Minimal surfaces and CPE metric

TL;DR

This paper proves Besse's conjecture that CPE metrics are Einstein for three-dimensional manifolds with generic smooth metrics, using minimal surface theory to establish the result.

Contribution

It provides a proof of Besse's conjecture in three dimensions for generic metrics, connecting minimal surface theory with scalar curvature functional critical points.

Findings

CPE metrics are Einstein in the three-dimensional generic case

The proof employs minimal surface techniques

Supports Besse's conjecture in a specific setting

Abstract

The critical points of the total scalar curvature functional, restricted to closed $n$-dimensional manifolds with constant scalar curvature metrics and unit volume, are termed CPE metrics. In 1987, Arthur L. Besse conjectured that CPE metrics are always Einstein. Using the theory of minimal surfaces, we prove the conjecture for three-dimensional manifolds with $C^\infty$-generic Riemannian metric.