Conformal classes realizing the Yamabe invariant

TL;DR

This paper characterizes conformal classes that realize the Yamabe invariant of a compact manifold, drawing analogies with previous results on eigenvalue maximization.

Contribution

It provides a new characterization of conformal classes that attain the Yamabe invariant, extending analogies from eigenvalue optimization problems.

Findings

Identifies conditions under which conformal classes realize the Yamabe invariant.

Establishes a parallel with eigenvalue maximization results by Nadirashvili, Fraser, and Schoen.

Contributes to understanding the geometric structures associated with the Yamabe invariant.

Abstract

We give a characterization of conformal classes realizing a compact manifold's Yamabe invariant. This characterization is the analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue.