Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature

TL;DR

This paper addresses the problem of finding conformal metrics with zero scalar curvature and prescribed boundary mean curvature on compact manifolds, resolving open cases and establishing new solvability conditions.

Contribution

It introduces new methods to solve the boundary curvature problem, extending previous results and resolving most remaining open cases.

Findings

Resolved most remaining open cases in boundary mean curvature problem

Established new solvability conditions for conformal scalar-flat metrics

Constructed local test functions to achieve these results

Abstract

Let $(M, g)$ be a compact Riemannian manifold with boundary $\partial M$. Given a function $f$ on $\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$ on $\partial M$. Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions.