Compactness and Non-compactness for the Yamabe Problem on Manifolds With Boundary

TL;DR

This paper investigates the compactness of solutions to the Yamabe problem on manifolds with boundary, establishing conditions for compactness and non-compactness based on dimension and boundary properties.

Contribution

It proves compactness for solutions when the boundary is umbilic and dimension is at most 24, and provides counter-examples for higher dimensions, advancing understanding of boundary effects.

Findings

Compactness holds for n ≤ 24 with umbilic boundary

Counter-examples show non-compactness for n ≥ 25

Weyl Vanishing Theorem established under these conditions

Abstract

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.