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

TL;DR

This paper establishes a conformal lower bound for weighted Dirac eigenvalues on manifolds with boundary, linking it to the relative Yamabe constant, and characterizes cases of equality as hemispheres with Killing spinors.

Contribution

It provides a new lower bound for weighted Dirac eigenvalues on manifolds with boundary and characterizes the equality case as hemispheres with Killing spinors.

Findings

Lower bound of eigenvalues using the relative Yamabe constant

Equality case characterized by hemispheres and Killing spinors

Generalizations to broader settings

Abstract

For the weighted Dirac eigenproblem on a compact spin manifold with the chiral boundary condition \begin{equation*} \left\{ \begin{array}{ll} D\varphi = \lambda f\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*} \lambda^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.