(2 λs ) 2 λ  , t ( s ) = t 0 + 1 2 λ  s − sin(2 λs ) 2 λ  + n X i =1  ( A i ( x 0 ) i + B i ( y 0 ) i )  1 − cos(2 λs ) 2 λ  − ( B i ( x 0 ) i − A i ( y 0 ) i )  sin(2 λs ) 2 λ  . AN ISOPERIME TRIC INEQUALITY IN THE HEISENBERG GR OUP H n 7 In ca s e x 0 = y 0 = 0, we o bta in x i ( s ) = A i  sin(2 λs ) 2 λ  + B i  1 − cos(2 λs ) 2 λ  , y i ( s ) = − A i  1 − cos(2 λs ) 2 λ  + B i  sin(2 λs ) 2 λ  , and so ˙ x i , ˙ y i , i = 1 , . . . , n , can b e expressed in terms of x i , y i in the following wa y ˙ x i ( s ) = λ sin(2 λs ) 1 − cos(2 λs ) x i ( s ) + λ y i ( s ) , (2.9) ˙ y i ( s ) = − λ x i ( s ) + λ sin(2 λs ) 1 − cos(2 λs ) y i ( s ) . (2.10) 2.5. The spheres S λ . F or any λ > 0 , p ∈ H n , consider the hypers urface S λ,p defined by S λ,p := [ v ∈ H p , | v | =1 γ v p,λ ([0 , π /λ ]) . If p is tra nslated to the p o int q = (0 , − π / (4 λ 2 )), then S λ := S λ,q is the union of the graphs ass o ciated to the functions f and − f , where f ( z ) = 1 2 λ 2 { λ | z | p 1 − λ 2 | z | 2 + a r ccos ( λ | z | ) } , | z | 6 1 λ . The hypersurface S λ is compact and homeo morphic to a (2 n )-dimensional sphere. Its singular set cons ists of the tw o p oints ± (0 , π / (4 λ 2 )) on the t -axis , c alled the p oles . It is known that the spheres S λ are C 2 but not C 3 around the singular p oints. These hypersurfaces were conjectured to b e the (smo oth) s o lutions to the isop erimetr ic prob- lem in H 1 by P . Pansu [14]. It was proven in [1 6] that the hypersurface s S λ are the only co mpact hypers urfaces of revolution with c o nstant mea n curv a ture λ in H n . W e shall deno te by B λ the to po logical c lo sed ball enclose d by S λ . It is well known that the spheres S λ consist of the union of segments of ge o desics of curv ature λ and length π /λ starting fro m a given p oint p ∈ H n . Lemma 2.3. The char acteristic curves in S λ ar e the ge o desics of curvatur e λ joining the p oles. Pr o of. Since S ± λ is the gr aph of the function ± f , the inner unit normal to S λ is prop ortiona l to n X i =1  ∂ f ∂ x i − y i  X i +  ∂ f ∂ y i + x i  Y i  − T on S − λ , and prop o rtional to − n X i =1  ∂ ( − f ) ∂ x i − y i  X i +  ∂ ( − f ) ∂ y i + x i  Y i  + T on S − λ . Let ν λ be the horizontal unit no r mal to S λ . F ro m ∂ f ∂ x i = − λ | z | (1 − λ 2 | z | 2 ) − 1 / 2 x i , ∂ f ∂ y i = − λ | z | (1 − λ 2 | z | 2 ) − 1 / 2 y i , we hav e that J ( ν λ ) is given by n X i =1  − λx i − (1 − λ 2 | z | 2 ) 1 / 2 | z | y i  Y i +  λy i − (1 − λ 2 | z | 2 ) 1 / 2 | z | x i  X i 8 M. RITOR ´ E on S + λ , and by n X i =1  − λx i + (1 − λ 2 | z | 2 ) 1 / 2 | z | y i  Y i +  λy i + (1 − λ 2 | z | 2 ) 1 / 2 | z | x i  X i on S − λ , wher e | z | 2 = P n i =1 ( x 2 i + y 2 i ). On the other hand, the tangent vector to a hor izontal g e o desic o f curv ature λ leav- ing from (0 , − π / (4 λ 2 )), is g iven b y ˙ γ ( s ) = P n i =1 ˙ x i ( s ) X i + ˙ y i ( s ) Y i , where ˙ x i ( s ), ˙ y i ( s ) satisfy (2.9) a nd (2.10). A direct computation shows (2.11) λ sin(2 λs ) 1 − cos(2 λs ) = ± (1 − λ 2 | z | 2 ) 1 / 2 | z | , where the plus sign is chosen in cas e s ∈ [0 , π / (2 λ )], that is, when γ ( s ) ∈ S − λ , and the minus sign if s ∈ [ π / (2 λ ) , π /λ ], when γ ( s ) ∈ S + λ . Replacing the v alue of λ sin(2 λs ) / (1 − cos(2 λs )) in e q uations (2.9) and (2 .10) by using (2 .1 1), we conclude that ˙ γ is equal to J ( ν λ ), i.e., γ is a c haracter istic curve of S λ .  Lemma 2.4. L et Σ ⊂ H n b e a C 2 c omp act hyp ersurfac e with a finite numb er of iso- late d singular p oints. Assu m e that Σ has c onstant me an curvatur e. Then Σ is volume- pr eserving ar e a-stationary. In p articular, the spher es S λ ar e volume-pr eserving ar e a- stationary. Pr o of. Let U b e a vector field inducing a volume-preserving v ar iation of Σ. Let u =  U, N  . By the first v ar iation of volume (2.3) we have ´ Σ u d Σ = 0. By the first v ariation of area [16, Lemma 3.2], we hav e A ′ (0) = − ˆ Σ div Σ  u ( ν H ) ⊤  d Σ , since u has mea n zero a nd div Σ ν H is co nstant. T o ana lyze the a bove integral, w e co nsider op en balls B ε ( p i ) of radius ε > 0 cen- tered at the p oints p 1 , . . . , p k of the singular set Σ 0 . By the divergence theor em in Σ, we hav e , fo r Σ ε = Σ \ S k i =1 B ε ( p i ), − ˆ Σ ε div Σ  u ( ν H ) ⊤  d Σ = k X i =1 ˆ ∂ B ε ( p i ) u  ξ i , ( ν H ) ⊤  d ( ∂ B ε ( p i )) , where ξ i is the inner unit normal vector to ∂ B ε ( p i ) in Σ, and d ( ∂ B ε ( p i )) is the Rie- mannian volume element of ∂ B ε ( p i ). Note also that     k X i =1 ˆ ∂ B ε ( p i ) u  ξ i , ( ν H ) ⊤  d ( ∂ B ε ( p i ))     6  sup Σ | u |  k X i =1 V 2 n − 1 ( ∂ B ε ( p i )) , where V 2 n − 1 ( ∂ B ε ( p i )) is the Riemannian (2 n − 1)-volume of ∂ B ε ( p i ). Observe that the function | div Σ ( u ( ν ⊤ H )) | is b ounded from ab ov e by (sup Σ | u | ) | div Σ ν H −| N H | | div Σ N | + |∇ Σ u | , which is a lso bo unded. So we ca n apply the dominated co nv erg ence theorem and the fact that V 2 n − 1 ( ∂ B ε ( p i )) → 0, when ε → 0, to prove that A ′ (0) = 0.  2.6. H - regular surfaces ( [1] , [4] ). Let Ω ⊂ H n be an o p e n set. Then C 1 H (Ω) is the set of co ntin uous re al functions in Ω such that ∇ H f is contin uous, [4, def. 5 .7], wher e ∇ H f is defined by ∇ H f := n X i =1 X i ( f ) X i + Y i ( f ) Y i . AN ISOPERIME TRIC INEQUALITY IN THE HEISENBERG GR OUP H n 9 F ollowing [4, def. 6.1] we say that Σ ⊂ H n is an H -regula r hypers ur face if, for every p ∈ Σ, ther e is an op en set Ω containing p , and a function f ∈ C 1 H (Ω) such that Σ ∩ Ω = { q ∈ Ω : f ( q ) = 0 } , ∇ H f ( q ) 6 = 0 . F rom [4, Thm. 6.5], we know that if Σ is an H -re gular hypersurfac e defined lo cally by a function f ∈ C 1 H (Ω) with no nv anishing horizo ntal gradient ∇ H f , and we let E := { q ∈ Ω : f ( q ) < 0 } , then E is a set o f lo ca lly finite p er imeter in Ω and ν H = − ∇ H f / |∇ H f | . W e hav e the following Lemma 2.5. L et Σ b e an H -r e gular hyp ersurfac e, and let ν H b e its horizontal u nit nor- mal ve ctor. If γ : I → H n is an inte gr al cu rve of J ( ν H ) with γ (0) ∈ Σ , t hen γ ( I ) ⊂ Σ . Pr o of. Lo cally Σ = { p ∈ Ω : f ( p ) = 0 } , for some op en set Ω ⊂ H n , and some f ∈ C 1 H (Ω) with non v anishing ho rizontal g radient. It is enough to prove the r esult in Σ ∩ Ω. If γ is any C 1 curve then (2.12) d ds f ( γ ( s )) =  ∇ H f ( γ ( s )) , ˙ γ ( s )  . Let us chec k this equa lity . The C 1 H function f is Pansu-differentiable in its do ma in of definition by [4, Thm. 5.10 ]. By the results in [4, § 5] this implies tha t, for p in a fixed s ma ll neighbor ho o d o f a , we have (2.13) f ( p ) − f ( a ) −  ∇ H f ( a ) , π a ( a − 1 · p ))  = || a − 1 · p || ∞ E ( a, p ) , where the function E ( a, p ) satisfies (2.14) lim p → a E ( a, p ) = 0 , || · || ∞ is the homogeneo us nor m defined in (2.2), and the pro jection π a is defined, as in [4, Def. 2.1 9], by π a ( z , t ) := n X i =1 x i ( X i ) a + y i ( Y i ) a . Hence, tak ing a = γ ( s ), p = γ ( s + h ), g ( s ) = ( f ◦ γ )( s ), we ge t from (2.13) (2.15) g ( s + h ) − g ( s ) h =  ∇ H f ( γ ( s )) , π γ ( s ) ( γ ( s ) − 1 · γ ( s + h )) h  + || γ ( s ) − 1 · γ ( s + h ) || ∞ h E ( γ ( s ) , γ ( s + h )) . Since γ is a hor iz o ntal curve, it is immediate to chec k that (2.16) lim h → 0 π γ ( s ) ( γ ( s ) − 1 · γ ( s + h )) h = ˙ γ ( s ) . On the other hand, since the norm || · || ∞ is globally equiv alent to the Carnot-Ca- rath´ eodo r y distance, there is a constant C > 0 s o that || γ ( s ) − 1 · γ ( s + h ) || ∞ 6 C d ( γ ( s ) , γ ( s + h )). As γ is a C 1 horizontal curve we have d ( γ ( s ) , γ ( s + h )) 6 || ˙ γ || ∞ h . So we hav e that || a − 1 · γ ( s + h ) || ∞ /h is bo unded and (2.17) lim h → 0 || γ ( s ) − 1 · γ ( s + h ) || ∞ h E ( γ ( s ) , γ ( s + h )) = 0 , by (2.1 4). T ak ing limits when h go es to 0 in (2.6 ), a nd using (2.1 6) and (2.17), we get (2.12). 10 M. RITOR ´ E Assume now that γ is an integral curve of J ( −∇ H f / |∇ H f | ) with γ (0) ∈ Σ. Since ∇ H f and J ( ∇ H f ) are orthogo nal, equation (2.1 2) implies d ds f ( γ ( s )) = 0 . W e co nclude that f ◦ γ is a constant function a nd, since f ( γ (0)) = 0, we have f ◦ γ ≡ 0 in Ω, and so γ ( I ) ⊂ Σ. Since ν H = − ∇ H f / |∇ H f | on Σ , γ in an integral curve of J ( ν H ) contained in Σ.  3. Proof o f the isoperimetric inequal ity W e shall denote by D r := { ( z , 0) : | z | 6 r } the closed E uclidean disk of radius r > 0 contained in the E uclidean h yp erplane Π 0 := { t = 0 } , and by C r := { ( z , t ) : | z | 6 r } the vertical cylinder ov er D r . The vertical t -ax is { (0 , t ) : t ∈ R } will b e denoted by L . F or a ny set B ⊂ H n , define B + := B ∩ { ( z , t ) ∈ H n : t > 0 } , B − := B ∩ { ( z , t ) ∈ H n : t 6 0 } . W e recall that to any finite p erimeter set E ⊂ H n we may add its density one po int s and r emov e its densit y zero p oints, without c hanging the per imeter and the volume of E . W e shall alwa ys nor malize a finite p erimeter set in this wa y . Theorem 3.1. L et E ⊂ H n b e a finite p erimeter set su ch that D ⊂ E ⊂ C , wher e D = D r , C = C r , for some r > 0 . Then (3.1) | ∂ E | > | ∂ B µ | , wher e B µ is the b al l with | B µ | = | E | . Equality holds in (3.1) if and only if B µ = E . Pr o of. It can b e ea sily proven tha t E ± := E ∩ ( H n ) ± are finite p erimeter s ets. The reduced b o undary ∂ ∗ E + of E + is co nt ained in ( ∂ ∗ E ∩ { t > 0 } ) ∪ int( D ), where int( D ) is the in terior of D inside Π 0 . W e choose tw o families of functions. F or 0 < ε < 1 we consider smo oth functions ϕ ε , dep ending on the Euc lide a n distance to the vertical axis L , so that 0 6 ϕ ε 6 1, and ϕ ε ( p ) = 0 , d ( p, L ) 6 ε 2 , ϕ ε ( p ) = 1 , d ( p, L ) > ε, |∇ ϕ ε ( p ) | 6 2 /ε, ε 2 6 d ( p, L ) 6 ε. Again for 0 < ε < 1 we co nsider smo oth functions ψ ε , dep ending on the distance to the Euclide a n hyperplane Π 0 , so that 0 6 ψ ε 6 1, and ψ ε ( p ) = 1 , d ( p, Π 0 ) 6 ε − 1 / 2 , ψ ε ( p ) = 0 , d ( p, Π 0 ) > ε − 1 / 2 + 1 , |∇ ψ ε ( p ) | 6 2 , ε − 1 / 2 6 d ( p, Π 0 ) 6 ε − 1 / 2 + 1 . Let λ := 1 /r . T he n the ball B λ satisfies B λ ∩ Π 0 = D . T ranslate vertically the closed halfspheres S + λ to get a foliation of C . Let X b e the vector field on C \ L given by the ho rizontal unit nor ma l to the leav es of the foliatio n. By (2.4), o n C \ L we hav e div X = − n λ. W e consider the horizontal vector field ψ ε ϕ ε X , which ha s compact supp ort on H n . ˆ E + div( ψ ε ϕ ε X ) dv = ˆ E + ψ ε ϕ ε div X dv + ˆ E +  ∇ ( ψ ε ϕ ε ) , X  dv . AN ISOPERIME TRIC INEQUALITY IN THE HEISENBERG GR OUP H n 11 Observe that lim ε → 0 ˆ E + ψ ε ϕ ε div X dv = − nλ | E + | , by Lebesg ue’s Dominated Conv ergence Theorem since ψ ε ϕ ε div X is unifor mly b ounded, E + has finite volume, a nd lim ε → 0 ψ ε ϕ ε = 1. On the other hand lim ε → 0 ˆ E +  ∇ ( ψ ε ϕ ε ) , X  dv = 0 , since  ϕ ε ∇ ψ ε , X  is b ounded and co nv er g es p oint wise to 0, a nd lim ε → 0 ˆ E + |  ψ ε ∇ ϕ ε , X  | dv 6 lim ε → 0 ˆ ( H n ) + ψ ε |∇ ϕ ε | dv = 0 . The last equa lit y is eas ily chec ked taking classica l cilindr ical co o rdinates in H n = R 2 n +1 . So we co nclude (3.2) lim ε → 0 ˆ E + div( ψ ε ϕ ε X ) dv = − nλ | E + | . By applying the Divergence Theorem for finite p erimeter s ets [4] to E + and to the vector field ψ ε ϕ ε X we hav e ˆ E + div( ψ ε ϕ ε X ) dv = − ˆ D  ψ ε ϕ ε X , N D  dD − ˆ ∂ ∗ E ∩{ t> 0 }  ψ ε ϕ ε X , ν H  d | ∂ E | , where N D is the Riemannian unit nor mal to D p ointing into ( H n ) + , dD is the Rie- mannian area element on D , and ν H is the inner horizo n tal unit normal to ∂ ∗ E . T aking limits when ε → 0, we get from (3.2) and inequa lity  X , ν H  6 1 , (3.3) − nλ | E + | > − ˆ D  X , N D  dD − ˆ ∂ ∗ E ∩{ t> 0 } d | ∂ E | , with equa lity if a nd only if, | ∂ E | -a.e., X = ν H on ∂ E ∩ ( H n ) + . W e may replace E + by B + λ in the previous reaso ning to obtain (3.4) − nλ | ∂ B + λ | = − ˆ D  X , N D  dD − ˆ ∂ ∗ B λ ∩{ t> 0 } d | ∂ B λ | . Hence fro m (3.3) a nd (3.4) w e obtain (3.5) ˆ ∂ ∗ E ∩{ t> 0 } d | ∂ E | > ˆ ∂ B λ ∩{ t> 0 } d | ∂ B λ | + nλ  | E + | − | B + λ |  , with equa lity if a nd only if, | ∂ E | -a.e., X = ν H on ∂ E ∩ ( H n ) + . W e consider now the foliation o f C by vertical translations of the clo sed halfspheres S − λ . Let Y be the vector field o n C \ L g iven by the hor izontal unit normal to the leav es o f the folia tion. By applying the previous argument we get a similar estimate (3.6) ˆ ∂ ∗ E ∩{ t< 0 } d | ∂ E | > ˆ ∂ B λ ∩{ t< 0 } d | ∂ B λ | + nλ  | E − | − | B − λ |  , with equa lity if a nd only if, | ∂ E | -a.e., Y = ν H on ∂ E ∩ ( H n ) − . Hence, adding (3.5 ) and (3.6) , and taking into acco unt | ∂ E | > | ∂ E | ( H n \ Π 0 ), that ∂ B λ ∩ Π 0 do not contribute to the p erimeter o f E and B λ , and tha t E ∩ Π 0 and B λ ∩ Π 0 do not contribute to the volume of E a nd B λ , we get (3.7) | ∂ E | > | ∂ B λ | + nλ  | E | − | B λ |  , with equality if and only if, | ∂ E | -a.e, X = ν H on ∂ E ∩ ( H n ) + and Y = ν H on ∂ E ∩ ( H n ) − and | ∂ E | = | ∂ E | ( H n \ Π 0 ). 12 M. RITOR ´ E Let f ( ρ ) := nρ | E | + | ∂ B ρ | − nρ | B ρ | . By Lemmae 2.2 and 2.4, the sphere S ρ is a c r it- ical po int of A − nρV , with ρ fixed, for any v aria tion. So we hav e A ( S ρ ) ′ − n ρ V ( B ρ ) ′ = 0, where primes indicate the deriv ative with res pe c t to ρ . Hence we hav e f ′ ( ρ ) = n ( | E | − | B ρ | ) . Since the function ρ 7→ | B ρ | is str ictly decreasing and ta kes its v alues in the interv a l (0 , + ∞ ), we o btain that f ( ρ ) is an strictly co nv ex function with a unique minim um µ for which | E | = | B µ | . Hence we obtain from (3.7) (3.8) | ∂ E | > f ( λ ) > f ( µ ) = | ∂ B µ | , which implies (3.1). Assume now that equality ho lds in (3.8). Then, since f is str ictly convex, λ = µ . By [12, Thm 1.2 ], ∂ E \ L is an H -re g ular hypers ur face. By Lemma 2.5, the integral curves of J ( ν H ), star ting from po ints in ∂ E \ L , are contained in ∂ E \ L . Observe now that ∂ D ⊂ ( ∂ E \ L ) ∩ S µ . F or e very p ∈ ∂ D , co nsider the integral curve γ p : I p → H n of J ( ν H ), where I p is the maxima l interv al fo r which γ p is defined. The tr ace γ p ( I p ) is contained in ∂ E \ L . Such a c ur ve is also a n integral curve of J ( X ) in ( H n ) + and an integral curve of J ( Y ) in ( H n ) − , and so it is co nt ained in the sphere S µ . In fact, it is par t of a characteristic curve o f S µ . Since ∂ E \ L is foliated by int egral curves o f J ( ν H ), it is easy to chec k that ∂ E ⊃ S p ∈ ∂ D γ p ( I p ) = S µ \ ( S µ ) 0 = S µ \ L . Here ( S µ ) 0 is the singular set of the C 2 hypersurface S µ , as defined in § 2.3. This implies tha t S µ ⊂ ∂ E . W e claim that B µ ⊂ E . T o prov e this we shall s how that B µ \ L ⊂ E reasoning by co nt radiction. If B µ \ L is not contained in E , as S µ = ∂ B µ ⊂ ∂ E , ther e is a p oint p in the interior of B µ \ L so that p 6∈ E . The Euclidean orthogona l pro jection p ′ of p ov er t = 0 lies in D ⊂ E . Hence there is a p oint q in the E uc lidea n segment [ p, p ′ ] ⊂ int( B µ ) that be longs to ∂ E \ L . As ∂ E \ L is H -regular , the p erimeter o f ∂ E in a small ball contained in int( B µ \ L ) a nd centered at q is p ositive, and so | ∂ E | > | ∂ B µ | , which contradicts our ass umption that equa lit y holds in (3.8 ). This implies B µ ⊂ E . As | B µ | = | E | we obtain B µ = E by the no rmalization of E .  References 1. Lui gi A mbrosio, F rancesco Serra Cassano, and Davide V ittone, Intrinsic r e gular hyp ersurfac es in Heisenb e r g gr oups , J. Geom. 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