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

TL;DR

This paper investigates the stability and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundary, showing conditions for compactness and sequences that blow up under perturbations.

Contribution

It establishes conditions under which the solution set remains compact or blows up when perturbing boundary mean curvature and scalar curvature functions.

Findings

Solution set is compact when perturbing mean curvature from below and scalar curvature with controlled maximum.

Existence of blow-up sequences when perturbing mean curvature from above or with large positive maximum scalar curvature.

Provides a detailed analysis of stability and blow-up phenomena for doubly perturbed Yamabe problems.

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.