An existence theorem of conformal scalar-flat metrics on manifolds with boundary

TL;DR

This paper proves an existence theorem for conformal scalar-flat metrics with constant mean curvature boundary on compact manifolds with boundary, completing previous results in the case of umbilic boundaries.

Contribution

It establishes the existence of such metrics on manifolds with umbilic boundary, addressing previously unresolved cases of the Yamabe-type problem.

Findings

Existence of conformal scalar-flat metrics with constant mean curvature boundary proven for umbilic boundary cases.

Completes the classification of solutions for the Yamabe-type problem on manifolds with boundary.

Advances understanding of geometric structures on manifolds with boundary.

Abstract

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this problem.