Brown-York mass and positive scalar curvature II - Besse's conjecture and related problems

TL;DR

This paper explores the connection between Besse's conjecture on Einstein manifolds and the positive mass theorem for Brown-York mass, advancing understanding of critical metrics with constant scalar curvature.

Contribution

It establishes a link between Besse's conjecture and the positive mass theorem, providing new insights into the geometry of CPE and V-static manifolds.

Findings

Connection between Besse's conjecture and positive mass theorem established

Geometric structure of CPE manifolds analyzed using positive mass techniques

Discussion of related results for V-static metrics included

Abstract

The Besse's conjecture was posted on the well-known book Einstein manifolds by Arthur L. Besse, which describes the critical point of Hilbert-Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse's conjecture and positive mass theorem for Brown-York mass. With the aid of positive mass theorem, we investigate the geometric structure of the so-called CPE manifolds, which provides us further understanding of Besse's conjecture. As a related topic, we also have a discussion of corresponding results for V-static metrics.