Critical Point Equation on three-dimensional manifolds and the Besse Conjecture

TL;DR

This paper proves the Besse conjecture for three-dimensional compact manifolds and establishes the rigidity of Miao-Tam critical metrics under certain boundary conditions.

Contribution

It provides the first proof of the Besse conjecture in three dimensions and demonstrates the rigidity of Miao-Tam critical metrics on compact manifolds with boundary.

Findings

Besse conjecture is resolved in three dimensions

Rigidity of Miao-Tam critical metrics is established

Results apply to compact manifolds with smooth boundary

Abstract

In this paper, we present the resolution of the Besse conjecture on a three dimensional compact manifold. We also prove the rigidity of the Miao-Tam critical metric on a three dimensional compact manifold with a smooth boundary.