Solvability of the Dirac Equation and geomeric applications

Transcription

1 Solvability of the Dirac Equation and geomeric applications Qingchun Ji and Ke Zhu May 21, KAIST

2 We study the Dirac equation by Hörmander s L 2 -method. On surfaces: Control solvability of the Dirac equation by a curvature integral (compact case). Automatic solvability in weighted L 2 space (noncompact case). On compact Riemannian manifolds: A new proof of Bär s theorem comparing the first eigenvalue of the Dirac operator with that of a Yamabe type operator.

3 On noncompact Riemannian manifolds with cylindrical ends: Solvability in L 2 space with suitable exponential weights, allowing mild negativity of the curvature. We also improve the above results when the Dirac bundle has a Z 2 -grading (which always exists when the underlying manifold is even dimensional), and discuss applications to holomorphic curves and instantons.

4 Let (M, g) be a Riemannian manifold, dim M = n. Clifford bundle Cl (M) = Σ r=0 r T C M/ v 2 + v v. Dirac bundle S M is a bundle of left Cl (M)-modules with compatible metric, and connection satisfying (Unitary) e s, e s = s, s, for any unit vector e T x M and any s, s S x, (Leibniz) (X s) = ( X) s + X ( s), for smooth vector field X on M and smooth section s of S Dirac operator D : Γ (S) Γ (S), Ds = Σ n i=1 e i ei s.

5 Examples: Title M = R 2 = e 1, e 2, Cl 2 = S = C C = 1, e 1 e 2 e 1, e 2, and [ ] 0 D = z. z 0 [ ] 0 M = H, Cl 4 = H 2X2 q, S = H H and D = q 0 where q = x 0 + i x 1 + j x 2 + x 3, {1, i, j, k} is the standard basis of H.

6 M is a spin manifold, and D is the Atiyah-Singer Dirac operator. There is no obstruction to the existence of Dirac bundles. (M, g) is a Riemannian manifold, S = even (M) odd (M) is a Dirac budle and the Dirac operator is given by the Hodge-de Rham operator D = d + δ.

7 Dirac equation Du = f. Solutions of Du = 0 are called harmonic spinors. On compact manifolds, D is self-adjoint, so solvability vanishing of harmonic spinors. Theorem (Lichnerowicz 62). If (M, g) is a compact spin manifold with scalar curvature S g > 0, then Du = 0 only has trivial solution. But S g > 0 condition is not always necessary.

8 Theorem (Hitchin 74). If two metrics g and g on M are conformally equivalent, then D = Ψ D Ψ 1, where Ψ : S S is a spinor bundle isomorphism. Corollary. ker D = ker D. So D and D have the same solvability, but S g > 0 S g > 0! It would be ideal to relax the pointwise positive scalar curvature condition by more flexible, global conditions (e.g. integral, eigenvalue etc.).

9 . Title D 2 = + R, where R is a section of Hom(S, S), such that for any smooth section s of S, R (s) = 1 2 n i,j=1 e i e j R ei,e j (s), where R V,W is the curvature operator on S. On spin manifolds, R = 1 4 S g. = Lichnerowicz s theorem. Notation. For any x M, let λ S (x) = minimal eigenvalue of R (x). λ S (x) is a Lipshitz function on M.

10 Every Dirac bundle S over an orientable, even-dimensional Riemannian manifolds has a natural Z 2 grading, i.e. a parallel decomposition S = S + S such that Cl i S j S ij, i, j = +,.

11 A Z 2 grading structure induces a splitting of the Dirac operator [ ] 0 D D = D + 0 where D ± : Γ(M, S ± ) Γ(M, S ). The curvature R Hom (S, S) splits as [ ] R + 0 R = 0 R where R ± Hom (S ±, S ± ).

12 Theorem (Hijazi 86, Bär 92) Let S be a Dirac bundle on a compact Riemannian manifold (M, g). 1 If dim M = 2, genus= 0, and D is the classic Dirac operator, then ( λ min D 2) 4π Vol (M).

13 2 If dim M 3, then λ min ( D 2) n n 1 λ min (L), where L = n 1 n 2 + λ S is a Yamabe type operator on M.

14 In noncompact case, we have the following fundamental result. Theorem (Gromov-Lawson 83). Let (M, g) be a complete Riemannian manifold, S be a Dirac bundle over (M, g). Assume that λ S α > 0 for some positive constant α holds on M, then Du = f is always solvable in L 2 (M, S).

15 Theorem 1 (Ji-Zhu 2014). Let (M, g) be a 2-dimensional Riemannian manifold. Suppose there exists a C 2 function ϕ : M R such that ϕ + 2λ S 0 on M. (1) 1 For any section f of S, if f 2 M ϕ+2λ S e ϕ <, then there exists a section u of S such that Du = f and u 2 e ϕ f 2 e ϕ. (2) ϕ + 2λ S M 2 If M is noncompact, then for any f L 2 loc (M, S), there exists a section u L 2 loc (M, S) such that Du = f. M

16 Corollary 1. We have λ min (D 2) 2 λ S (x). Vol (M) M Particularly, M λ S (x) > 0 = Du = f is always solvable in L 2 (M, S). The Z 2 -graded version is also ture: λ S > 0 = D + u = f is always solvable. Corollary 2. On any noncompact surface, the Poisson equation d u = f is always solvable in L 2 loc.

17 Theorem 2 (Ji-Zhu 2014). Let (M, g) be a compact Riemannian manifold, dim M 3. Then λ min ( D 2) where L ± = n 1 n 2 + λ S ±. n n 1 min { λ min ( L + ), λ min ( L )}, Corollary 2 and Theorem 2 refine Bär s result(note for classic Dirac operator on surfaces, λ S = 1 2Gauss curvature, so we recover Bär s result by using Gauss-Bonnet formula ).

18 On noncompact Riemannian manifolds, we have Theorem 3 (Ji-Zhu 2014). Let (M, g) be a Riemannian manifold with cylindrical ends, dim M 3. Suppose for some compact subset K, M\K is contained in the cylindrical ends, and there exist a constant α > 0 such that λ S α on M\K. (3) Then there exists a constant β > 0 with the following significance: when λ S β on K, (4)

19 we have for any section f L 2 δ (M, S), there exists a section u W 1,2 δ (M, S) such that Du = f and u W 1,2 δ (M,S) C f L 2 δ (M,S). (5) Note we allow mild negativity of the curvature.

20 1. Holomorphic curves. Let Σ be an immersed holomorphic curve in a Kähler manifold X. Let N Σ/X be the normal bundle, and D = 2 ( + ( ) ) be the Dolbeault Dirac operator on S = S + S = N Σ/X 0,1 ( N Σ/X ). From Corollary 1 we have Corollary 3. If λ S > 0, then Σ is Fredholm regular.

21 1 When dim C X = 2, λ S > 0 is the well-known Chern number condition (Hofer-Lizan-Sikorav 97) c 1 ( NΣ/X ) 2g 1 to ensure transversality of Σ. This is because Σ S = N Σ/X 0,1 (Σ), λ S = c 1 ( NΣ/X ) + χ (Σ).

22 2 When Σ = S 2, Title N Σ/X = m i=1 L i, where L i are holomorphic line bundles, then Σ is transversal if and only if c 1 (L i ) 1 for any i. Remark. Corollary 3 generalizes the "automatic transversality" of holomorphic curves Σ in Kähler manifolds X (symplectic manifold case is a work in progress).

23 2. Instantons Title Theorem (McLean 98) Let A be an instanton in a G 2 manifold X. Then its normal bundle N A/X is a Dirac bundle, and the deformation of A is governed by the Dirac operator D. Corollary 4. λ min (L) > 0 = A is rigid, where L = n 1 n 2 + λ S. In particular, if the normal bundle N A/X has positive curvature then A is rigid.

24 Let X be a symplectic manifold with a compatible almost complex structuren J. For a J-holomorphic curve u : Σ X, its linearized operator L : Γ (u (TX)) Γ ( u (TX) 0,1 (Σ) ) is a real Cauchy-Riemann operator of the form L = L 0 + α,

25 where L 0 is a complex linear Cauchy-Riemann operator on the complex bundle u (TX) Σ, and α is a real linear homomorphism coming from the Nijinhuis tensor of J. We are generalizing Corollary 3 to almost complex J case, to give transversality criteria of Σ via a curvature integral condition similar to M λ S (x) > 0.

26 (I). Weighted L 2 -estimate and solvability criteria Lemma Let L 2 ϕ (M, S) be the L 2 space with measure e ϕ dvol g, and D ϕ be its formal adjoint of D : L 2 ϕ (M, S) L 2 ϕ (M, S), then D ϕs = e ϕ D ( e ϕ s ) = ϕ s + Ds. Proposition 1 (Ji-Zhu 2014). For any smooth section s of S with compact support and any C 2 function ϕ : M R, for ε > 0 we have M D ϕs 2 e ϕ C M [ ( ϕ 1 2 ) ( ) ] ϕ 2 + 2λ S s 2 e ϕ n ε

27 Proposition 2 (Solvability criteria, Ji-Zhu 2014) Let ϕ : M R be a C 2 function and ε > 0 be a constant such that ( ϕ 1 2 ) ( ) ϕ 2 + 2λ S 0 on M. (6) n ε For any section f L 2 ϕ(m, S), if I ϕ (f ) := M f 2 ϕ ( 1 n) 2 ( ) ε ϕ 2 e ϕ <, (7) + 2λ S

28 then there exists a section u L 2 ϕ(m, S) such that where C = Du = f and u 2 ϕ 1 C I ϕ (f ), n 2(n 1)+(n 2)ε.

29 (II). Constructing the weights 1. dim M = 2(Theorem 1) compact case. Fact: ϕ such that ϕ + 2λ S > 0 on M M λ S > 0. noncompact case Exhaustion sequence for M. Use it to construct the weight ϕ (similar to Hörmander s construction).

30 2. dim M 3, compact case (Theorem 2). Idea: let A = ( 1 n) 2 ( ) ε > 0, and v = e Aϕ. Then ϕ A ϕ 2 + 2λ S = v 1 A + 2λ }{{ S v. } Yamabe So the existence of ϕ can be reduced to positive eigenfunction of L = A + 2λ S (Nodal domain theorem first eigenfunction> 0).

31 3. Cylindrical case (Theorem 3). Glue two weights as ϕ = γ (1 ρ) log η + ρh, where η > 0 is the first eigenfunction of the Dirichlet problem on K, and h is a linear function in the cylindrical direction.

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