Extremal metrics for Laplace eigenvalues in perturbed conformal classes

TL;DR

This paper demonstrates that extremal metrics for Laplace eigenvalues in a conformal class can be approximated in nearby classes, utilizing perturbed harmonic maps with constant density.

Contribution

It introduces a method to find extremal metrics in neighboring conformal classes based on existing extremal metrics, involving perturbed harmonic maps.

Findings

Existence of extremal metrics implies nearby extremal metrics in perturbed classes.

Perturbed harmonic maps with constant density are constructed as part of the proof.

The approach applies to many cases within the studied framework.

Abstract

We prove that in many cases the existence of an extremal metric for some Laplace eigenvalue in a conformal class allows to find extremal metrics in conformal classes close by. As a consequence and as part of the arguments we obtain perturbed harmonic maps with constant density.