Conformal Deformation to Scalar Flat Metrics with Constant Mean Curvature on the Boundary in Higher Dimensions

TL;DR

This paper advances the understanding of conformal metrics with zero scalar curvature and constant boundary mean curvature in higher dimensions, building on and extending previous work through local test functions and the positive mass theorem.

Contribution

It refines existing results on the Yamabe problem with boundary by addressing remaining cases using new local test functions and a reduction to the positive mass theorem.

Findings

Resolved most remaining cases of the problem

Reduced the problem to the positive mass theorem

Extended previous methods by Brendle and others

Abstract

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test function construction, we are able to seattle the most cases left by Escobar's and Marques's works. Moreover, we reduce the remaining case to the positive mass theorem. In this proof, we use the method developed in previous works by Brendle and by Brendle and the author.