Christian Baer
We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.
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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.
Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 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 a…
We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2- or 3-manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.
We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, \emph{dual-Dirac maps}, defines a \emph{dual-Dirac category} which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions …
A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction method.
We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the extrinsic Dirac operator and a generalized MIT Bag Dirac operator. This extends some results previously known for the Euclidean spaces to the case of general spin geometry.
Given two unitary involutions $σ_{1}$ and $σ_{2}$ satisfying $G σ_{i} = - σ_{i} G$ on $ker B$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with $σ_{1}$ and $σ_{2}$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This…
Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel tra…
We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension $n = 7$.
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature $S \not= 0$ that carry a non-trivial solution of the Einstein-Dirac equation.
We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.
In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with (resp.without) boundary
We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.
A Note on the Spectrum of Magnetic Dirac Operators
本文研究完备黎曼流形上磁Dirac算子及其带势版本的谱，给出了谱为极大或离散的充分条件，并证明了其可具有稠密特征值。
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.
Complex Dirac structures with constant real index on flag manifolds
本文通过李代数的根刻画极大旗流形上具有常实指标的不变复Dirac结构，并在$B$-变换下给出完全分类。
We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for the limiting case of the estimates.
In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.
In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on $Spin^c$ manifolds without boundary. The limiting case is then studied and an example is given.
We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study of add…
In this paper, we get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace-Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov.
In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.
We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe several Weitzenbock type formulae. Our method is based on work of P. Gauduchon applied to two almost complex manifolds naturally associated to a given metric contact manifold.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
In Kähler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.
We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them.
An integer valued topological index of a Dirac operator is introduced for a pair of a 4n+2 dimensional open Spin^c manifold and a section of the determinant line bundle satisfying some property. We show a relation between the index and an index of a Dirac operator of its characteristic submanifold, by a localization of K-class.
We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.
The notion of a Dirac submanifold of a Poisson manifold was studied by Xu (arXiv:math.SG/0110326). We give an interpretation of Xu's definition in terms of a general notion of tensor fields soldered to a normalized submanifold. Then, this interpretation is used to define Dirac submanifolds of a Jacobi manifold. Several properties and examples are discussed.
We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,Λ,E)$ for which 1 is an admissible function and Jacobi quotient manifolds of $M$. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.
We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.
Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the $η$-series in terms of values of Riemann-Hurwitz zeta functions, and the $η$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\Z_2^…
Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds
本文在紧致CR流形上建立Kohn-Dirac算子的新特征值估计，刻画等式情形为CR扭量旋量的存在，并分类此类流形，构造例子。
Conformal perturbations of dirac operators and general Kastler-Kalau-Walze type theorems for even dimensional manifolds with boundary
本文证明了带边与无边偶维紧流形上Dirac算子共形扰动的一般Kastler-Kalau-Walze型定理。
In this paper, we describe the group SpinT (n) and give some properties of this group. We construct SpinT spinor bundle S by means of the spinor representation of the group SpinT (n) and define covariant derivative operator and Dirac operator on S. Finally, Schrodinger-Lichnerowicz-type formula is derived by using these operators.
The Rescaled Dirac operator fDh and the noncommutative residue for 6-dimensional manifolds
本文计算了6维紧流形上重标度Dirac算子fDh的非交换留数，证明了有边界情形的Kastler-Kalau-Walze型定理，并给出若干重要特例。
Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to th…
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $μ:M\to \mathfrak{g}^*$, the manifolds $M/G$, $μ^{-1}(0)$ and $μ^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $μ^{-1}(0)$ and $μ^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.
We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point $m$ of a Dirac manifold $M$, there is a well-defined transverse Poisson structure to the pre-symplectic leaf $P$ through $m$. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and we prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted D…
In this paper we present a summarizing description of the connection between Dirac operators on conformally flat manifolds and automorphic forms based on a series of joint work with John Ryan over the last fifteen years. We also outline applications to boundary value problems.
A description of time-dependent Mechanics in terms of Lagrangian submanifolds of Dirac manifolds (in particular, presymplectic and Poisson manifolds) is presented. Two new Tulczyjew triples are discussed. The first one is adapted to the restricted Hamiltonian formalism and the second one is adapted to the extended Hamiltonian formalism.
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.
The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary
本文将定向偶数维 Riemannian 流形上 Hodge-Dirac 算子的谱 Einstein 双线性泛函结果推广至带边界的 4 维情形，并证明了带边流形上相关的 Dabrowski-Sitarz-Zalecki 型定理。
We prove $R$-bisectoriality and boundedness of the $H^\infty$-functional calculus in $L^p$ for all $1<p<\infty$ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on $k$-forms.
This paper studies Dirac operators on end-periodic spin manifolds of dimension at least 4. We provide a sufficient condition for such an operator to be Fredholm for a generic end-periodic metric; this condition is shown to be necessary in dimension 4. We make use of end-periodic Dirac operators to give an analytical interpretation of an invariant of non-orientable smooth 4-manifolds due to Cappell and Shaneson. From this interpretation we show that some exotic non-orientable 4-manifolds do not admit a metric of positive scalar curvature.
This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversality criteria for holomorphic curves in Kähler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a $Z_2$-grading. On Riemannian man…
We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Dirac equation and in dimension n=3 the two equations essentially coincide. It turns out that any Sasakian manifold with Ricci tensor related in some special way to the metric tensor as well as to…
In a previous paper we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we show that the only manifolds in the limit case, i.e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation and show that a nontrivial solution defines a parallel spinor on the associated hyperkaehler manifold.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boundary.
In this paper we investigate the properties of a semi-linear problem on a spin manifold involving the Dirac operator, through the construction of Rabinowitz-Floer homology groups. We give several existence results for sub-critical and critical non-linearities as application of the computation of the different homologies.
A note on coupled Dirac-harmonic maps
本文提出一种从 $S^1\ imes M$ 型流形构造耦合狄拉克-调和映射的方法，其中 $M$ 为带满足范数条件的调和旋量对的闭自旋流形。
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.
We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.
We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.
In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds.
Spectrum of the Dirac operator on Compact Riemannian Manifolds
本文研究紧黎曼流形上Dirac算子的特征值问题，通过外在估计推导出任意连续n个特征值之和的不等式，并应用于欧氏空间、单位球面和射影空间中的紧子流形，同时给出曲率条件下的普适界及Atiyah-Singer Laplacian特征值和的上界估计。
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