Proof of the remaining cases of the Yamabe boundary problem

TL;DR

This paper completes the proof of the boundary Yamabe problem by addressing remaining cases using advanced bubble techniques and topological arguments, confirming the problem's resolution.

Contribution

It solves the last open cases of the boundary Yamabe problem by combining bubble constructions with algebraic topological methods.

Findings

All cases of the boundary Yamabe problem are now resolved.

The paper extends bubble techniques to manifolds with boundary.

It confirms the boundary Yamabe problem as fully solved.

Abstract

In this paper, we solve the remaining cases of the boundary Yamabe problem introduced by Escobar in 1992. Indeed, using the bubbles of Brendle-Chen, which are an adaptation to manifolds with boundary of the original ones introduced by Brendle for the study of the Yamabe flow on closed Riemannian manifolds of dimension greater or equal to 6, and the algebraic topological argument of Bahri-Coron, we solve the cases left open after the work of Brendle-Chen. Thus combining our work with the ones of Brendle-Chen and Escobar, we have that the boundary Yamabe problem is a done deal.