Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with umbilic boundary

TL;DR

This paper investigates the stability of the Yamabe boundary problem on manifolds with umbilic boundary under perturbations of mean and scalar curvatures, revealing conditions for stability and instability.

Contribution

It provides new results on the stability and blow-up behavior of solutions to the doubly perturbed Yamabe boundary problem on manifolds with umbilic boundary.

Findings

Stable under perturbations from below for mean and scalar curvatures.

Unstable under perturbations from above for either curvature.

Establishes conditions for compactness and blow-up of solutions.

Abstract

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.