Holomorphic Spinors and the Dirac Equation

In the first part of this paper, the closed spin Kähler manifolds of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator, are characterized by holomorphic spinors. In the second part, the space of holomorphic spinors on a Kähler–Einstein manifold is described by eigenspinors of the square of the Dirac operator and vanishing theorems for holomorphic spinors are proved.     

The first Dirac eigenvalues on manifolds with positive scalar curvature

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

     

On Higher Eta-Invariants and Metrics of Positive Scalar Curvature

Let N be a closed connected spin manifold admitting one metric ofpositive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving ₁(N) and dim N, for N to admit an infinite number of metrics of positive scalar curvature that are nonbordant. Mathematics Subject Classifications (2000) 55N22, 19L41.     

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.     

The Hijazi inequality on conformally parabolic manifolds

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.     

Dirac Operators on Hermitian Spin Surfaces

We prove the conformal invariance of the dimension of thekernel of any of the self-adjoint Dirac operators associated to thecanonical Hermitian connections on Hermitian spin surface. In the caseof a surface of nonnegative conformal scalar curvature we estimate thefirst eigenvalue of the self-adjoint Dirac operator associated to theChern connection and list the surfaces on which its kernel isnontrivial.     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Non-negative scalar curvature

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C ^* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.     

Green functions for the Dirac operator under local boundary conditions and applications

In this article, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in some cases, solves the Yamabe problem on manifolds with boundary. Finally, using the MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.     

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.     

Quasi-homogeneous Hilbert Modules

This paper is to study the quasihomogeneous Hilbert modules and generalize a result of Arveson [3] which relates the curvature invariant to the index of the Dirac operator. This work was partially supported by NKBRPC (#2006CB805905) and SRFDP.     

Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2,3 and r = 2^[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.      

Dirac eigenvalue estimates on surfaces

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. For higher genus g the estimate is given by The corresponding estimate also holds for the -spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in . Received: 15 May 2001; in final form: 11 September 2001 / Published online: 1 February 2002     

Remark on the limit case of positive mass theorem for manifolds with inner boundary

In (Comm. Math. Phys. 188 (1997) 121–133) Herzlich proved a new positive mass theorem for Riemannian 3-manifolds (N,g) whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass m(g) must be strictly positive). We prove that the mass is bounded from below by a positive constant c(g), m(g)c(g), and the equality m(g)=c(g) holds only if, outside a compact set, (N,g) is conformally flat and the scalar curvature vanishes. The constant c(g) is uniquely determined by the metric g via a Dirac-harmonic spinor.     

Higher eta-invariants

We define the higher eta-invariant of a Dirac-type operator on a nonsimply-connected closed manifold. We discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary. We give applications to questions of positive scalar curvature for manifolds with boundary, and to a Novikov conjecture for manifolds with boundary.Partially supported by the Humboldt Foundation and NSF grant DMS-9101920.