Some Conformally Flat Spin Manifolds, Dirac Operators and Automorphic Forms

Dirac Type Operators for Arithmetic Subgroups of Generalized Modular Groups

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.

调和旋量的Calderón问题

The Calderón problem for harmonic spinors

本文证明具有MIT边界条件的Dirac算子的边界共轭映射唯一确定紧致旋量流形的共形类，并推广到扭曲Dirac算子及实解析酉联络的局部规范等价性。

Magnetic Dirac-harmonic maps

We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physics. In two dimensions it is conformally invariant. We call critical points of this functional magnetic Dirac-harmonic maps. We study geometric and analytic properties of magnetic Dirac-harmonic maps including their regularity and the removal of isolated singularities.

The higher spin Dirac operators

There is a certain family of conformally invariant first order elliptic operators on Riemannian spin manifold which include Dirac operator as its first and simplest member. Their general definition is given and their basic properties are described. A special attention is paid to the Rarita-Schwinger operator the second simplest operator in the row. Its basic properties are described in more details. In the last part indices of discussed operators are computed.

Harmonic, Monogenic and Hypermonogenic Functions on Some Conformally Flat Manifolds in $R^n$ arising from Special Arithmetic Groups of the Vahlen Group

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, …

Bergman and Calderón projectors for Dirac operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g=\bar{g}/ρ^2$ where $ρ$ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$. We show that $({\rm I…

Monogenic Functions in Conformal Geometry

Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.

带边紧曲面上 Dirac-调和映射的唯一性

Uniqueness of Dirac-harmonic maps from a compact surface with boundary

本文研究带边紧曲面上Dirac-调和映射的唯一性，通过证明单位圆盘上小能量弱Dirac-调和映射的能量凸性，首次得到共形于单位圆盘的曲面上小能量任意边值Dirac-调和映射的唯一性结果。

0-椭圆算子的边值问题

Boundary value problems for 0-elliptic operators

本文针对带边界流形上具有无限维核的半-Fredholm 0-椭圆算子，提出 symbolic 0-calculus，并在椭圆边界条件下构造左右拟基本解。