Quantitative Stability for Minimizing Yamabe Metrics with minimal boundary

TL;DR

This paper establishes a quantitative relationship between near-minimizers of the Yamabe energy and actual minimizing Yamabe metrics on compact manifolds with boundary, providing explicit stability bounds.

Contribution

It introduces a quantitative stability result for minimizing Yamabe metrics with boundary, linking energy deficits to metric closeness within conformal classes.

Findings

Closeness of conformal metrics is controlled by the Yamabe energy deficit.

Provides explicit power-law bounds for stability.

Extends stability analysis to manifolds with boundary.

Abstract

In this paper, we investigate the stability of minimizing Yamabe metrics on compact manifolds with boundary, in the sense introduced by Escobar. We show that if a function nearly minimizes the Yamabe energy, then the associated conformal metric is quantitatively close to a minimizing Yamabe metric within its conformal class. Moreover, this closeness is controlled by an appropriate power of the Yamabe energy deficit.