A new extrinsic estimate for the spectrum of the Dirac operator

Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

Upper bounds for the first eigenvalue of the Dirac operator on surfaces

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow {\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics.

The Dirac operator on untrapped surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.

Hypersurfaces with small extrinsic radius or large $λ_1$ in Euclidean spaces

We prove that hypersurfaces of $\R^{n+1}$ which are almost extremal for the Reilly inequality on $λ_1$ and have $L^p$-bounded mean curvature ($p>n$) are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that when a supplementary $L^q$ bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when $q>n$, but not necessarily diffeomorphic to a sphere when $q…

On eigenvalue estimates for the Dirac operator

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

局部可约黎曼流形上Dirac-Witten算子的特征值估计

Eigenvalue estimate for the Dirac-Witten operator on locally reducible Riemannian manifolds

本文研究局部可约类空子流形上Dirac-Witten算子的特征值问题，利用内蕴与外蕴表达式给出最优下界，并分析了极限情形。

Lower bounds for the eigenvalue estimates of the submanifold Dirac operator

We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction.

欧几里得空间中子流形的谱伯恩斯坦定理

Spectral Bernstein theorems for submanifolds in Euclidean spaces

本文研究欧几里得空间中子流形的本质谱，探讨了有限总均曲率、外向距离梯度收敛性、外向体积增长或曲率pinching等几何条件，并证明了当子流形的二阶基本形式满足L^p有界时，其本质谱为[0, +∞)。

紧黎曼流形上Dirac算子的谱

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。

Dirac Operator on Embedded Hypersurfaces

New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.

A generalized MIT Bag operator on spin manifolds in the non-relativistic limit

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.

The spectrum of the twisted Dirac operator on Kahler submanifolds of the comlex projective space

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding \CP^d \rightarrow \CP^n in order to test the sharpness of the upper bounds

A relation between the Ricci tensor and the spectrum of the Dirac operator

Using Weitzenböck techniques on any compact Riemannian spin manifold we derive a general inequality depending on a real parameter and joining the spectrum of the Dirac operator with terms depending on the Ricci tensor and its first covariant derivatives. The discussion of this inequality yields vanishing theorems for the kernel of the Dirac operator $D$ and new lower bounds for the spectrum of $D^2$ if the Ricci tensor satisfies certain conditions.

A brief note on the spectrum of the basic Dirac operator

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on $M$. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric.

The symmetry of Spin^c Dirac spectrums on Riemannian product manifolds

It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq 1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or $(e^{…

Surgery and the Spectrum of the Dirac Operator

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.

The spectrum of basic Dirac operators

This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-like metric. We explain the result by Habib-R. that the spectrum of such an operator is independent of the choice of bundle-like metric, provided that the transverse geometric structure is fixed. We discuss consequences, which include defining a new version of the exterior derivat…

The Dirac spectrum on manifolds with gradient conformal vector fields

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 spectrum of twisted Dirac operators on compact flat manifolds

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^…

On the Dirac spectrum of homogeneous 3-spheres

We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$ which are isospectral for the associated classical Dirac operator $D$ must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of $D^2$. We show analogous results for left-invariant metrics on $\operatorname{SO}(3)=S^3/\{\pm1\}$ for each of its two spin structures.

Properties of the Dirac spectrum on three dimensional lens spaces

We present a spectral rigidity result for the Dirac operator on lens spaces. More specifically, we show that each homogeneous lens space and each three dimensional lens space $L(q;p)$ with $q$ prime is completely characterized by its Dirac spectrum in the class of all lens spaces.

关于磁Dirac算子谱的一个注记

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。

$L^p$-spectrum of the Dirac operator on products with hyperbolic spaces

We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on $p$. As a first example where $p$-independence fails we compute explicitly the $L^p$-spectrum for the hyperbolic space and its product with compact spaces.

Dependence on the spin structure of the Dirac spectrum

The theme is the influence of the spin structure on the Dirac spectrum of a spin manifold. We survey examples and results related to this question.

The point spectrum of the Dirac operator on noncompact symmetric spaces

In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory we show that $D$ has point spectrum iff the $\hat A$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible then $M = U(p,q)/U(p) \times U(q)$ with $p+q$ odd, and $Spec_p(D) = \{0\}$.

关于 Dirac 算子的 $L^p$ 谱

On the $L^p$ Spectrum of the Dirac operator

本文研究开流形上 Dirac 算子及其平方的 $L^p$ 谱，通过 $p$ 无关条件、$L^1$ 谱和广义 Weyl 准则，扩展了 $L^2$ 谱极大的流形类别。

关于磁狄拉克算子的谱

On the spectrum of the magnetic Dirac operator

本文研究了磁狄拉克算子的谱性质，通过特征值估计和具体计算，探索了其在闭合黎曼流形上的数学特性。