Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces

TL;DR

This paper explores the minimization of Dirac eigenvalues on surfaces with fixed area and conformal class, revealing links to harmonic maps into complex projective spaces and providing new proofs for known eigenvalue minimization results.

Contribution

It establishes a connection between critical Dirac eigenvalues and harmonic maps into complex projective spaces, and demonstrates minimization results for Dirac eigenvalues on tori and spheres.

Findings

First nonzero Dirac eigenvalue minimized by flat metric on many tori

New geometric proof of Bär's theorem for the sphere

Connection between Dirac eigenvalues and harmonic maps into complex projective spaces

Abstract

Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of B\"ar's theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.