Connected sums of closed Riemannian manifolds and fourth order conformal invariants

CONNECTED SUM S OF C LOSED RIEMAN NIAN MANIFOLDS AND F OUR TH ORDER CONF ORMAL INV ARIANTS D a vid Raske A b st r ac t. In this note we take some initi al steps i n the inv estigation of a fourth order analogue of the Y amabe problem in conformal geometry . The Paneitz cons tants and the P aneitz in v arian ts considered are believed to be very helpful to understand the topology of the unde rlined manifolds. W e calculate how those quan tities c hange, analogous to ho w t he Y amabe cons tants and the Y amab e inv ariants do, under the connected sum operations. 1. Introduc tion Let ( M , g ) b e a connected compact Riemannian manifold without b oundary of dimension n ≥ 5. Let (1.1) Q [ g ] = − n − 4 4( n − 1) ∆ R + ( n − 4)( n 3 − 4 n 2 + 16 n − 16) 16( n − 1) 2 ( n − 2) 2 R 2 − 2( n − 4) ( n − 2) 2 | Ric | 2 b e the so-called Q -curv ature, where R is the scalar curv ature, Ric i s the Ricci curv ature. And l et (1.2) P [ g ] = ( − ∆) 2 − div g (( ( n − 2) 2 + 4 2( n − 1)( n − 2) Rg − 4 n − 2 Ric g ) d ) + Q [ g ] b e the so-cal l ed the Paneitz-Branson op erat or. It is known that (1.3) P [ g ] u = Q [ g u ] u n +4 n − 4 whic h is called the Paneitz-Branson equat i on, where g u = u 4 n − 4 g ( cf. [P] [Br] [XY] [DHL] [DMA] ). W e consider the equation (1.3) a s a fourth o r der analo gue of the w ell-kno wn scalar curv at ure eq uation (1.4) L [ g ] v = R [ g v ] v n +2 n − 2 , T yp eset by A M S -T E X 1 2 CONNECTED SUM where (1.5) L [ g ] = − 4( n − 1) n − 2 ∆ + R is the so-called conformal Laplacia n and g v = v 4 n − 2 g . Th e well-kno wn Y amab e problem in conformal geometry is to find a metric, in a giv en class of conformal metrics, w hi ch is of constan t scalar curv a t ure, i.e. to solve L [ g ] v = Y v n +2 n − 2 on a given manifold ( M , g ) for some p osit iv e function v and a constan t Y . The affirmative resolution t o t he Y amab e problem was giv en in [Sc] after other notable w orks [Y a] [T r] [Au]. In fact, it was prov en that there exists a so-called Y amab e metric g v in the cl a ss [ g ] which is a minimizer for the so-called Y amab e functional Y ( v ) = R M ( v L [ g ] v ) dv g ( R M v 2 n n − 2 dv g ) n − 2 n . In chapter one w e i n v estigate a fourth o rder analogue of the Y amab e problem. Let C + ∞ ( M ) b e the space of smo oth non-negative functions on M . Similar to the Y amabe problem, we define the Paneitz functional (1.6) ℘ g ( u ) = R M ( uP [ g ] u ) dv g ( R M u 2 n n − 4 dv g ) n − 4 n for u ∈ C + ∞ ( M ) and the Panei tz c onstant asso cia t ed with ( M , [ g ]) (1.7) λ ( M , [ g ]) = inf u ∈ C + ∞ ( M ) ℘ ( u ) . It i s clear that λ ( M , [ g ]) is a conformal in v aria n t of the conformal class [ g ] b ecause of the conformall y cov arian t prop erty of the Paneitz-Branson op erator: (1.8) P [ g w ] u = w − n +4 n − 4 P [ g ]( w · u ) where g w = w 4 n − 4 g ∈ [ g ]. T o describ e the differen tial structure of M , we define (1.9) λ ( M ) = sup [ g ] λ ( M , [ g ]) . W e wil l refer to λ ( M ) as the Panei tz Invariant of the manifold M as t he counter part of Y amab e inv arian t. In [Gi] , Gil-Medrano studied the Y amab e constan t for a connected sum of tw o closed manifolds. One interesting consequence of connected sum results in [Gi] is that every compact mani fol d without b oundary admit s a conformal class of metri cs whose Y amab e constan t is v ery negative. In Section 2 of Chapter One we calculate as Gil-Medrano did in [Gi] to ve rify that DA VID RASKE 3 Theorem 1.1. L et ( M 1 , g 1 ) and ( M 2 , g 2 ) b e two c omp act Riemannian m anifo lds of d i mension n ≥ 5 . Then, f or e ach ǫ > 0 , ther e is a c onformal class [ g ] of metrics on M 1 # M 2 such that (1.10) λ ( M 1 # M 2 , [ g ]) < min { λ ( M 1 , [ g 1 ]) , λ ( M 2 , [ g 2 ]) } + ǫ and ther e exists a c onformal class [ h ] of metrics on M 1 # M 2 such that (1.11) λ ( M 1 # M 2 , [ h ]) < 2 − n − 4 n ( λ ( M 1 , [ g 1 ]) + λ ( M 2 , [ g 2 ])) + ǫ. Due t o the works of Sc ho en and Y au [SY] (see a lso [GL]), one knows that there is some top o logical constrain t for a ma ni fol d to p ossess a metric of p osi t ive Y amab e constan t. Therefore it is interesting to see ho w the Y amab e in v aria nt is effected b y connected sum. It w as pro v en in [Ko] [SY] [ GL] that the Y amab e inv arian t of connected sum of t w o ma nifolds with p ositive Y amab e i n v ariants is still positi ve. More precisely , Koba y ashi in [ Ko] show ed t hat the Y amab e in v a ri an t o f connected sum of t w o manifolds is greater t han or equal to the smaller of the Y amab e in v ariants of the tw o. In Section 3 of Chapter 1 we obtain an analogue for the Paneitz in v ari an t. Theorem 1.2. If M 1 and M 2 ar e c omp act manif o lds of dimensi on n ≥ 5 , then (1.12) λ ( M 1 # M 2 ) ≥ min { λ ( M 1 ) , λ ( M 2 ) } . The p o sitivity of Paneitz inv arian t i n dimension higher t han 4 should b e a top o- logical constraint, a s indicated by successful researc hes in [CY] (references therein) for fourth order analogue of ho w Gaussian curv a ture influences the geomet ry of surfaces i n di mension 4. Another testing ground i s to consider closed lo call y con- formally flat manifolds. Then the recen t w orks i n [CHY] [G] i ndicate to us that the p ositivi t y of fourth order curv at ure i s indeed very informative ab out the top ology of t he underlined manifol ds. W e would also lik e to men tion the w ork by Xu and Y ang in [XY] where they demonstrated that posit ivity o f the P aneitz-Branson op er- ator is stable under the pro cess of taking connected sums of t w o cl osed Riemannian manifolds. In Sectio n 1 of Chapter 1 we discuss some preliminary facts a b out the P aneitz functional. In Sectio n 2 w e calculate and verify Theorems 1.1. In Section 3 we pro v e Theorem 1.2. 2. Prelimina ries Recall that the Y amab e constan t of an y closed manifold of dimension greater than 2 i s a finite n um b er and the l argest p ossible Y amabe constant is realized 4 CONNECTED SUM and only real i zed b y the Y amab e constan t of the standard round sphere in eac h dimension. The difficult part is t o sho w that the round sphere is the only one that has the l argest Y amab e constan t , which was the last step in the resol uti on of Y amab e problem sol ved b y Schoen i n [Sc] based on a p ositiv e mass theorem of Sc ho en and Y au . W e observe that, b y (1.3) , (2.1) Z M ( uP [ g ] u ) dv g = Z M uQ [ g u ] u n +4 n − 4 dv g = Z M Q [ g u ] u 2 n n − 4 dv g = Z M Q [ g u ] dv g u , where g u = u 4 n − 4 g ∈ [ g ]. Hence Z M ( uP [ g ] u ) dv g = Z M (( ( n − 4)( n 3 − 4 n 2 + 16 n − 16) 16( n − 1) 2 ( n − 2) 2 R 2 − 2( n − 4) ( n − 2) 2 | Ric | 2 ) dv ) [ g u ] ≤ ( ( n − 4)( n 3 − 4 n 2 + 16 n − 16) 16( n − 1) 2 ( n − 2) 2 Z M ( R 2 ) dv ) [ g u ] When w e consider a Y amabe metric g u , i .e. (2.2) R M ( Rdv )[ g u ] v ol( M , g u ) n − 2 n = Y v ol( M , g u ) 2 n ≤ n ( n − 1)vol ( S n , g 0 ) 2 n , w e ha v e (2.3) R M ( uP [ g ] u ) dv g v ol( M , g u ) n − 4 n ≤ ( n − 4)( n 3 − 4 n 2 + 16 n − 16) 16( n − 1) 2 ( n − 2) 2 Y 2 v ol( M , g u ) 4 n ≤ ( n − 4)( n 3 − 4 n 2 + 16 n − 16) 16( n − 1) 2 ( n − 2) 2 ( n ( n − 1)) 2 v ol( S n , g 0 ) 4 n = R S n ( Qdv )[ g 0 ] v ol( S n , g 0 ) n − 4 n = λ ( S n , [ g 0 ]) . Consequen tly we obta in Lemma 2.1. L et ( M n , g ) b e a close d R iemannian manifold of dimensi o n gr e at than 4 with nonne g a ti ve Y amab e c onstant. Then (2.4) λ ( M n , [ g ]) ≤ λ ( S n , [ g 0 ]) and the e quality holds if and only if ( M , g ) is c onf ormal ly e quivalent to the s tandar d r ound s p h e r e ( S n , g 0 ) . On the other hand, b y some ch oices of t esti ng functions similar to the ones used to estimate the Y amab e functional, w e get DA VID RASKE 5 Lemma 2.2. L et ( M n , g ) b e a close d R iemannian manifold of dimensi o n gr e at than 4 . Then (2.5) −∞ < λ ( M n , [ g ]) ≤ λ ( S n , [ g 0 ]) , wher e g 0 is the standar d r ound metric on the spher e S n . Pr o of. The Paneitz constan t is easi l y seen to b e b ounded from t he b elow. Because, b y ( 1 .2), (2.6) Z M ( uP [ g ] u ) dv = Z M | ∆ u | 2 dv + a n Z M R |∇ u | 2 dv − 4 n − 4 Z M Ric( ∇ u, ∇ u ) dv + Z M Qu 2 dv , where a n = ( n − 2) 2 + 4 2( n − 1)( n − 2) . It suffices to estimate (2.3 ) for nonnegative functions suc h that Z M u 2 n n − 4 dv = 1 . Hence, By Ho l der inequali t y , (2.7) Z M ( uP [ g ] u ) dv ≥ Z M | ∆ u | 2 dv − C 1 Z M |∇ u | 2 dv − C 2 Z M u 2 dv ≥ Z M | ∆ u | 2 dv − C 1 Z M ( − ∆ u ) udv − C 2 Z M u 2 dv ≥ 1 2 Z M | ∆ u | 2 dv − 1 2 C 2 1 Z M u 2 dv − C 2 Z M u 2 dv ≥ − ( 1 2 C 2 1 + C 2 )( Z M u 2 n n − 4 dv ) n − 4 n v ol( M , g ) 4 n ≥ − ( 1 2 C 2 1 + C 2 )v ol( M , g ) 4 n . for some constants C 1 , C 2 > 0 dep ending on ( M n , g ). T o estimate the upper b ound w e c ho ose to w orks in a geo desic normal co ordinate in v ery small g eo desic ball B 2 ǫ ⊂ M and transplan t the rescaled round sphere metric. Let B 2 ǫ (0) ⊂ R n and (2.8) g ij ( x ) = δ ij + O ( | x | 2 ) , ∀ x ∈ B 2 ǫ (0) . 6 CONNECTED SUM Let (2.9) u ǫ ( x ) =    ( 2 ǫ 3 ǫ 6 + | x | 2 ) n − 4 2 ∀ x ∈ B ǫ (0) 0 ∀ x / ∈ B 2 ǫ (0) b e a smo oth nonnegative function on M . Then it is easily calculated that (2.10) Z M ( u ǫ P [ g ] u ǫ ) dv = Z B ǫ (0) | ∆ u ǫ | 2 dx + o (1) = Z R n | ∆( 2 ǫ 3 ǫ 6 + | x | 2 ) n − 4 2 | 2 dx + o (1) = Z R n | ∆( 2 1 + | x | 2 ) n − 4 2 | 2 dx + o (1) and (2.11) Z M u 2 n n − 4 ǫ dv = Z B ǫ (0) u 2 n n − 4 ǫ dx + o (1) = Z R n ( 2 ǫ 3 ǫ 6 + | x | 2 ) n dx + o (1) = Z R n ( 2 1 + | x | 2 ) n dx + o (1) . Therefore (2.12) ℘ ( u ǫ ) = R M ( u ǫ P [ g ] u ǫ ) dv ( R M u 2 n n − 4 ǫ dv ) n − 4 n = R R n | ∆ s | 2 dx ( R R n s 2 n n − 4 dx ) n − 4 n + o (1) , where s = ( 2 1+ | x | 2 ) n − 4 2 . Th us, take ǫ → 0, we arrive a t (2.13) λ ( M , [ g ]) ≤ λ ( S n , [ g 0 ]) . One interes ting question would b e whether ( M , g ) i s conformally equiv alen t t o ( S n , g 0 ) when λ ( M , [ g ]) = λ ( S n , [ g 0 ]) without assuming the Y amab e constan t of ( M , g ) is nonnegative. In o ther w ords one w ould b e in terested in searc hing for some analogue of a p ositiv e mass theorem of Sc ho en and Y au here if it make a n y sense. DA VID RASKE 7 3. Conne cted S ums and the P aneit z Constant In this se ction we will cal culat e the P aneitz fun ctional o n a connected sum of t w o closed manifolds and v erify Theorem 1 . 1. Let ( M , g ) b e a closed manifold of dimension higher than 4 . Fix a p oin t p ∈ M and let (3.1) f δ = ( 0 ∀ x ∈ B δ ( p ) 1 ∀ x ∈ M \ B 2 δ ( p ) b e a family o f smo oth functions. W e ma y ask (3.2)            0 ≤ f δ ≤ 1 |∇ f δ | < C 0 δ | ∆ f δ | < C 0 δ 2 for some num b er C 0 > 0. First we calculate Lemma 3.1. L et ( M , g ) b e a close d manifold of dimensi on gr e ater than 4 . L et u ∈ C + ∞ ( M ) b e gi ven. Then u δ = f δ u ∈ C + ∞ ( M ) and (3.3) ℘ g ( u δ ) = ℘ g ( u ) + o (1) as δ → 0 Pr o of. W e simply cal culate, for a fixed δ > 0, by (2.6 ) and ( 3.2), (3.4) Z M ( u δ P [ g ] u δ ) dv = Z M | ∆ u δ | 2 dv + a n Z M R |∇ u δ | 2 dv − 4 n − 4 Z M Ric( ∇ u δ , ∇ u δ ) dv + Z M Qu 2 δ dv = Z M ( uP [ g ] u ) dv + o (1) and (3.5) Z M u 2 n n − 4 δ dv = Z M u 2 n n − 4 dv + o (1) , as δ → 0. 8 CONNECTED SUM No w let us consider the connected sum of tw o closed Riemannian manifolds. Let ( M 1 , g 1 ) a nd ( M 2 , g 2 ) b e tw o compact Riemannian manifolds without b oundary of dimension n ≥ 5 . F or x 1 ∈ M 1 and x 2 ∈ M 2 , let B δ 1 ( x 1 ) ⊂ M 1 and B δ 2 ( x 2 ) ⊂ M 2 b e geo desic balls resp ectively . T o make the connected sum one simpl y to take off the op en bal ls B 1 2 δ 1 ( x 1 ) and B 1 2 δ 2 ( x 2 ) from M 1 and M 2 , identify ∂ B 1 2 δ 1 ( x 1 ) with ∂ B 1 2 δ 2 ( x 2 ). Hence (3.6) M 1 # M 2 = h ( M 1 \ B 1 2 δ 1 ( x 1 )) [ ( M 2 \ B 1 2 δ 2 ( x 2 )) i / { ∂ B 1 2 δ 1 ( x 1 ) ∼ ∂ B 1 2 δ 2 ( x 2 ) } . W e may construct a metric g on the connected sum M 1 # M 2 suc h that g agrees with g 1 on M 1 \ B δ 1 ( x 1 ) and g 2 on M 2 \ B δ 2 ( x 2 ). Noti ce that top ological ly M 1 # M 2 do es not dep end o n t he v alue of δ i when they are sufficien tly small . No w let us calculate and estimate the Paneitz functional o n the connected sum. Theorem 3.2. L et ( M 1 , g 1 ) and ( M 2 , g 2 ) b e two close d R iemannian manifolds of dimension n ≥ 5 . Then for e ach ǫ > 0 , ther e is a c onformal structur e [ g ] on M 1 # M 2 such that (3.7) λ ( M 1 # M 2 , [ g ]) < min { λ ( M 1 , [ g 1 ]) , λ ( M 2 , [ g 2 ]) } + ǫ. A lternatively, we may find a c onformal structur e [ g ] on M 1 # M 2 such that (3.8) λ ( M , [ g ]) < λ ( M 1 , [ g 1 ]) + λ ( M 2 , [ g 2 ])2 − n − 4 n + ǫ. Pr o of. Let us assume that λ ( M 1 , [ g 1 ]) ≤ λ ( M 2 , [ g 2 ]) and ǫ > 0 fixed. By t he definition of the P aneitz constan t, we know that there i s a real num b er δ > 0 and a smo oth function u δ ∈ C + ∞ ( M ) suc h that u δ v anishes on a geo desic ball B δ ( x 1 ) of radius δ and cen tered at x 1 ∈ M 1 and suc h that ℘ g ( u δ ) < λ ( M 1 , [ g 1 ]) + ǫ. Let g b e a metric on M = M 1 # M 2 whic h agrees with g 1 , when restricted to M 1 \ B δ ( x 1 ). And define the function f u δ on M 1 # M 2 as follows: ( f u δ = u δ on M 1 \ B δ ( x 1 ) f u δ = 0 elsewhere . W e t hen hav e it that ℘ g ( f u δ ) = R M (∆ f u δ 2 + a n R |∇ f u δ | 2 − 4 n − 2 Ric ( ∇ f u δ , ∇ f u δ ) + Q f u δ 2 ) dv ( R M f u δ 2 n n − 4 dv ) n n − 4 . DA VID RASKE 9 Recalling t hat u δ v anishes on B δ ( x 1 ) we see that ℘ g ( f u δ ) = ℘ g 1 ( u δ ) < λ ( M 1 , [ g 1 ]) + ǫ. Consequen tly , λ ( M , [ g ]) < λ ( M 1 , [ g 1 ]) + ǫ = min( λ ( M 1 , [ g 1 ]) , λ ( M 2 , [ g 2 ])) + ǫ. W e will no w pro ceed to pro v e (3.8). Fi rst noti ce that Lemma 3 . 1 can b e use t o sa y that for an y fixed ǫ > 0 , x 1 ∈ M 1 , x 2 ∈ M 2 , we can find tw o p ositive reals δ 1 , δ 2 and smo ot h functions u δ 1 , u δ 2 , where u δ i ∈ C ∞ ( M i ) , wit h the foll owing prop erties: ( u δ 1 = 0 on B δ 1 ( x 1 ) ℘ g 1 ( u δ 1 ) < λ ( M 1 , [ g 1 ]) + ǫ 1 and ( u δ 2 = 0 on B δ 2 ( x 2 ) ℘ g 2 ( u δ 2 ) < λ ( M 2 , [ g 2 ]) + ǫ 1 , where ǫ 1 = 2 − n +4 /n ǫ . Also, notice that we can assume w i thout loss of generalit y that the L 2 n n − 4 ( M ) norms of u δ 1 and u δ 2 are normalized. Using the same reasoning as in the proof of (3.7), a metric g on M 1 # M 2 can be constructed suc h that g agrees with g i when restri ct ed t o M i \ B δ i ( x i ). Let us consider no w the function e u on M = M 1 # M 2 give n b y (3.9) e u =      u δ 1 on M 1 \ B δ 1 ( x 1 ) u δ 2 on M 2 \ B δ 2 ( x 1 ) 0 elsewhere then ℘ g ( e u ) = R M 1 \ B δ 1 ( x 1 ) ((∆ e u ) 2 + a n R |∇ e u | 2 − 4 n − 4 Ric ( ∇ e u, ∇ e u ) + Q e u 2 ) dv ( R M 1 \ B δ 1 ( x 1 ) e u 2 n n − 4 dv + R M 2 \ B δ 2 ( x 2 ) e u 2 n n − 4 dv ) n n − 4 + R M 2 \ B δ 2 ( x 2 ) ((∆ e u ) 2 + a n R |∇ e u | 2 − 4 n − 2 Ric ( ∇ e u, ∇ e u ) + Q e u 2 ) dv ( R M 1 \ B δ 1 ( x 1 ) e u 2 n n − 4 dv + R M 2 \ B δ 2 ( x 2 ) e u 2 n n − 4 dv ) n n − 4 Using ( 3.9) we then obtain ℘ g ( e u ) = R M 1 \ B δ 1 ( x 1 ) ((∆ f u δ 1 ) 2 + a n R |∇ f u δ 1 | 2 − 4 n − 2 Ric ( ∇ f u δ 1 , ∇ f u δ 1 ) + Q f u δ 1 2 ) dv ( R M 1 \ B δ 1 ( x 1 ) f u δ 1 2 n n − 4 dv + R M 2 \ B δ 2 ( x 2 ) f u δ 2 2 n n − 4 dv ) n n − 4 + R M 2 \ B δ 2 ( x 2 ) ((∆ f u δ 2 ) 2 + a n R |∇ f u δ 2 | 2 − 4 n − 2 Ric ( ∇ f u δ 2 , ∇ f u δ 2 ) + Q f u δ 2 2 ) dv ( R M 1 \ B δ 1 ( x 1 ) f u δ 1 2 n n − 4 dv + R M 2 \ B δ 2 ( x 2 ) f u δ 2 2 n n − 4 dv ) n n − 4 10 CONNECTED SUM No w, recalling t he a b o v e stated prop erties of u δ 1 and u δ 2 , we may a lso assume Z M i \ B δ i ( x i ) u δ i 2 n n − 4 dv = 1 , and ℘ g i ( u δ i ) = Z M i \ B δ i ( x i ) (∆ f u δ i 2 + a n R |∇ f u δ i | 2 − 4 n − 2 Ric( ∇ f u δ i , ∇ f u δ i ) + Q f u δ i 2 ) dv < λ ( M i , [ g i ]) + ǫ 1 . Th us λ ( M , [ g ]) ≤ ℘ g ( e u ) < ( λ ( M 1 , [ g 1 ]) + λ ( M 2 , [ g 2 ]) + 2 ǫ 1 )2 − n − 4 n = ( λ ( M 1 , [ g 1 ]) + λ ( M 2 , [ g 2 ]))2 − n − 4 n + ǫ. 4. Conne cted S ums and the P aneit z In v ariants Koba y ashi in [Ko] sho w ed that the Y amab e inv arian t of connected sum of tw o manifolds is greater than o r equal to the smaller of the Y amab e in v ari a n ts of the t w o. The aim of this section is to generalize this result of Kobay ashi t o the case of compact manifolds of dimension n ≥ 5, and with the Y amab e in v arian t Y ( M ) replaced b y it’ s fourth order a nal ogue the P aneitz in v arian t λ ( M ). Namely , we ha ve Theorem 4.1. If M 1 and M 2 ar e close d manifo lds of dimensio n n ≥ 5 . If λ ( M 1 ) > 0 and λ ( M 2 ) > 0 then (4.1) λ ( M 1 # M 2 ) ≥ min { λ ( M 1 ) , λ ( M 2 ) } . W e will basically follow the approac h tak en b y Koba y ashi in [Ko]. First w e consider the P aneitz inv arian t on the disjoin t union of compact mani fol ds. T ake t w o n -manifolds wi t h conformal structures, say ( M 1 , [ g 1 ]) and ( M 2 , [ g 2 ]). W e wri te ( M , [ g ]) = ( M 1 , [ g 1 ]) F ( M 2 , [ g 2 ]) if M is the di sjoint union of M 1 and M 2 , and g i = { g | M i ; g ∈ [ g ] } for i = 1 , 2. Let u b e a smo ot h non-negative function on M . Since M is t he disjoint union of M 1 and M 2 it follo ws that w e can w ri te u = u 1 + u 2 , where u i = 0 on M j , where i 6 = j and where u i is a non-negativ e smo oth function on M i . If we assume that λ ( M i , [ g i ]) ≥ 0 for i = 1 , 2, t hen it can easily b e seen t hat λ ( M , [ g ]) = min { λ ( M 1 , [ g 1 ]) , λ ( M 2 , [ g 2 ]) } . DA VID RASKE 11 Due to Lemma 2. 2, we can assume that λ ( M 1 ) and λ ( M 2 ) are finite; a nd w e can use t he a b o v e equati o n to conclude that λ ( M ) = min { λ ( M 1 ) , λ ( M 2 ) } . Let M b e a compact manifold of dimension n ≥ 5, and p 1 and p 2 t w o p oints of M . W e tak e off t w o small balls around p 1 and p 2 , and t hen attac h a handle instead, the handle b eing to p ological l y the pro duct of a line segmen t and S n − 1 . The new manifold obtained i n this w a y wil l b e denoted b y M . Let M 1 and M 2 b e Riemannian manifolds and l et M 1 F M 2 denote the disjoint union of M 1 and M 2 . If M = M 1 F M 2 and p 1 and p 2 are taken from M 1 and M 2 resp ectiv ely , then M = M 1 # M 2 . Therefore w e see t hat in o rder to prov e Theorem 4.1 it suffices to sho w λ ( M ) ≥ λ ( M ) . Pr o of of The or em 4.1. Let ǫ b e an arbitrary p o sitive n um b er, whic h wi l l b e fixed throughout. First, we take a metri c g on M suc h t hat (4.2) λ ( M , [ g ]) > λ ( M ) − ǫ. Due to con tin uity considerations we ma y assume that [ g ] is conformally flat around the p oin ts p 1 and p 2 . Then there is a function γ ∈ C ∞ ( M \ { p 1 , p 2 } ) and g ∈ [ g ] suc h t hat e g = e γ g i s a complete metric of M \ { p 1 , p 2 } and that eac h of the tw o ends i s i sometric to the half infinite cy l inder [ 0 , ∞ ) × S n − 1 (1). F or con v enience, we write ( M \ { p 1 , p 2 } , e g ) = [0 , ∞ ) × S n − 1 (1) [ ( f M , e g ) [ [0 , ∞ ) × S n − 1 (1) , where f M i s the complemen t o f the tw o cylinders. W e can gl ue ( f M , e g ) and [0 , l ] × S n − 1 (1), al ong their b oundaries to get a smo oth Riemannian manifol d ( M , g l ), where M is as men tioned in the b eginning of the section: (4.3) ( M , g l ) = ( f M , e g ) [ [0 , l ] × S n − 1 (1) . W e t hen hav e λ ( M , [ g l ]) = inf f > 0 R M ((∆ f ) 2 + a n R |∇ f | 2 − 4 n − 2 Ric( ∇ f , ∇ f ) + Qf 2 ) dv ( R M f 2 n n − 4 dv ) n n − 4 , So, t ak e a p o sitive function f l ∈ C ∞ ( M ) suc h t hat (4.4) Z M ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 Ric( ∇ f l , ∇ f l ) + Q f 2 ) dv < λ ( M , [ g l ]) + 1 l + 1 and (4.5) Z M f l 2 n n − 4 dv = 1 . 12 CONNECTED SUM Lemma 4.2. Ther e i s a se ction, say { t l } × S n − 1 , in the cylindri c a l p art of M such that Z { t l }× S n − 1 ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 R ic ( ∇ f l , ∇ f l ) + Qf 2 ) dv < B l , wher e B is a c onstant indep endent of l . Pr o of. Using (4 . 4) we ha ve it that Z S n − 1 × [0 ,l ] ((∆ f ) 2 + a n R |∇ f | 2 − 4 n − 2 Ric( ∇ f , ∇ f ) + Qf 2 ) dv < λ ( M , [ g l ]) + 1 1 + l − Z f M ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 Ric( ∇ f l , ∇ f l ) + Qf l 2 ) dv . It fol l o ws t hen that it suffices to demonstrate that there exi sts a constan t D , i nde- p enden t of l , suc h that Z f M ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 Ric( ∇ f l , ∇ f l ) + Qf l 2 ) dv > D. T ow ards this end, we first not ice that we can rewrite (4.3 ) as follows : ( M , g l ) = ( f M 1 , e g 1 ) [ [0 , l ] × S n − 1 (1) [ ( f M 2 , e g 2 ) , where ( f M i , e g i ), i ∈ { 1 , 2 } , is conformal to ( M i , g i ) \ ( B i ( p i ) , δ ) , where B i ( p i ) is a small ball cen tered at p i and δ is the Euclidean metric. No w, noti ng that a n R + 4 n − 4 Ric is a strictly p ositiv e op erator o n the cylindrical comp onen t of M and that Q i s a strictly p ositive function on the cy lindrical comp onen t, w e see that w e can write ( f M i , e g i ) = ( N i , h i ) [ ( N i ′ , h i ′ ) where ( N ′ 1 , h 1 ′ ) T ([0 , l ] S S n − 1 ) = S n − 1 × { 0 } ; ( N ′ 2 , h 2 ′ ) T ([0 , l ] S S n − 1 ) = S n − 1 × { l } ; h i ′ is conformall y flat; a n R h i ′ − 4 n − 2 Ric h i ′ is a p ositive op erator p oint wise on N i ′ ; and Q h i ′ is p ositive on N i ′ . In geometri c terms we can think o f ( N ′ i , h ′ i ) as a small part of the nec ks of the connected sum M adjacen t to the cy l indrical comp o- nen t. W e will no w use this refined decomp osition o f M to decompose f l ; that i s, we write f l = f l, 1 + f c,l + f 2 ,l , where f 1 ,l is supp orted on f M 1 ; f 2 ,l is supp orted on f M 2 ; and f c,l is su pp orted on N 1 ′ S ([0 , l ] × S n − 1 ) S N 2 ′ . F urthermore w e assume t hat f 1 ,l , f 2 ,l , and f c,l v anish smo othly at some nonzero distance aw a y from the b ound- aries o f their respective supp orts. W e will no w see that the energies R M f 1 ,l P g f 1 ,l dv , DA VID RASKE 13 R M f 2 ,l P g f 2 ,l dv , and R M f c,l P g f c,l dv are all b o unded b elo w b y a constan t indep en- den t of l . First not ice that f c,l P g f c,l ≥ 0 on M , and hence the last integral li sted ab o ve is nonzero. Now, notice that due to our assumption that f i,l , i ∈ { 1 , 2 } , v anish near the bo undaries of their respective sup p orts, w e can extend f i,l to a smo oth, non-negativ e function f ′ i,l on M i , by defining f ′ i,l to b e zero o n M i \ f M i . Lemma 2.1 then provides us wi th the existence of negative constan ts D i suc h that R M i f i,l P g f i,l f i,l dv ≥ D i ( R M i f i,l 2 n n − 4 dv ) n n − 4 ≥ D i . Since D i is determined strictly b y the conformal structure of ( M i , g i ), the ab o v e bounds are indep enden t of l . Putting these three energy estimates together w e ha ve i t that t here exists a constan t D suc h that Z f M ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 Ric( ∇ f l , ∇ f l ) + Qf l 2 ) dv > D. As a consequence w e hav e it that there is a t l ∈ [0 , l ] suc h t hat Z { t l }× S n − 1 ((∆ f l ) 2 + a n R |∇ f l | 2 − 4 n − 2 Ric( ∇ f l , ∇ f l ) + Qf l 2 ) dv < ( λ ( M , C l ) + 1 1 + l + D ) /l , whic h gives us Lemma 4. 1 with B = ( λ ( M ) + 1 + B 1 ). No w w e cut off M o n the section { t 1 × S n − 1 } , and attac h t wo half-infinite cy linders to it, so ( M , \{ p 1 , p 2 } , g ) reapp ears. But this t ime we describ e it as follows: ( M , \{ p 1 , p 2 } , g ) = [0 , ∞ ) × S n − 1 (1) [ ( M − { t 1 } × S n − 1 , g l ) [ [0 , ∞ ) × S n − 1 (1) . W e think of the function f l as defined on M − {{ t l } × S n − 1 } , and extend it to the whole space M − { p 1 , p 2 } as fol l o ws: Let F l b e Lipsc hitz function of M − { p 1 , p 2 } suc h that F l = f l on M − { t l } × S n − 1 and F l ( t, x ) = ( (1 − t ) e f l ( x ) for ( t, x ) ∈ [0 , 1] × S n − 1 ; 0 for ( t, x ) ∈ [1 , ∞ ] × S n − 1 , where e f l = f l | { t l }× S n − 1 ∈ C ∞ ( S n − 1 ). No w it easy to see from (4.4) and (4 . 6) that Z M \{ p 1 ,p 2 } ((∆ F l ) 2 + a n R |∇ F l | 2 − 4 n − 2 Ric( ∇ F l , ∇ F l ) + QF 2 ) dv < λ ( M , [ g l ]) + B l , 14 CONNECTED SUM where B is a constant indep enden t of l . Ob viously from (4.5) we get Z M \{ p 1 ,p 2 } F l 2 n n − 4 dv > 1 . Therefore, we hav e (4.9) inf R M \{ p 1 ,p 2 } ((∆ F ) 2 + a n R |∇ F | 2 − 4 n − 2 Ric( ∇ F , ∇ F ) + Q F 2 ) dv ( R M \{ p 1 ,p 2 } F 2 n n − 4 dv ) n n − 4 ≤ λ ( M ) , where the infim um is taken o v er all nonnegativ e Lipsc hitz functions F w i th compact suppor t . 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