Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A conformal lower bound of weighted Dirac eigenvalues on manifolds with boundary

Mingwei Zhang

Abstract

For the weighted Dirac eigenproblem on a compact spin manifold with the chiral boundary condition \begin{equation*} \left\{ \begin{array}{ll} D\varphi = λf\varphi & \text{in } M, \\ \mathbf{B}\varphi = 0 & \text{on } \partial M, \end{array} \right. \end{equation*} we first give a lower bound of the eigenvalue using the relative Yamabe constant \begin{equation*} λ^2 \geq \frac{n}{4(n-1)} Y(M,\partial M,[g]), \end{equation*} then prove that equality holds if and only if (up to a conformal transformation) $M$ is a hemisphere and $\varphi$ is a Killing spinor. More generalizations are studied.

A conformal lower bound of weighted Dirac eigenvalues on manifolds with boundary

Abstract

For the weighted Dirac eigenproblem on a compact spin manifold with the chiral boundary condition \begin{equation*} \left\{ \begin{array}{ll} D\varphi = λf\varphi & \text{in } M, \\ \mathbf{B}\varphi = 0 & \text{on } \partial M, \end{array} \right. \end{equation*} we first give a lower bound of the eigenvalue using the relative Yamabe constant \begin{equation*} λ^2 \geq \frac{n}{4(n-1)} Y(M,\partial M,[g]), \end{equation*} then prove that equality holds if and only if (up to a conformal transformation) is a hemisphere and is a Killing spinor. More generalizations are studied.
Paper Structure (15 sections, 12 theorems, 126 equations)

This paper contains 15 sections, 12 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spin manifolds and the Dirac operator
  4. The boundary Dirac operator
  5. The chiral boundary operator
  6. The weighted eigenvalue problem
  7. The integral Schrödinger-Lichnerowicz formula
  8. The relative Yamabe constant
  9. A family of conformal invariants
  10. Proof of Theorem \ref{['main_thm']}
  11. Generalizations
  12. Generalizations of the Dirac equation
  13. Generalizations of the boundary condition
  14. An application
  15. The regularity

Key Result

Theorem 1.1

Let $(M^n,g)$$(n\geq2)$ be a compact spin manifold with boundary $\partial M$ and $f$ a non-zero function. If $(M,g)$ admits a non-trivial solution $\varphi$ to intro_0-form_eq, where $\varphi\in L^p$ with $p>\frac{n}{n-1}$, then Moreover, the following statements are equivalent:

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: Conformal invariance
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 19 more