Solvability of the Dirac Equation and geomeric applications Qingchun Ji and Ke Zhu May 21, KAIST
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.
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.
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.
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.
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.
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.
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.).
. 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.
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 = +,.
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 ± ).
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).
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.
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).
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
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.
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 ).
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)
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.
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.
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).
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).
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.
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,
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.
(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 ) ( 1 + 1 ) ] ϕ 2 + 2λ S s 2 e ϕ n ε
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 ) ( 1 + 1 ) 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 ( ) 1 + 1 ε ϕ 2 e ϕ <, (7) + 2λ S
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.
(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).
2. dim M 3, compact case (Theorem 2). Idea: let A = ( 1 n) 2 ( ) 1 + 1 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).
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.
Title M. Atiyah, I. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968) 546 604. C. Bär, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), no. 1, 39-46. J.P. Demailly, Estimations L 2 pour l opérateur d unfibré vectoriel holomorphe semipositif au dessus d une variété kählérienne complète, Ann. Sci. École Norm. Sup.15 (1982) 457-511.
S. Donaldson, Floer Homology Groups in Yang-Mills Theory, Cambridge Tracts in Mathematics 147, Cambridge University Press, Cambridge, 2002. D. Gayet, Smooth moduli spaces of associative submanifolds, preprint, arxiv:1011.1744. M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. No. 58 (1983), 83-196.
K. Habermann, L. Habermann, Introduction to Symplectic Dirac Operators, Lecture Notes in Mathematics, Vol. 1887, Springer-Verlag, Berlin, 2006. R. Harvey, B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157. O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162. N. Hitchin, Harmonic Spinors, Adv. Math. 14 (1974), 1 55.
H. Hofer, V. Lizan, J. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997), no. 1, 149 159. L. Hörmander, L 2 -estimates and existence theorems for the operator, Acta Math. 113, 1965, 89-152. L. Hörmander, A history of existence theorems for the Cauchy-Riemann complex in L 2 spaces, J. Geom. Analysis 13, No.2 (2003), 329-357. Q. Ji and K. Zhu,, arxiv: 1407.6936
Q. Ji, K. Zhu, Fredholm regular criteria of holomorohic curves by curvature integrals, in preparation B. Lawson, M. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. A. Lichnerowicz, Laplacien sur une variété riemanniennne et spineure, Atti Accad. Naz. dei Lincei, Rendiconti 33 (1962), 187-191.
R. Lockhart, R. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409 447. N.C. Leung, X. Wang, K. Zhu, Thin instantons in G 2 -manifolds and Seiberg-Witten invariants, J. Diff. Geom. 95 (2013) 419-481. R. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), no. 4, 705-747. Y.-T. Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively
twisted plurigenera for manifolds not necessarily of general type, Complex geometry (Göttingen, 2000), 223 277, Springer, Berlin, 2002. Y.-T. Siu, Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles, Finite or Infinite Dimensional Complex Analysis and, in: Adv. Complex Anal. Appl., vol. 2, Kluwer Acad. Publ., Dordrecht, 2004, pp. 45-83. Y.-T. Siu, Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles. Sci. China Math. 54 (2011), no. 8, 1767 1802.
C. Taubes, L 2 moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology, I. International Press, Cambridge, MA, 1993. C. Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv., 85 (2010), no. 2, 347 407.