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

Abstract

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.

Keywords

• non umbilic boundary
• yamabe problem
• compactness
• blow up analysis