Quantitative Stability for Minimizing Yamabe Metrics

TL;DR

This paper establishes a quantitative relationship between near-minimizers of the Yamabe energy and actual Yamabe metrics on closed manifolds, demonstrating a quadratic stability estimate that can fail in specific cases.

Contribution

It provides the first quantitative stability result for Yamabe minimizers and constructs an example where the quadratic estimate does not hold.

Findings

Near-minimizers are close to Yamabe metrics in a quadratic manner.

The quadratic stability estimate generally holds for generic cases.

An explicit example shows the quadratic estimate can fail.

Abstract

On any closed Riemannian manifold of dimension $n\geq 3$, we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.