Conformally critical metrics and optimal bounds for Dirac eigenvalues on spin surfaces

TL;DR

This paper investigates the minimization of Dirac eigenvalues within conformal classes on spin surfaces, establishing existence criteria and deriving isoperimetric inequalities for the sphere.

Contribution

It provides new criteria for the existence of minimizers and characterizes the conformal spectrum of the Dirac operator on surfaces, especially the sphere.

Findings

Established a criterion for the existence of minimizers.

Derived isoperimetric inequalities for the Dirac operator on the sphere.

Characterized the conformal spectrum of the Dirac operator on surfaces.

Abstract

We study the minimization problem for eigenvalues of the Dirac operator within a fixed conformal class on a closed spin Riemannian manifold. We establish a criterion for the existence of a minimizer for this variational problem, focusing specifically on the case of closed surfaces. Furthermore, we apply our results to derive isoperimetric inequalities for the Dirac operator on the two-dimensional sphere, providing a complete characterization of its conformal spectrum.