# Extremal metrics for Laplace eigenvalues in perturbed conformal classes on products

## TL;DR

This paper proves that small perturbations of product conformal classes on manifolds with a sphere admit metrics that extremize Laplace eigenvalues, using perturbed harmonic maps with constant density.

## Contribution

It introduces a method to find extremal metrics for Laplace eigenvalues in perturbed conformal classes on product manifolds.

## Key findings

- Existence of extremal metrics in perturbed conformal classes.
- Construction of perturbed harmonic maps with constant density.
- Extension of extremal metric theory to product manifolds.

## Abstract

In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments we obtain perturbed harmonic maps with constant density.