Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

TL;DR

This paper proves the existence of conformal metrics with constant scalar curvature and boundary mean curvature on compact manifolds, extending previous results and exploring behavior in positive Yamabe cases.

Contribution

It establishes existence results for such metrics in specific dimensions and topologies, and analyzes the boundary mean curvature behavior in positive Yamabe scenarios.

Findings

Existence of conformal metrics with prescribed curvatures in dimensions 6 and 7.

Existence results for spin manifolds and other cases.

Sequences of metrics with unbounded boundary mean curvature in positive Yamabe cases.

Abstract

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+\infty$.