Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

TL;DR

This paper proves that the set of solutions to the Yamabe problem on manifolds with boundary remains compact when the mean curvature term is linearly perturbed negatively, extending known results to perturbed cases.

Contribution

It demonstrates the compactness of solution sets for the Yamabe problem under linear negative perturbations of the boundary mean curvature, for both umbilic and non-umbilic boundaries.

Findings

Solution set remains compact under perturbation

Results apply to both umbilic and non-umbilic boundaries

Extends known compactness results to perturbed Yamabe problems

Abstract

Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term with a negative smooth function, the set of solutions of Yamabe problem is still a compact set.