aa r X i v : . [ m a t h . DG ] N ov A SURGERY FORMULA FOR THE SECOND YAMABEINVARIANT
Abstract.
Let (
M, g ) be a compact Riemannian manifold of dimension n ≥ g on M , we let λ ( g ) be the second eigenvalue of the Yamabeoperator L g := n − n − ∆ g + Scal g . Then, the second Yamabe invariant isdefined as σ ( M ) := sup inf h ∈ [ g ] λ ( h )Vol( M, h ) /n . where the supremum is taken over all metrics g and the infimum is taken overthe metrics in the conformal class [ g ]. Assume that σ ( M ) >
0. In the spiritof [4], we prove that if N is obtained from M by a k -dimensional surgery(0 ≤ k ≤ n − n depending only on n suchthat σ ( N ) ≥ min( σ ( M ) , Λ n ). We then give some topological conclusions ofthis result. S. El Sayed October 18, 2017
Contents
1. Introduction 12. Joining manifolds along a submanifold 52.1. Surgery on manifolds 53. The constants Λ n,k n,k
64. Limit spaces and limit solutions 75. L -estimates on W S -bundles 86. Main Theorem 96.1. Construction of the metric g θ Introduction
Definition of the Yamabe operator L g , eigenvalues of L g , smooth Yamabeinvariant σ ( M ) [email protected] Subject Classification. 35J60 (Primary), 35P30, 57R65, 58J50, 58C40 (Secondary).Key words and phrases. Yamabe operator, second Yamabe invariant, surgery, α -genus. Let (
M, g ) be a compact Riemannian manifold of dimension n ≥
3. We denote thescalar curvature by Scal g . Let us define µ ( M, g ) := inf ˜ g ∈ [ g ] Z M Scal ˜ g dv ˜ g (Vol ˜ g ( M )) − ( n − /n and σ ( M ) := sup g µ ( M, g )where, in the definition of µ ( M, g ), the infimum runs over all the metrics g ′ in theconformal class [ g ] of g and where, in the definition of σ ( M ), the supremum is takenover all the Riemannian metrics g on M . The number µ ( M, g ), also denoted by µ ( g ) if no ambiguity, is called the Yamabe constant while σ ( M ) is called the Yamabeinvariant . The Yamabe constant played a crucial role in the solution of the Yamabeproblem solved between 1960 and 1984 by Yamabe, Tr¨udinger, Aubin and Schoen.This problem consists in finding a metric e g conformal to g such that the scalarcurvature Scal e g of e g is constant. For more information, the reader may refer to[17, 13, 7]. An important geometric meaning of µ ( M, g ) and σ ( M ) is contained inthe following well known result: Proposition 1.1.
Let M be a compact differentiable manifold of dimension n ≥ • if g is a Riemannian metric on M , the conformal class [ g ] of g contains ametric of positive scalar curvature if and only if µ ( M, g ) > • M carries a metric g with positive scalar curvature if and only if σ ( M ) > M isthe procedure of constructing from M a new manifold N := M \ S k × B n − k ∪ S k × S n − k − ¯ B k +1 × S n − k − , by removing the interior of S k × B n − k and gluing it with ¯ B k +1 × S n − k − along theboundaries. Gromov-Lawson and Schoen-Yau proved in [12] and [19] the following Theorem 1.2.
Let M be a compact manifold of dimension n ≥ σ ( M ) >
0. Assume that N is obtained from M by a surgery of dimension k (0 ≤ k ≤ n − σ ( N ) > Corollary 1.3.
Every manifold M of dimension n ≥ σ ( M ).These works were generalized by B. Ammann, M. Dahl and E. Humbert in [4] wherethey proved in particular Theorem 1.4. If N is obtained from M by a surgery of dimension 0 ≤ k ≤ n − σ ( N ) ≥ min( σ ( M ) , Λ n ) , where Λ n is a positive constant depending only on n .As a corollary, they obtained the following SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 3
Corollary 1.5.
Let M be a simply connected compact manifold of dimension n ≥
5, then one of this assumptions is satisfied(1) σ ( M ) = 0 (which implies that M is spin);(2) σ ( M ) ≥ α n , where α n is a positive constant depending only on n .Now, let us define the Yamabe operator or conformal Laplacian L g := a ∆ g + Scal g , where a = n − n − and where ∆ g is the Laplace-Beltrami operator. The operator L g is an elliptic differential operator of second order whose spectrum is discrete:Spec( L g ) = { λ ( g ) , λ ( g ) , · · · } , where λ ( g ) < λ ( g ) ≤ · · · are the eigenvalues of L g . The variational characteriza-tion of λ i ( g ) is given by λ i ( g ) = inf V ∈ Gr i ( H ( M )) sup v ∈ V \{ } R M vL g v dv g R M v dv g , where Gr i ( H ( M )) stands for the i -th dimensional Grassmannian in H ( M ) . Oneimportant property of the eigenvalues of L g is that their sign is a conformal invari-ant equal to the sign of the Yamabe constant (see [10]). Consequently, a compactmanifold M possesses a metric with positive λ if and only if it admits a positivescalar curvature metric.Now, if µ ( M, g ) ≥ , it is easy to check that µ ( M, g ) = inf e g ∈ [ g ] λ ( e g )Vol( M, e g ) n , (1)where [ g ] is the conformal class of g and λ is the first eigenvalue of the Yamabe op-erator L g . Inspired by these definitions, one can define the second Yamabe constant and the second Yamabe invariant by µ ( M, g ) = inf e g ∈ [ g ] λ ( e g )Vol( M, e g ) n , and σ ( M ) = sup g µ ( M, g ) . The second Yamabe constant µ ( M, g ) or µ ( g ) if no ambiguity was introducedand studied in [6] when µ ( M, g ) ≥
0. This study was enlarged in [10] where westarted to investigate the relationships between the sign of the second eigenvalueof the Yamabe operator L g and the existence of nodal solutions of the equation L g u = ǫ | u | N − u, where ǫ = − , , +1. The present paper establishes a surgeryformula for σ ( M ) in the spirit of Theorem 1.4. More precisely, our main result isthe following Theorem 1.6.
Let M be a compact manifold of dimension n ≥ σ ( M ) >
0. Assume that N is obtained from M by a surgery of dimension0 ≤ k ≤ n −
3, then we have σ ( N ) ≥ min( σ ( M ) , Λ n ) , where Λ n is a positive constant depending only on n . A SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT
Note that B¨ar and Dahl in [8] proved a surgery formula for the spectrum of theYamabe operator with interesting topological consequences.The proof of Theorem 1.6 is inspired by the one of Theorem 1.4 but some newdifficulties arise here. Let us recall the strategy: first, we fix a metric g on M suchthat µ ( M, g ) is close to σ ( M ). Then the goal is to construct on N a sequence ofmetrics g θ such that lim inf θ → µ ( N, g θ ) ≥ min( µ ( M, g ) , Λ n )where Λ n > n (see Theorem 6.1). Surprisingly, if µ ( M, g ) = 0,we are not able to prove Theorem 6.1 directly. So the first step is to show that onecan assume that µ ( M, g ) = 0 (see Paragraph 6.1.1). Here, we use exactly the samemetrics than in [4] and use many of their properties established in [4]. The proofconsists in studying the first and second eigenvalues λ ( u N − θ g θ ) and λ ( u N − θ g θ )of L u N − θ g θ where u θ is such that µ ( g θ ) = λ ( u N − θ g θ )Vol u N − θ g θ ( M ) /n , or in other words, u θ is such that the metric u N − θ g θ achieves the infimum in thedefinition of µ ( N, g θ ). Two main difficulties arise in this situation: • Contrary to what happened in [4], we could not show that λ ( u N − θ g θ ) and λ ( u N − θ g θ ) are bounded. • The proof of Theorem 1.4 was consisting in obtaining some good “limitequations“. The difficulty here is to ensure thatlim θ λ ( u N − θ g θ ) = lim θ λ ( u N − θ g θ ) . The way to overcome these difficulties is to proceed in two steps: the first one isto show that λ ( u N − θ g θ ) >
0. In a second step, we are able to get the desiredinequality.Let us now come back to Theorem 1.6. Standard cobordism techniques allow todeduce the following corollary
Corollary 1.7.
Let M be a compact, spin, connected and simply connected man-ifold of dimension n ≥ n ≡ , , , | α ( M ) | ≤
1, then σ ( M ) ≥ α n , where α n is a positive constant depending only on n and α ( M ) is the α -genus of M (see Section 7).When M is not spin, the conclusion of the corollary still holds but is a direct ap-plication of Corollary 1.5 and the fact that σ ( M ) ≥ σ ( M ). Note that: • In dimensions 1 , α ( M ) ∈ Z / Z and hence the condition on the α -genus | α ( M ) | ≤ n ≡ , σ ( M ) ≥ α n , for some α n > n . • In dimensions 0 mod 8, when M is spin, α ( M ) = ˆ A ( M ) , where ˆ A is the ˆ A -genus. SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 5
Hence if M is simply connected (not necessarily spin) connected of dimension n ≡ | ˆ A | ≤ σ ( M ) ≥ α n , where α n is a positive constant depending only on n . • In dimensions 4 mod 8, when M is spin, we have α ( M ) = ˆ A ( M ). When M is spin and ˆ A ( M ) ≤
2, we get that | α ( M ) | ≤ M of dimension n ≥ n ≡ | ˆ A | ≤
2, we obtain that σ ( M ) ≥ α n , where α n is a positive constant depending only on n . Acknowledgements:
I would like to thank Emmanuel Humbert for his en-couragements, support and remarks along this work. I am also very grateful toBernd Ammann, Mattias Dahl, Romain Gicquaud and Andreas Hermann for theirremarks and their suggestions.2.
Joining manifolds along a submanifold
Surgery on manifolds.Definition 2.1.
A surgery on a n -dimensional manifold M is the procedure ofconstructing a new n -dimensional manifold N = ( M \ f ( S k × B n − k )) ∪ ( B k +1 × S n − k − ) / ∼ , by cutting out f ( S k × B n − k ) ⊂ M and replacing it by B k +1 × S n − k − , where f : S k × B n − k → M is a smooth embedding which preserve the orientation and ∼ meansthat we paste along the boundary. Then, we construct on the topological space N a differential structure and an orientation that makes a differentiable manifold suchthat the following inclusions M \ f ( S k × B n − k ) ⊂ N, and B k +1 × S n − k − ⊂ N preserve the orientation. We say that N is obtained from M by a surgery ofdimension k and we will denote M k → N. Surgery can be considered from another point of view. In fact, it is a special caseof the connected sum: We paste M and S n along a k -sphere. In this section wedescribe how two manifolds are joined along a common submanifold with trivializednormal bundle. Strictly speaking this is a differential topological construction, butsince we work with Riemannian manifolds we will make the construction adapted tothe Riemannian metrics and use distance neighborhoods defined by the metrics etc.Let ( M , g ) and ( M , g ) be complete Riemannian manifolds of dimension n . Let W be a compact manifold of dimension k , where 0 ≤ k ≤ n . Let ¯ w i : W × R n − k → T M i , i = 1 ,
2, be smooth embeddings. We assume that ¯ w i restricted to W × { } maps to the zero section of T M i (which we identify with M i ) and thus gives anembedding W → M i . The image of this embedding is denoted by W ′ i . Furtherwe assume that ¯ w i restrict to linear isomorphisms { p } × R n − k → N ¯ w i ( p, W ′ i forall p ∈ W i , where N W ′ i denotes the normal bundle of W ′ i defined using g i . We A SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT set w i := exp g i ◦ ¯ w i . This gives embeddings w i : W × B n − k ( R max ) → M i for some R max > i = 1 ,
2. We have W ′ i = w i ( W × { } ) and we define the disjointunion ( M, g ) := ( M ∐ M , g ∐ g ) , and W ′ := W ′ ∐ W ′ . Let r i be the function on M i giving the distance to W ′ i . Then r ◦ w ( p, x ) = r ◦ w ( p, x ) = | x | for p ∈ W , x ∈ B n − k ( R max ). Let r be the function on M definedby r ( x ) := r i ( x ) for x ∈ M i , i = 1 ,
2. For 0 < ǫ we set U i ( ǫ ) := { x ∈ M i : r i ( x ) < ǫ } and U ( ǫ ) := U ( ǫ ) ∪ U ( ǫ ). For 0 < ǫ < θ we define N ǫ := ( M \ U ( ǫ )) ∪ ( M \ U ( ǫ )) / ∼ , and U Nǫ ( θ ) := ( U ( θ ) \ U ( ǫ )) / ∼ where ∼ indicates that we identify x ∈ ∂U ( ǫ ) with w ◦ w − ( x ) ∈ ∂U ( ǫ ). Hence N ǫ = ( M \ U ( θ )) ∪ U Nǫ ( θ ) . We say that N ǫ is obtained from M , M (and ¯ w , ¯ w ) by a connected sum along W with parameter ǫ .The diffeomorphism type of N ǫ is independent of ǫ , hence we will usually write N = N ǫ . However, in situations when dropping the index causes ambiguities, wewill keep the notation N ǫ . For example the function r : M → [0 , ∞ ) gives acontinuous function r ǫ : N ǫ → [ ǫ, ∞ ) whose domain depends on ǫ . It is also goingto be important to keep track of the subscript ǫ on U Nǫ ( θ ) since crucial estimateson solutions of the Yamabe equation will be carried out on this set.The surgery operation on a manifold is a special case of taking connected sumalong a submanifold. Indeed, let M be a compact manifold of dimension n andlet M = M , M = S n , W = S k . Let w : S k × B n − k → M be an embeddingdefining a surgery and let w : S k × B n − k → S n be the canonical embedding. Since S n \ w ( S k × B n − k ) is diffeomorphic to B k +1 × S n − k − we have in this situationthat N is obtained from M using surgery on w , see [16, Section VI, 9].3. The constants Λ n,k Definition of Λ n,k . In this paragraph, we define some constants Λ n,k in thesame way than in [4]. The only difference is that the functions we considered arenot necessarily positive. More precisely, let (
M, h ) be a Riemannian manifold ofdimension n ≥
3. For i = 1 , ( i ) the set of C functions v (notnecessarily positive) solution of the equation L h v = µ | v | N − v, where µ ∈ R . We assume that v satisfies • v , • k v k L N ( M ) ≤ , • v ∈ L ∞ ( M ) , SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 7 together with • v ∈ L ( M ) , for i = 1 ,or • µ k v k N − L ∞ ( M ) ≥ ( n − k − ( n − n − , for i = 2 . For i = 1 ,
2, we set µ ( i ) ( M, h ) := inf v ∈ Ω ( i ) ( M,h ) µ ( v ) . If Ω ( i ) ( M, h ) is empty, we set µ ( i ) = ∞ . Definition 3.1.
For n ≥ ≤ k ≤ n −
3, we defineΛ ( i ) n,k := inf c ∈ [ − , µ ( i ) ( H k +1 c × S n − k − ) , and Λ n,k := min(Λ (1) n,k , Λ (2) n,k ) , where H k +1 c := ( R k × R , η k +1 c = e ct ξ k + dt )When considering only positive functions v , B. Ammann, M. Dahl and E. Humbertproved in [4] that these constants are positive. It is straightforward to see that thepositivity of v has no role in their proof and hence it remains true that Λ n,k > Limit spaces and limit solutions
Lemma 4.1.
Let M be an n -dimensional manifold. let ( g θ ) be a sequence ofmetrics which converges toward a metric g in C on all compact K ⊂ M when θ →
0. Assume that v θ is a sequence of functions such that k v θ k L ∞ ( M ) is boundedand k L g θ v θ k L ∞ ( M ) tends to 0. Then, there exists a smooth function v solution ofthe equation L g v = 0such that v θ tends to v in C on each compact set K ⊂⊂ V . Proof:
Let
K, K ′ be compact sets of M such that K ′ ⊂ K , we have − g ijθ (cid:0) ∂ i ∂ j v θ − Γ kij ∂ k v θ (cid:1) + n − n −
1) Scal g θ v θ = f θ → . Using Theorem 9.11 in [11], one easily checks that k v θ k H ,p ( K ′ ,g ) ≤ C ( k L g θ v θ k L p ( K,g θ ) + k v θ k L p ( K,g θ ) ) . It follows that v θ is bounded in H ,p ( K ′ , g ) for all p ≥
1. Using Kondrakov’stheorem, there exists v K ′ such that v θ tends to v K ′ in C ( K ′ ) . Taking an increasingsequence of compact sets K m such that ∪ m K m = M , ( v θ ) converges to v m on C ( K m ) , we define v := v m on K m . Using the diagonal extraction process, wededuce that v θ tends to v in C on any compact set and that v verifies the same A SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT
Yamabe equation as v θ . Since for each compactly supported smooth function ϕ ,we have Z M L g θ ϕv θ dv g θ → Z M L g ϕvdv g , and k L g θ v θ k L ∞ ( M ) → , we obtain that L g v = 0 in the sense of distributions. Using standard regularitytheorems, v is smooth. 5. L -estimates on W S -bundles
We suppose that the product P := I × W × S n − k − is equipped with a metric g WS of the form g WS = dt + e ϕ ( t ) h t + σ n − k − and we mean by W S -bundle this product, where h t is a smooth family of metricson W and depending on t and ϕ is a function on I . Let π : P → I be the projectiononto the first factor and F t = π − ( t ) = { t } × W × S n − k − , and the metric inducedon F t is defined by g t := dt + e ϕ ( t ) h t + σ n − k − . Let H t be the mean curvature of F t in P , it is given by the following H t = − kn − ϕ ′ ( t ) + e ( h t ) , with e ( h t ) := tr h t ( ∂ t h t ) . The derivative of the element of volume of F t is ∂ t dv g t = − ( n − H t dv g t . From the definition of H t , when t → h t is constant, we obtain that H t = − kn − ϕ ′ ( t ) . Definition 5.1.
We say that the condition ( A t ) is verified if the following assump-tions are satisfied:1 . ) t h t is constant , . ) e − ϕ ( t ) inf x ∈ W Scal h t ( x ) ≥ − n − k − a, . ) | ϕ ′ ( t ) | ≤ , . ) 0 ≤ − kϕ ′′ ( t ) ≤ ( n − n − k − . ( A t )Similarly, for the condition B t , we should have another assumptions to verify1 . ) t ϕ ( t ) is constant,2 . ) inf x ∈ F t Scal g WS ( x ) ≥ Scal σ n − k − = ( n − k − n − k − , . ) ( n − e ( h t ) + n − ∂ t e ( h t ) ≥ − ( n − k − . ( B t ) SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 9
Theorem 5.2.
Let α, β ∈ R such that [ α, β ] ⊂ I. We suppose also that one of theconditions ( A t ) and ( B t ) is satisfied. We assume that we have a solution v of theequation L g WS v = a ∆ g WS v + Scal g WS v = µu N − v + d ∗ A ( dv ) + Xv + ǫ∂ t v − sv (2)where s, ǫ ∈ C ∞ ( P ), A ∈ End( T ∗ P ), and X ∈ Γ( T P ) are perturbation termscoming from the difference between G and g WS . We assume that the endomorphism A is symmetric and that X and A are vertical, that is dt ( X ) = 0 and A ( dt ) = 0.Such that µ k u k N − L ∞ ( P ) ≤ ( n − k − ( n − n − . (3)Then there exists c > α , β , and ϕ , such that if k A k L ∞ ( P ) , k X k L ∞ ( P ) , k s k L ∞ ( P ) , k ǫ k L ∞ ( P ) , k e ( h t ) k L ∞ ( P ) ≤ c then Z π − (( α + γ,β − γ )) v dv g WS ≤ k v k L ∞ n − k − g α ( F α ) + Vol g β ( F β )) , where γ := √ n − k − .Remark that we should have β − α > γ to obtain our result and note that thistheorem gives us an estimate of k v k L . For the proof of this Theorem, we mimic exactly the proof of Theorem 6.2 in [4].The only difference is that we consider here a nodal solution (and not a positivesolution) of the equation L g WS v = µu N − v + d ∗ A ( dv ) + Xv + ǫ∂ t v − sv. Other details are exactly the same.6.
Main Theorem
Theorem 1.6 is a direct corollary of
Theorem 6.1.
Let (
M, g ) be a compact Riemannian manifold of dimension n ≥ µ ( M, g ) > N be obtained from M by a surgery of dimension0 ≤ k ≤ n −
3. Then there exists a sequence of metrics g θ such thatlim inf θ → µ ( N, g θ ) ≥ min( µ ( M, g ) , Λ n ) , where Λ n > n .Indeed, to get Theorem 1.6, it suffices to apply Theorem 6.1 with a metric g suchthat µ ( M, g ) is arbitrary closed to σ ( M ). The conclusion easily follows since µ ( N, g θ ) ≤ σ ( M ). This section is devoted to the proof of Theorem 6.1.6.1. Construction of the metric g θ . Modification of the metric g . For a technical reason, we will need in the proofof Theorem 6.1 that µ ( g ) = 0. To get rid of this difficulty, we need the followingproposition: Proposition 6.2.
There exists on M a metric g ′ arbitrary close to g in C suchthat µ ( g ′ ) = 0.Indeed, let us assume for a while that Theorem 6.1 is true if µ ( g ) = 0 and let ussee that the result remains true if µ ( g ) = 0. A first observation is that if g ′ is closeenough to g in C , then as one can check, µ ( g ′ ) is close to µ ( g ). Let us consider ametric g ′ given by Proposition 6.2 close enough to g so that µ ( g ′ ) > µ ( g ) − ǫ > ǫ . From Theorem 6.1 applied to g ′ , we obtain a sequence ofmetrics g θ on N such thatlim inf θ → µ ( N, g θ ) ≥ min( µ ( M, g ′ ) , Λ n ) ≥ min( µ ( M, g ) − ǫ, Λ n ) . Letting ǫ tend to 0, we obtain Theorem 6.1. It remains to prove Proposition 6.2. Proof of Proposition 6.2:
At first, in order to simplify notations, we willconsider g as a metric on M ∐ S n and equal to the standard metric g = σ n on S n . Since µ ( g ) = 0, we can assume that Scal g = 0, possibly making a conformalchange of metrics. Let us consider a metric h for which Scal h is negative andconstant and whose existence is given in [7]. Consider the analytic family of metrics g t := th + (1 − t ) g . Since the first eigenvalue λ t of L g t is simple, the function t → λ t is analytic (see for instance Theorem VII.3.9 in [14]). Since λ = 0 and λ <
0, itfollows that for t arbitrary close to 0, λ t = 0. Proposition 6.2 follows since µ ( g t )has the same sign than λ t .6.1.2. Definition of the metric g θ . As explained above, we will use the same con-struction as in [4]. Consequently, we give the definition of g θ without additionalexplanations. The reader may refer to [4] for more details. We keep the same nota-tions than in Section 2. Let h be the restriction of g to the surgery sphere S ′ ⊂ M and h be the restriction of the standard metric σ n = g on S n to S ′ ⊂ S n . Define S ′ := S ′ ∐ S ′ and h := h ∐ h on S ′ . In the following, r denotes the distancefunction to S ′ in ( M ∐ S n , g ∐ σ n ). In polar coordinates, the metric g has the form g = h + ξ n − k + T = h + dr + r σ n − k − + T (4)on U ( R max ) \ S ′ ∼ = S ′ × (0 , R max ) × S n − k − . Here T is a symmetric (2 , S ′ which is the error term measuring the fact that g is not in generala product metric (at least near S ′ ). We also define the product metric g ′ := h + ξ n − k = h + dr + r σ n − k − , (5)on U ( R max ) \ S ′ so that g = g ′ + T . As in [4], we have | T ( X, Y ) | ≤ Cr | X | g ′ | Y | g ′ , | ( ∇ U T )( X, Y ) | ≤ C | X | g ′ | Y | g ′ | U | g ′ , | ( ∇ U,V ) T ( X, Y ) | ≤ C | X | g ′ | Y | g ′ | U | g ′ | V | g ′ , for X, Y, U, V ∈ T x M and x ∈ U ( R max ). We define T := T | M and T := T | S n . Wefix R ∈ (0 , R max ), R < F : M \ S ′ → R SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 11 such that F ( x ) = ( , if x ∈ M \ U ( R max ) ∐ S n \ U ( R max ); r ( x ) − , if x ∈ U i ( R ) \ S ′ .Next we choose a sequence θ = θ j of positive numbers tending to 0. For any θ wethen choose a number δ = δ ( θ ) ∈ (0 , θ ) small enough to suit with the argumentsbelow. For any θ > δ there is A θ ∈ [ θ − , ( δ ) − ) and asmooth function f : U ( R max ) → R depending only on the coordinate r such that f ( x ) = ( − ln r ( x ) , if x ∈ U ( R max ) \ U ( θ );ln A θ , if x ∈ U ( δ ),and such that (cid:12)(cid:12)(cid:12)(cid:12) r dfdr (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) dfd (ln r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ , and (cid:13)(cid:13)(cid:13)(cid:13) r ddr (cid:18) r dfdr (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ = (cid:13)(cid:13)(cid:13)(cid:13) d fd (ln r ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ → θ →
0. Set ǫ = e − A θ δ that we assume smaller than 1 and use this ǫ to construct M as in Section 2. On U Nǫ ( R max ) = ( U ( R max ) \ U ( ǫ )) / ∼ we define t by t := ( − ln r + ln ǫ, on U ( R max ) \ U ( ǫ );ln r − ln ǫ, on U ( R max ) \ U ( ǫ ).One checks that • r i = e | t | +ln ǫ = ǫe | t | ; • F ( x ) = ǫ − e −| t | for x ∈ U ( R ) \ U N ( θ ), or equivalently if | t | + ln ǫ ≤ ln R and hence F g = ǫ − e − | t | ( h + T ) + dt + σ n − k − on U ( R ) \ U N ( θ ); • and f ( t ) = ( −| t | − ln ǫ, if ln θ − ln ǫ ≤ | t | ≤ ln R max − ln ǫ ;ln A θ , if | t | ≤ ln δ − ln ǫ .We have | df /dt | ≤ k d f /dt k L ∞ →
0. Now, we choose a cut-off function χ : R → [0 ,
1] such that χ = 0 on ( −∞ , − | dχ | ≤
1, and χ = 1 on [1 , ∞ ). Finally, wedefine g θ := F g i , on M i \ U i ( θ ); e f ( t ) ( h i + T i ) + dt + σ n − k − , on U i ( θ ) \ U i ( δ ); A θ χ ( t/A θ )( h + T ) + A θ (1 − χ ( t/A θ ))( h + T )+ dt + σ n − k − , on U Nǫ ( δ ).Moreover, the metric g θ can be written as g θ := g ′ θ + e T t on U N ( R ) , where g ′ θ is the metric without error term and it is equal to g ′ θ = e f ( t ) e h t + dt + σ n − k − , where the metric e h t is given by e h t := χ ( tA θ ) h + (1 − χ ( tA θ )) h , and e T t is the error term and his expression is given by the following e T t := e f ( t ) ( χ ( tA θ ) T + (1 − χ ( tA θ )) T ) . We further have the following properties of the error term e T t | e T ( X, Y ) | ≤ Cr | X | g ′ θ | Y | g ′ θ , |∇ e T t | g ′ θ ≤ Ce − f ( t ) , |∇ e T t | g ′ θ ≤ Ce − f ( t ) , where ∇ is the Levi-Civita connection with respect to the metric g ′ θ , for all X , Y ∈ T x N and x ∈ U N ( R ) . A preliminary result.
In order to prove Theorem 6.1, we will start by prov-ing the following results.
Theorem 6.3. Part 1: let ( u θ ) be a sequence of functions which satisfy L g θ u θ = λ θ | u θ | N − u θ , such that R N | u θ | N dv g θ = 1 and λ θ → θ → λ ∞ , where λ ∞ ∈ R . Then, at least oneof the two following assertions is true(1) λ ∞ ≥ Λ n , where Λ n > n ;(2) there exists a function u ∈ C ∞ ( M ∐ S n ), u ≡ S n , u M solutionof L g u = λ ∞ | u | N − u, with Z M | u | N dv g = 1such that for all compact sets K ⊂ M ∐ S n \ S ′ (note that K can alsobe considered as a subset of N ), F n − u θ tends to u in C ( K ), where F isdefined in Section 6.1. Moreover, we have(a) the norm L of u θ is bounded uniformly in θ ;(b) lim b → lim sup θ → sup U N ( b ) u θ = 0;(c) lim b → lim sup θ → R U N ( b ) u Nθ dv g θ = 0 . Part 2: let u θ be as in Part 1 above and assume that Assertion 2) is true. Let v θ be a sequence of functions which satisfy L g θ v θ = µ θ | u θ | N − v θ , such that R N v Nθ dv g θ = 1 , µ θ → µ ∞ where µ ∞ < µ ( S n ). Then, there exists afunction v ∈ C ∞ ( M ∐ S n ), v ≡ S n , v M solution of L g v = µ ∞ | u | N − v with Z M | v | N dv g = 1and such that for all compact sets K ⊂ M ∐ S n \ S ′ , F n − v θ tends to v in C ( K ) . Moreover,(1) the norm L of v θ is bounded uniformly in θ ;(2) lim b → lim sup θ → sup U N ( b ) v θ = 0;(3) lim b → lim sup θ → R U N ( b ) v Nθ dv g θ = 0 . SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 13
Proof of Theorem 6.3 Part 1.
Let ( u θ ) be a sequence of functions whichsatisfy L g θ u θ = λ θ | u θ | N − u θ , such that R N | u θ | N dv g θ = 1 and λ θ → θ → λ ∞ , where λ ∞ ∈ R . We proceed exactlyas in [4] where here, the manifold M is S n equiped with the standard metric σ n ,and where W is the sphere S k . The only difference will be that u θ may now havea changing sign. Remark . In the proof of the main theorem in [4], it was proven that λ ∞ > −∞ . Here, we made the assumption that λ ∞ has a limit. Without this assumption, onecould again prove that λ ∞ > −∞ but the point here is that there is no reason why λ ∞ should be bounded from above contrary to what happened in [4].The argument of Corollary 7.7 in [4] still holds here and shows thatlim inf θ k u θ k L ∞ ( N ) > . (7)Several cases are studied: Case I. lim sup θ → k u θ k L ∞ ( N ) = ∞ .Set m θ := k u θ k L ∞ ( N ) and choose x θ ∈ N such that u θ ( x θ ) = m θ . After taking asubsequence, we can assume that lim θ → m θ = ∞ . We have to study the followingtwo subcases. Subcase I.1.
There exists b > x θ ∈ N \ U N ( b ) for an infinite numberof θ . Subcase I.2.
For all b > x θ ∈ U N ( b ) for θ sufficiently small. Case II.
There exists a constant C such that k u θ k L ∞ ( N ) ≤ C for all θ . Subcase II.1.
There exists b > θ → λ θ sup U N ( b ) u θN − ! < ( n − k − ( n − n − . Subsubcase II.1.1. lim sup b → lim sup θ → sup U N ( b ) u θ > Subsubcase II.1.2. lim b → lim sup θ → sup U N ( b ) u θ = 0. Subcase II.2. λ θ sup U N ( b ) u θN − ≥ ( n − k − ( n − n − λ ∞ ≥ µ ( S n ). The proof still holdswhen u θ has a changing sign. In Subsubcase II.1.1 and Subcase II.2, we obtainthat λ ∞ ≥ Λ n,k where Λ n,k is a positive number depending only on n and k . Thedefinition of Λ n,k in [4] is the infimum of energies of positive solutions of the Yam-abe equation on model spaces (see Section 3). This definition has to be slightlymodified to allow nodal solutions. As explained in Section 3 the proof that Λ n,k > In Subcases I.1, I.2, II.1.1 and II.2, we then get that λ ∞ ≥ Λ n , whereΛ n := min k ∈{ , ··· ,n − } { Λ n,k , µ } . In particular, Assertion 1) of part 1 in Theorem 6.3 is true. So let us examineSubsubcase II.1.2. The assumption of Subcase II.1 allows to obtain as in [4] that Z N u θ dv g θ ≤ C. (8)for some C >
0. The assumptions of Subcase II.1.2 are thatsup N ( u θ ) ≤ C (9)and that lim sup b → lim sup θ → sup U N ( b ) u θ = 0 . (10) Step 1.
We prove that lim b → lim sup θ → R U N ( b ) | u θ | N dv g θ = 0.Let b >
0. We have, by Relation (8) Z U N ( b ) | u θ | N dv g θ ≤ A sup U N ( b ) | u θ | N − , where A is a positive number which does not depend on b and θ. The claim thenfollows from (10).
Step 2. C convergence on all compact sets of M ∐ S n \ S ′ . Let (Ω j ) j be an increasing sequence of subdomains of ( M ∐ S n \ S ′ ) with smoothboundary such that S j Ω j = M ∐ S n \ S ′ , Ω j ⊂ Ω j +1 . The norm k u θ k L ∞ ( N ) isbounded, then so is k u θ k L ∞ (Ω j +1 ) . Using standard results on elliptic regularity (formore details, see for example [11]), we see that the sequence ( u θ ) is bounded inthe Sobolev space H ,p (Ω ′ j ) ∀ p ∈ (1 , ∞ ) where Ω ′ j is any domain such that Ω j ⊂ Ω ′ j ⊂ Ω ′ j ⊂ Ω j +1 . The Sobolev embedding Theorem implies that ( u θ ) is boundedin C ,α (Ω j ) for any α ∈ (0 , u θ ) converges to functions e u j ∈ C (Ω j ) and such that e u j | Ω j − = e u j − . We define e u = e u j on Ω j . By taking a diagonal subsequence of u θ , we get that u θ tends to e u in C on anycompact subset of M ∐ S n \ S ′ and by C -convergence of the functions u θ , thefunction e u satisfies the equation L g θ e u = λ ∞ | e u | N − e u on M ∐ S n \ S ′ . (11)We recall that g θ = F g = ( F n − ) n − g on U N ( b ). By conformal invariance of theYamabe operator we obtain for all vL F g v = F − n +22 L g ( F n − v ) . Now we set u = F n − e u. SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 15
We obtain L g u = F n +22 L F g e u = F n +22 λ ∞ | e u | N − e u = λ ∞ | u | N − u. This shows that u is a solution on ( M ∐ S n \ S ′ , g ) of the following equation L g u = λ ∞ | u | N − u. Moreover, using Step 1 and the fact that R N u Nθ dv g θ = 1, the function u satisfies Z M ∐ S n u N dv g = Z M ∐ S n \ S ′ e u N dv g = lim b → lim θ → Z U N ( b ) u Nθ dv g θ = 1 . Step 3.
Removal of the singularityThe next step is to show that u is a solution on all M ∐ S n of L g u = λ ∞ | u | N − u. (12)To prove this fact, we will show that for all ϕ ∈ C ∞ ( M ∐ S n ), we have Z M ∐ S n L g uϕ dv g = Z M ∐ S n λ ∞ | u | N − uϕ dv g . First, we have Z M ∐ S n uL g ϕ dv g = Z M ∐ S n uL g ( ϕ − χ ǫ ϕ + χ ǫ ϕ ) dv g = Z M ∐ S n uL g ( χ ǫ ϕ ) dv g + Z M ∐ S n uL g ((1 − χ ǫ ) ϕ ) dv g , where (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ ǫ = 1 if d g ( x, S ′ ) < ǫ,χ ǫ = 0 if d g ( x, S ′ ) ≥ ǫ, | dχ ǫ | < ǫ . Since (1 − χ ǫ ) is compactly supported in M ∐ S n \ S ′ , we have Z M ∐ S n uL g ((1 − χ ǫ ) ϕ ) dv g = Z M ∐ S n ( L g u )(1 − χ ǫ ) ϕ dv g → Z M ∐ S n L g uϕ dv g = Z M ∐ S n λ ∞ | u | N − uϕ dv g . Then, it remains to prove that Z M ∐ S n uL g ( χ ǫ ϕ ) dv g → . We have L g ( χ ǫ ϕ ) = C n ∆( χ ǫ ϕ ) + Scal g ( χ ǫ ϕ )= C n ∆ χ ǫ ϕ + C n ∆ ϕχ ǫ + Scal g ( χ ǫ ϕ ) − h∇ χ ǫ , ∇ ϕ i = χ ǫ L g ϕ + C n (∆ χ ǫ ) ϕ − h∇ χ ǫ , ∇ ϕ i . According to Lebesgue Theorem, it holds that Z M ∐ S n uχ ǫ L g ϕ dv g → (cid:12)(cid:12)(cid:12)(cid:12)Z M ∐ S n uL g ( χ ǫ ϕ ) dv g (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ Z C ǫ u dv g (13) ≤ Cǫ (cid:18)Z C ǫ u dv g (cid:19) (Vol( Supp ( C ǫ ))) , (14)where C ǫ = { x ∈ M ∐ S n ; ǫ < d ( x, S ′ ) < ǫ } = U N (2 ǫ ) \ U N ( ǫ ).In addition, we get from (8) that Z N e u dv F g < + ∞ , which implies that Z C ǫ e u dv F g < + ∞ . Let us compute Z C ǫ e u dv g θ = Z C ǫ (cid:16) F n − (cid:17) nn − F − ( n − u dv g = Z C ǫ F u dv g < + ∞ . We recall that F = r on C ǫ . Coming back to (13), we deduce (cid:12)(cid:12)(cid:12)(cid:12)Z M uL g ( χ ǫ ϕ ) dv g (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ (cid:18)Z C ǫ u F F dv g (cid:19) (Vol( C ǫ )) ≤ Cǫ × ǫ × ǫ n − k = Cǫ n − k − . Since k ≤ n − , we have n − k − > , which implies that Z M ∐ S n uL g ( χ ǫ ϕ ) dv g → . Finally, we get that u is a solution on M ∐ S n of the equation L g u = λ ∞ | u | N − u. Step 4.
We have either u ≡ S n either λ ∞ ≥ µ ( S n ) . SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 17
Note that the function u verifies Z M ∐ S n | u | N dv g ≤ . (15)Since Z M ∐ S n | u | N dv g = Z M ∐ S n | e u | N dv g θ ≤ Z N | e u | N dv g θ ≤ lim θ → Z N | u θ | N dv g θ = 1 . Assume that u S n .Setting w = u | S n and using equations (12) and (15), we have µ ( S n ) ≤ Y ( w ) = λ ∞ R S n w N dv g (cid:0)R S n w N dv g (cid:1) n − n = λ ∞ (cid:18)Z S n w N dv g (cid:19) n ≤ λ ∞ . Then we obtain that λ ∞ ≥ µ ( S n ) and hence, the conclusion 1) of Theorem 6.3 Part1 is true.6.2.2. Proof of Theorem 6.3 Part 2.
We consider a function v θ satisfying L g θ v θ = µ θ | u θ | N − v θ , (16)with Z N | v θ | N dv g θ = 1 . A first remark is the following: as in Lemma 7.6 of [4], we observe that U N ( b ) is a W S -bundle for any b >
0. Since u θ satisfieslim b → lim sup θ → sup U N ( b ) u θ = 0 . Then, for b small enough, we have µ θ k u θ k N − U N ( b ) ≤ ( n − k − ( n − n − . We then can apply Theorem 5.2 on U N ( b ) and the proof of Lemma 7.6 of [4] showsthat there exists numbers c , c > θ such that Z N | v θ | dv g θ ≤ c k v θ k L ∞ ( N ) + c . (17)As a consequence, we get thatlim inf θ → k v θ k L ∞ ( N ) > . Indeed, assume that lim θ → k v θ k L ∞ ( N ) = 0 . By Equation (17), we have1 = Z N | v θ | N dv g θ ≤ k v θ k N − L ∞ ( N ) Z N | v θ | dv g θ ≤ k v θ k N − L ∞ ( N ) ( c k v θ k L ∞ ( N ) + c ) → , as θ →
0. This gives the desired contradiction. In the rest of the proof, we willstudy several cases. In what follows, only Subcase II.1.2 will be a big deal: SubcasesI.1, I.2 and II.1 will be excluded by arguments mostly contained in [4]. So we willjust give few explanations for these cases.
Case I. lim sup θ → k v θ k L ∞ ( N ) = ∞ .Set m θ := k v θ k L ∞ ( N ) and choose x θ ∈ N with v θ ( x θ ) = m θ . After taking asubsequence we can assume that lim θ → m θ = ∞ . Subcase I.1.
There exists b > x θ ∈ N \ U N ( b ) for an infinite numberof θ .By taking a subsequence we can assume that there exists ¯ x ∈ M ∐ S n \ U ( b ) suchthat lim θ → x θ = ¯ x . We define ˜ g θ := m n − θ g θ . For r >
0, [4] tells that for θ smallenough, there exists a diffeomorphismΘ θ : B n (0 , r ) → B g θ ( x θ , m − n − θ r )such that the sequence of metrics (Θ ∗ θ (˜ g θ )) tends to the flat metric ξ n in C ( B n (0 , r )),where B n (0 , r ) is the standard ball in R n centered in 0 with radius r . We let˜ u θ := m − θ u θ , ˜ v θ := m − θ v θ and we have L ˜ g θ ˜ v θ = λ θ ˜ u N − θ ˜ v θ = λ θ m N − θ u θN − ˜ v θ . Since k u θ k L ∞ ( N ) ≤ C, it follows that k L ˜ g θ ˜ v θ k L ∞ ( N ) tends to 0. Applying Lemma4.1, we obtain a solution v R n L ξ n v = 0. SinceScal ξ n = 0, v is harmonic and admits a maximum at x = 0. As a consequence, v isconstant equal to v (0) = 1. This is a contradiction, since k v k L N ≤ . Subcase I.2.
For all b > x θ ∈ U N ( b ) for θ sufficiently small.We proceed as in Subcase I.2 in [4]. As in Subcase I.1 above, we get from Lemma4.1 a function v which is harmonic on R n and admits a maximum at x = 0. Thisis again a contradiction. Case II.
There exists a constant C such that k v θ k L ∞ ( N ) ≤ C for all θ .By (17), there exists a constant A independent of θ such that k v θ k L ( N,g θ ) ≤ A . (18)We split the treatment of Case II into two subcases. Subcase II.1. lim sup b → lim sup θ → sup U N ( b ) v θ > v which is a solution of L G c v = 0 on R k +1 × S n − k − , G c for some c ∈ [ − ,
1] where G c = e cs ξ k + ds + σ n − k − . In Subcases I.1 and I.2, we used the fact that λ θ m N − θ SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 19 tends to 0 to show that at the limit L G c v = 0. Here, the argument is different: firstwe set α := lim sup b → lim sup θ → sup U N ( b ) v θ >
0. Then, we can suppose thatthere exists a sequence of positive numbers ( b i ) and ( θ i ) such thatsup U N ( b i ) v θ i ≥ α , for all i . To simplify, we write θ for θ i and b for b i . Take x ′ θ ∈ U N ( b θ ) such that v θ ( x ′ θ ) ≥ α . For r, r ′ >
0, we define U θ ( r, r ′ ) := B e h tθ ( y θ , e − f ( t θ ) r ) × [ t θ − r ′ , t θ + r ′ ] × S n − k − . As in [4], the function v is obtained as the limit of v θ on each U θ ( r, r ′ ) (with r, r ′ > L G c v = 0 follows from the observation thatsup U θ ( r,r ′ ) | u θ | = 0 , hence | u θ | N − v θ → U θ ( r, r ′ ) . Subcase II.2. lim b → lim sup θ → sup U N ( b ) v θ = 0.By the same method than in Subsection 6.2.1, we obtain that there is a function v solution of the following equation L g v = µ ∞ | u | N − v, such that Z N v N dv g ≤ . Suppose that v S n , then we have µ ( S n ) ≤ Y ( v ) = µ ∞ R S n u N − v dv g ( R S n v N dv g ) N = 0since u ≡ S n . This is a contradiction. This proves that v S n . By thesame argument than in Part 1, we have R M | v | N dv g = 1. We finally obtain that thefunction v satisfies all the desired conclusions of Theorem 6.3 Part 2.6.3. Proof of Theorem 6.1.
Let ( g θ ) the sequence of metrics defined on N as inSection 6.1. Step 1:
For θ small enough, we show that if λ k ( M, g ) > ⇒ λ k ( N, g θ ) > , where λ k is the k th eigenvalue associated to the Yamabe equation. Remark . Note that this step implies that the existence of a metric with positive λ k is preserved by surgery of dimension k ∈ { , · · · , n − } . This is an alternativeproof of a result already contained in [8]. We proceed by contradiction and we suppose that λ k ( N, g θ ) ≤ . Let u θ be aminimizing solution of the Yamabe problem. By referring to [10], there existsfunctions v θ, = u θ , v θ, , · · · , v θ,k solution of the following equation on NL g θ v θ,i = λ θ,i u N − θ v θ,i , where λ θ,i = λ i ( N, u N − θ g θ ) , such that Z N v θ,iN dv g θ = 1 and Z N u θN − v θ,i v θ,j dv g θ = 0 for all i = j. By conformal invariance of the sign of the eigenvalues of the Yamabe operator (see[10]), we have λ θ,i = λ i ( N, u N − θ g θ ) ≤ . Moreover, by construction, it is easy to check that λ θ, = µ θ where µ θ = µ ( N, g θ )is the Yamabe constant of the metric g θ . The main theorem in [4] implies thatlim θ → λ θ, = lim θ → µ θ > −∞ . It follows that there exists a constant C > − C ≤ λ θ, ≤ · · · ≤ λ θ,k ≤
0. Then, for all i , λ θ,i is bounded and byrestricting to a subsequence we can assume that λ ∞ ,i := lim θ → λ θ,i exists. Parts1) and 2) of Theorem 6.1 give the existence of functions u = v , · · · , v k defined on M, with v i = 0 for all i such that F n − v θ,i tends to v i in C on each compact set K ⊂ M ∐ S n \ S ′ . The functions v i are solutions of the following equation L g v i = λ ∞ ,i u N − v i . Moreover, we have Z M | v i | N dv g ≤ b → lim sup θ → Z U Nǫ ( b ) | v θ,i | N dv g = 0 . Let us show that for all i = j , we get that Z M u N − v i v j dv g = 0 . Set e u θ = F n − u θ , and e v θ,i = F n − v θ,i . For b > i = j Z M \ U ( b ) u N − v i v j dv g = lim θ → R M \ U ( b )= N \ U Nǫ ( b ) e u N − θ e v θ,i e v θ,j dv g = lim θ → R M \ U ( b )= N \ U Nǫ ( b ) u N − θ v θ,i v θ,j dv g θ where we used dv g θ = F n dv g . Using now the fact that R N u N − θ v θ,i v θ,j dv g θ = 0,we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M \ U ( b ) u N − v i v j dv g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim θ → Z N \ U Nǫ ( b ) u N − θ v θ,i v θ,j dv g θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim θ → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U Nǫ ( b ) u N − θ v θ,i v θ,j dv g θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 21
We write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U Nǫ ( b ) u N − θ v θ,i v θ,j dv g θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z U Nǫ ( b ) u Nθ dv g θ ! N − N Z U Nǫ ( b ) | v θ,i | N dv g θ ! N Z U Nǫ ( b ) | v θ,j | N dv g θ ! N . Using the assertion lim b → lim sup θ → Z U Nθ ( b ) v Nθ,i dv g θ = 0 . we obtain that lim b → lim sup θ → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U Nǫ ( b ) u N − θ v θ,i v θ,j dv g θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . We get finally that (cid:12)(cid:12)(cid:12)(cid:12)Z M u N − v i v j dv g (cid:12)(cid:12)(cid:12)(cid:12) = lim b → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M \ U ( b ) u N − v i v j dv g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 for all i = j. We now write0 < λ k ( M, g ) ≤ sup ( α , ··· ,α k ) =(0 , ··· , F ( u, α v + · · · + α k v k )= sup ( α , ··· ,α k ) =(0 , ··· , R M ( α v + · · · + α k v k ) L g ( α v + · · · + α k v k ) dv g R M u N − ( α v + · · · + α k v k ) dv g = sup ( α , ··· ,α k ) =(0 , ··· , α R M v L g v dv g + · · · + α k R M v k L g v k dv g α R M u N − v dv g + · · · + α k R M u N − v k dv g = sup ( α , ··· ,α k ) =(0 , ··· , α λ ∞ , R M u N − v dv g + · · · + α k λ ∞ ,k R M u N − v k dv g α R M u N − v dv g + · · · + α k R M u N − v k dv g ≤ , since each λ ∞ ,i ≤
0. This gives the desired contradiction.
Remark . Note that, for i ≥ R M u N − v i dv g = 0 if M isnot connected. Step 2: Conclusion
Since µ ( M, g ) > , from Step 1, we get that µ ( N, g θ ) > . Assume µ ( N, g θ ) <µ ( S n ) (otherwise, we are done). Using [10] we construct a sequence ( v θ ) solution of L g θ v θ = µ ( N, g θ ) | v θ | N − v θ , such that Z N v Nθ dv g θ = 1 . By Theorem 6.3 Part 1), this holds that lim θ → µ ( N, g θ ) ≥ Λ n (and the conclusionof Theorem 6.1 is true) or there exists a function v solution on M of the equation: L g v = µ ∞ | v | N − v, with µ ∞ = lim θ µ ( N, g θ ) ≥ Z M | v | N dv g = 1 . This is what we assume until now.As explained in Paragraph 6.1.1, we can assume that µ ( g ) = 0. Case 1: µ ( g ) < M is connected (so is N ) and let us prove that v has a changing sign.We suppose by contradiction that v ≥ . The maximum principle gives that v > u be a positive solution of the Yamabe equation on M, i.e. L g u = µ ( g ) u N − . Since v >
0, we can write: L g v = µ ∞ |{z} ≥ | v | N − v = µ ∞ v N − . Multiplying the second equation by u and integrating, we get µ ( g ) |{z} < Z M u N − v dv g = Z M L g uv dv g = Z M uL g v dv g = µ ∞ |{z} ≥ Z M v N − u dv g . This gives a contradiction. Then v have a changing sign and this implies that µ ( M, g ) ≤ sup α,β F ( v, αv + + βv − ) = µ ∞ . If M is now disconnected, then the Yamabe minimizer u is positive on a connectedcomponent of M . If uv
0, the same proof holds. If uv ≡ µ ( M, g ) ≤ sup α,β F ( v, αu + βv ) = µ ∞ In any case, the conclusion of Theorem 6.1 is true.
Case 2: µ ( M, g ) > λ ( N, g θ ) >
0. In [10], it is established that the sign of the eigenvalues ofthe Yamabe operator is conformally invariant. Consequently, λ ( N, v N − θ g θ ) > µ = λ ( N, v N − θ g θ ) and let u θ be associated to µ . Since associated to thefirst eigenvalue of the Yamabe operator, u θ is positive on at least one connectedcomponent of N (and 0 on the other). In addition, u θ is a solution of the equation L g θ u θ = µ | v θ | N − u θ , such that Z N u Nθ dv g θ = 1 and Z N | v θ | N − u θ v θ dv g θ = 0 . Using Theorem 6.3 Step 2), there exists a function u solution on M of the followingequation L g u = µ ∞ , | v | N − u, where µ ∞ , := lim θ µ . Note that this limit exists after a possible extraction of asubsequence since 0 ≤ µ ≤ µ ( N, g θ ). Proceeding as in Step 1, we show that Z M | v | N − uv dv g = 0 . (19) SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 23
By maximum principle and since u θ > u > M . Then, u and v satisfy the equations L g u = µ ∞ , | v | N − u, and L g v = µ ∞ | v | N − v. These equations implies that µ ∞ , and µ ∞ are some eigenvalues of the generalizedmetric | v | N − g (see [10]). Since positive, u is associated to the first eigenvalue of L | v | N − g i.e. µ ∞ , = λ ( M, | v | N − g ). Hence, µ ∞ , ≤ µ ∞ .Finally, we obtain that µ ( M, g ) ≤ λ ( | v | N − g )Vol | v | N − g ( M ) n = µ ∞ since Vol | v | N − g ( M ) = Z M | v | N dv g = 1and since µ ∞ , ≤ µ ∞ are associated to two non proportional eigenfunctions in themetric | v | N − g (thanks to Relation (19)) where we recall that µ ∞ = lim θ → µ ( N, g θ ) . This proves Theorem 6.1.
Remark . The reason why we need µ ( g ) = 0 is the following. If µ ( g ) = 0, theproof of Case 1 clearly does not lead to a contradiction. So, we would like to applythe method used in Case 2 above. For this, we need that λ ( v N − θ g θ ) is bounded.When µ ( g ) >
0, this holds true since0 ≤ λ ( v N − θ g θ ) ≤ λ ( v N − θ g θ ) = µ ( N, g θ ) → µ ∞ . If µ ( g ) = 0, one cannot say nothing about the sign of λ ( v N − θ g θ ). In particular, ifit is negative, we were not able to prove that λ ( v N − θ g θ ) is bounded from aboveand the proof breaks down. 7. Some applications
In this section, we establish some topological applications of Theorem 1.6.7.1.
A preliminary result.
We have
Proposition 7.1.
Let V , M be two compact manifolds such that V carries a metric g with Scal g = 0 and σ ( M ) > , then σ ( V ∐ M ) ≥ min( µ ( g ) , σ ( M )) > . Proof: On V ∐ M , let G = λg + µh , where λ and µ are two positive constants andfor a small ǫ , h is a metric such that σ ( M ) ≤ µ ( M, h ) + ǫ . We haveSpec( L G ) = Spec( L λg ) ∪ Spec( L µh )= λ − Spec( L g ) ∪ µ − Spec( L h )= { λ − λ , λ − λ , · · · } ∪ { µ − λ ′ , µ − λ ′ , · · · } where λ i (resp. λ ′ i ) denotes the i -th eigenvalue of L g (resp. L h ). The assumptionwe made allows to claim that λ = 0, λ > λ ′ >
0. Hence, we deduce that λ ( L G ) = min { λ − λ , µ − λ ′ } .We know that Vol G ( V ∐ M ) = λ n Vol g ( V ) + µ n Vol h ( M ) . • For µ = 1 and λ → + ∞ , we have λ ( L G ) = λ − λ .λ ( L G )Vol G n ( V ∐ M ) = λ − λ (cid:16) C + λ Vol g n ( V ) (cid:17) → λ → + ∞ λ Vol g n ( V ) = µ ( g ) . • For λ = 1 and µ → + ∞ , in this case λ ( L G )) = µ − λ ′ . Hence λ ( L G )Vol G n g ( V ∐ M ) = µ − λ ′ (cid:16) C + µ Vol h n (cid:17) → µ → + ∞ λ ′ Vol h n = µ ( M, h ) ≥ σ ( M ) − ǫ. Finally we get that σ ( V ∐ M ) ≥ min( µ ( g ) , σ ( M )) . Remark . (1) It is known that if σ ( M ) > σ ( N ) >
0, then σ ( M ∐ N ) = min( σ ( M ) , σ ( N )) , where M ∐ N is the disjoint union of M and N . (see [4]).(2) Let V with σ ( V ) ≤
0, then for k ≥ σ ( V ∐ · · · ∐ V | {z } k times ∐ M ) ≤ . Indeed, let any metric g = g ∐ g ∐ · · · ∐ g k ∐ g n on V ∐ · · · ∐ V ∐ M . Let v i be functions associated to λ ( g i ) which is non-negative by assumption. Thefunctions ˜ v i = 0 ∐ · · · ∐ v i |{z} i th factor ∐ · · · ∐ L g ( ˜ v i ) = λ ( g i ) v i and thus are eigenfunctions of L g . This implies that λ k ( g ) ≤ k ≥ λ ( g ) ≤ | α ( M ) | ≤ M is obtained from a model manifold V ∐ N with a number of factors V (where V carries a scalar flat metric and σ ( N ) >
0) not larger than 1. We recallthat the α -genus is an homomorphism from the spin cobordism ring Ω Spin ∗ to thereal K -theory ring KO ∗ ( pt ) , α : Ω Spin ∗ → KO ∗ ( pt ) . It is important that α is a ring homomorphism, i.e. for any connected closed spinmanifolds M and N , α ( M ∐ N ) = α ( M ) + α ( N ) and α ( M × N ) = α ( M ) .α ( N ) . Noting that KO n ( pt ) vanishes if n = 3 , , , Z if n = 0 , Z / Z if n = 1 , α is exactlythe ˆ A -genus in dimensions 0-mod 8 and equal to ˆ A -genus in dimensions 4 mod 8.In [9], Proposition 3.5 says that in dimensions n = 0 , , , V such that α ( V ) = 1 and V carries a metric g such that Scal g = 0. • When α ( M ) = 0 then Thm A in [20] applies and σ ( M ) ≥ α n where α n dependingonly on n . SURGERY FORMULA FOR THE SECOND YAMABE INVARIANT 25
Theorem 7.3.
Let M be a spin manifold, if α ( M ) = 0, this is equivalent to theexistence of a manifold N cobordant to M such that the scalar curvature of N ,Scal g is positive.Remember that a cobordism is a manifold W with boundary whose boundary ispartitioned in two, W = M ∐ ( − N ). Theorem 7.4. If M is cobordant to N and if M is connected then M is obtainedfrom N by a finite number of surgeries of dimension 0 ≤ k ≤ n − . Proposition 7.5.
Let M be a spin, simply connected, connected manifold of di-mension n ≥
5, if n = 0 , , , | α ( M ) | ≤
1, then σ ( M ) ≥ α n , where α n is a positive constant depending only on n . Proof:
Proposition 3.5 in [9] gives us that for each n = 0 , , , n ≥
1, thereis a manifold V of dimension n such that V carries a metric g such that Scal g = 0and α ( V ) = 1. • First case: If α ( M ) = 0, then M is cobordant to a manifold N such that Scal g on N is positive. In this case we can obtained M from N by a finite number ofsurgeries of dimension k ≤ n −
3. Hence, by Corollary σ ( M ) ≥ c n with c n is apositive constant depending only on n . • Second case: If α ( M ) = 1, then α ( M ∐ ( − V )) = 0, so there exists a manifold N with Scal g > M ∐ ( − V ) is cobordant to N which is equivalent to saythat M is cobordant to V ∐ N . Consequently M can be obtained from V ∐ N by afinite number of surgeries of dimension k ≤ n −
3. Applying the main theorem ofthis paper, we get the desired result.
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