The Obata-V\'etois argument and its applications

TL;DR

This paper simplifies a mathematical argument to identify conditions under which certain conformally Einstein manifolds are actually Einstein, and applies these results to compute constants and prove inequalities on such manifolds.

Contribution

It introduces a simplified approach to the Obata-Vétois argument and extends it to identify new conditions for Einstein manifolds, with applications to Yamabe constants and Sobolev inequalities.

Findings

Identifies a specific interval for parameter a where conformally Einstein manifolds are Einstein.

Computes Yamabe-type constants for Einstein manifolds with nonnegative scalar curvature.

Proves sharp Sobolev inequalities and extremizes the functional determinant on certain manifolds.

Abstract

We simplify V\'etois' Obata-type argument and use it to identify a closed interval $I_n$, $n \geq 3$, containing zero such that if $a \in I_n$ and $(M^n,g)$ is a closed conformally Einstein manifold with nonnegative scalar curvature and $Q_4 + a\sigma_2$ constant, then it is Einstein. We also relax the scalar curvature assumption to the nonnegativity of the Yamabe constant under a more restrictive assumption on $a$. Our results allow us to compute many Yamabe-type constants and prove sharp Sobolev inequalities on closed Einstein manifolds with nonnegative scalar curvature. In particular, we show that closed locally symmetric Einstein four-manifolds with nonnegative scalar curvature extremize the functional determinant of the conformal Laplacian, partially answering a question of Branson and {\O}rsted.