aa r X i v : . [ m a t h . DG ] N ov Almost harmonic spinors
Nicolas Ginoux ∗ , Jean-Fran¸cois Grosjean † November 4, 2018
Abstract.
We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequenceof volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As anapplication, we compare the Dirac spectrum with the conformal volume.
Spineurs presque harmoniques
R´esum´e.
Nous montrons que, sur toute vari´et´e spinorielle compacte sans bord non diff´eomorphe `a la sph`erede dimension deux, il existe une suite de m´etriques riemanniennes de volume un pour laquelle la plus petitevaleur propre non nulle de l’op´erateur de Dirac tend vers z´ero. Comme application, nous comparons le spectrede l’op´erateur de Dirac avec le volume conforme.
Let M n be an n ( ≥ D g the spin Dirac operatorassociated to a Riemannian metric g . We denote by λ ( D g ) and λ +1 ( D g ) the smallest and the smallestpositive eigenvalue of D g respectively. It is well-known that the product λ +1 ( D g )Vol( M n , g ) n is scaling-invariant and bounded from below by a positive constant in any conformal class [1, Thm. 2.3]. One canask whether the infimum of λ +1 ( D g )Vol( M, g ) n on the space of all Riemannian metrics remains positive.This holds true if M is the 2-sphere S since it has only one conformal class; alternatively, it follows fromC. B¨ar’s estimate [6] valid for any Riemannian metric g on S : λ ( D g )Area( S , g ) ≥ π. (1)In this respect S is the only exception: Theorem 1.1
For any n ( ≥ -dimensional closed spin manifold M n not diffeomorphic to S there existsa sequence ( g p ) p ∈ N of Riemannian metrics on M n such that λ +1 ( D g p )Vol( M n , g p ) n −→ p →∞ . Therefore one can get the Dirac spectrum as close to 0 as one wants with fixed volume. Note however thatTheorem 1.1 does not prove the existence of non-zero harmonic spinors, i.e., that 0 is a Dirac eigenvalue.Theorem 1.1 is proved in Section 2. In Section 3 we apply it to compare the Dirac spectrum with theconformal volume.
Acknowledgment.
This work started during a stay of the first-named author at the Institut ´Elie Cartande Nancy, which he would like to thank for its hospitality and support. It is also a pleasure to thankBernd Ammann and Emmanuel Humbert for fruitful discussions and their critical reading of the paper. ∗ NWF I - Mathematik, Universit¨at Regensburg, D-93040 Regensburg,
E-mail:[email protected] † Institut ´Elie Cartan (Math´ematiques), Universit´e Henri Poincar´e Nancy I, B.P.239 F-54506 Vandoeuvre-L`es-NancyCedex,
E-mail: [email protected] Proof
The proof of Theorem 1.1 relies on a standard technique first used in the spinorial context by C. B¨ar [7]to show the existence of metrics with harmonic spinors. Namely we prove the result by gluing a modelmanifold admitting such a sequence and by studying the convergence of the spectrum on the connectedsum. Thus the proof is two-step.
Lemma 2.1 i) Theorem 1.1 holds true on the standard sphere S n for any n ≥ .ii) Theorem 1.1 holds true on the -torus T endowed with any of its spin structures.Proof : Both statements follow from elementary arguments. i ) For any n ≥ S n a metric e g with Ker( D e g ) = { } : for n ≡ n = 2 m ≥ n ≥ e g provides a smooth one-parameter-family of Riemannian metrics ( g t ) ≤ t ≤ with g = canand g = e g . Since the volume remains bounded and the Dirac spectrum depends continuously on themetric in the C -topology, we obtain the result. ii ) For a real parameter a > T := R / Γ, where Γ := Z · (cid:18) (cid:19) ⊕ Z · (cid:18) a (cid:19) , withinduced flat metric g a . It carries 4 spin structures, 3 of which can be deduced from each other by anorientation-preserving diffeomorphism of T (see e.g. [4]). Thus it suffices to prove the statement for twospin structures which cannot be obtained from each other by a diffeomorphism, for example for the spinstructure inducing a trivial covering on both factors and for the spin structure inducing a trivial coveringon the first factor and a non-trivial one on the second one. For the former spin structure the smallestpositive eigenvalue of D g a is π a and for the latter one it is π a . Since Area( T , g a ) = a we conclude thatin both situations λ +1 ( D g a )Area( T , g a ) −→ a →∞ (cid:3) In the second step we consider the dimensions n = 2 and n ≥ : Theorem 2.2 (C. B¨ar [8])
Let ( N n , g ) and ( N n , g ) be closed Riemannian spin manifolds of dimen-sion n ≥ . Let L > and η ≥ with ± ( L + η ) / ∈ (Spec( D g ) ∪ Spec( D g )) .Then for any ε > , there exists a Riemannian metric ˜ g on the connected sum e N n := N n ♯ N n suchthat the Dirac eigenvalues of N n · ∪ N n and ( e N n , ˜ g ) in ] − L − η, L + η [ differ at most by ε and that Vol( e N n , e g ) ≤ Vol( N n , g ) + Vol( N n , g ) + ε . Note that, as an easy consequence, Theorem 2.2 remains valid when replacing the eigenvalues of theDirac operator by those of its square. Fix now any Riemannian metric g on M n (with n ≥ p is anypositive integer, pick from Lemma 2.1 a Riemannian metric g p of volume one on S n with λ +1 ( D g p ) ≤ p .Setting L := λ +1 ( D g p ), ε := λ +1 ( D gp )2 and choosing η > L + η / ∈ (Spec( D g p ) ∪ Spec( D g )), Theorem2.2 implies the existence of a Riemannian metric e g p on e N n := M n ♯ S n such that at least one eigenvalueof D e g p lies in the interval [ λ +1 ( D gp )2 , λ +1 ( D gp )2 ] and that Vol( e N n , e g p ) ≤ Vol( M n , g ) + 1 + p . Since e N n isspin diffeomorphic to M n we conclude the proof of Theorem 1.1 for n ≥ n = 2 we perform an induction on the genus of the surface. On T Theorem 1.1 alreadyholds true by Lemma 2.1. Assume it to hold true for any closed oriented surface M ( γ ) of genus γ > M ( γ + 1) of genus γ + 1. The oriented surface can be obtained asthe connected sum of some M ( γ ) and T . Moreover, the spin structure induced on a circle bounding acompact oriented surface is always a non-trivial covering [11, p.91], in particular it itself bounds a disk.Therefore, every spin structure on M ( γ + 1) is induced by some spin structure on M ( γ ) and some on T . It would remain to prove the analog of Theorem 2.2 for surgeries of codimension 2, at least forconnected sums of surfaces. We conjecture this holds true, using arguments and techniques from [2].Actually much less is needed here: Thanks to Christian B¨ar for indicating to us the right reference. The argument given in the published version was wrong. Thanks also to Christian B¨ar. emma 2.3 Let ( M , g ) and ( M , g ) be any oriented closed Riemannian surfaces. Fix L > and η ≥ with ± ( L + η ) / ∈ (Spec( D g ) ∪ Spec( D g )) .Then for any ε > there exists a Riemannian metric ˜ g on M ♯ M such that, for any eigenvalue λ of D g or D g in ] − L − η, L + η [ , there exists an eigenvalue ˜ λ of D ˜ g such that | ˜ λ − λ | ≤ ε and Area( M ♯ M , ˜ g ) ≤ Area( M , g ) + Area( M , g ) + ε .Proof : The proof relies on a classical cut-off procedure for eigenvectors of D g and D g . We want toshow that dim(Ker( D g − λ Id)) + dim(Ker( D g − λ Id)) ≤ dim (cid:16) ⊕ µ ∈ [ λ − ε,λ + ε ] Ker( D e g − µ Id) (cid:17) . Fix p i ∈ M i , i = 1 ,
2, and some sufficiently small δ >
0. Consider the connected sum f M := M ♯ M obtained by gluing M \ B p ( δ ) and M \ B p ( δ ) along their boundary, where B p ( r ) denotes the open metric disc of center p and radius r . From [7, p.932] or [2, Sec. 3.1-3.2] there exists a smooth Riemannian metric ˜ g δ on f M whichcoincides with g i on M i \ B p i ( √ δ ) and such that Area( M ♯ M , ˜ g δ ) ≤ Area( M , g )+ Area( M , g )+ c ·√ δ ,where c > g and g . In particular we may choose δ > c · √ δ < ε . For i = 1 , χ i ∈ C ( f M , [0 , χ i | Mi \ Bpi ( √ δ ) := 1, χ i | Bpi ( δ ) := 0 and χ i ( x ) := 2 − ln( d ( x,p i ))ln( δ ) otherwise, where d ( x, p ) denotes the distance between x and p . Note that χ and χ are well-defined and continuous on the whole f M and that they can be smoothed out at both ∂B p i ( δ )and ∂B p i ( √ δ ) such that the L -norm of their gradient changes arbitrarily little. We keep denoting thecorresponding smooth functions by χ and χ . Consider the mapΦ : Ker( D g − λ Id) ⊕ Ker( D g − λ Id) −→ Γ(Σ f M )( ϕ , ϕ ) χ ϕ + χ ϕ , which is well-defined because of χ i | Bpi ( δ ) = 0 and injective by the unique continuation property (each Diraceigenvector vanishing on an open subset of a connected Riemannian spin manifold must vanish identically).Now from the min-max principle it suffices to show that k ( D ˜ gδ − λ ) ϕ k L2 k ϕ k L2 ≤ ε for all ϕ ∈ Im(Φ) \ { } .Since the subspaces Φ(Ker( D g − λ Id)) and Φ(Ker( D g − λ Id)) are L -orthogonal to each other (forsupp( χ ) ∩ supp( χ ) has zero measure), we can assume that ϕ ∈ Φ(Ker( D g − λ Id)) with k ϕ k L = 1.Using the formula D g ( f ϕ ) = df · ϕ + f D g ϕ , we compute: Z f M | ( D ˜ g δ − λ ) ϕ | v ˜ g δ = Z M | ( D ˜ g − λ ) χ ϕ | v ˜ g δ = Z M | dχ | | ϕ | v ˜ g δ ≤ C sup M ( | ϕ | ) Z √ δδ r ln( δ ) rdr ≤ − C ′ ln( δ ) , where C > B p ( √ δ ) \ B p ( δ ) and C ′ = 2 C · sup ϕ ∈ Ker( D g − λ Id) (sup M ( | ϕ | ) (note that C ′ < ∞ since Ker( D g − λ Id) is finite-dimensional). Wededuce that k ( D ˜ g δ − λ ) ϕ k −→ δ → (cid:3) The proof of Theorem 1.1 for n = 2 follows the lines of that for n ≥
3: given any Riemannian metric g on M ( γ ) and a positive p ∈ N , pick from Lemma 2.1 a Riemannian metric g p of unit area on T with λ +1 ( D g p ) ≤ p , whatever the spin structure of T is. Lemma 2.3 ensures the existence of a Riemannianmetric e g p on M ( γ + 1) = M ( γ ) ♯ T such that at least one eigenvalue of D e g p lies in the interval[ λ +1 ( D gp )2 , λ +1 ( D gp )2 ] and that Area( M ( γ + 1) , e g p ) ≤ Area( M ( γ ) , g ) + 1 + p . This proves the resultfor γ + 1 and concludes the proof of Theorem 1.1. This note was motivated by the study of the relationship between the Dirac spectrum and the so-calledconformal volume, which is the conformal invariant defined for any closed Riemannian manifold ( M n , g )3y V c ( M n , [ g ]) := inf N ∈ N (cid:16) inf ϕ ∈ Imm c ( M n , S N ) (cid:0) sup γ ∈ Conf( S N ) (Vol( M n , ( γ ◦ ϕ ) ∗ can)) (cid:1)(cid:17) , where Imm c ( M n , S N ) denotes the set of conformal immersions ( M n , g ) −→ ( S N , can) and Conf( S N )the group of conformal diffeomorphisms of ( S N , can). First introduced by P. Li and S.-T. Yau [12],it has been shown to be directly related to the Laplace spectrum since it provides an upper bound ofthe corresponding spectral invariant [12, 10]: (0 < ) λ (∆)Vol( M n , g ) n ≤ n V c ( M n , [ g ]) n . For the Diracoperator such a result cannot be expected because of sup g ∈ [ g ] (cid:0) λ +1 ( D g )Vol( M n , g ) n (cid:1) = ∞ , see [5, Thm. 1.1].However, one could reasonably conjecture that the conformal volume bounds λ ( D g )Vol( M, g ) n frombelow, provided the possible eigenvalue 0 is left aside. For M = S this is the case because of (1) and4 π = Area c ( S ) (see [12]). It is hopeless for any other manifold: Corollary 3.1
For any n ( ≥ -dimensional closed Riemannian spin manifold ( M n , g ) not diffeomorphicto S there exists no positive constant c ( M ) (depending only on M ) such that λ +1 ( D g )Vol( M, g ) n ≥ c ( M )V c ( M n , [ g ]) n . Proof : It is elementary to show that [12, Fact 2]V c ( M n , [ g ]) ≥ Vol( S n , can) , (2)whose r.h.s. does not depend on the metric g . We conclude with Theorem 1.1. (cid:3) Still there exists a subtle relationship between the Dirac spectrum and the conformal volume. Indeedby [1, Thm. 3.1 & 3.2] and [3, Thm. 1.1], inf g ∈ [ g ] (cid:16) λ +1 ( D g )Vol( M n , g ) n (cid:17) ≤ n Vol( S n , can) n , hencecombining with (2) one obtainsinf g ∈ [ g ] (cid:16) λ +1 ( D g )Vol( M n , g ) n (cid:17) ≤ n c ( M n , [ g ]) n . References [1] B. Ammann,
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