Blow-up phenomena for a boundary Yamabe problem with umbilic boundary

TL;DR

This paper investigates blow-up phenomena in a boundary Yamabe problem with umbilic boundary, establishing existence of positive solutions under certain geometric conditions in high dimensions.

Contribution

It extends the analysis of boundary Yamabe problems by addressing blow-up phenomena with non-zero Weyl tensor and negative scalar curvature in dimensions n≥8.

Findings

Existence of positive solutions for the boundary Yamabe problem when n≥8.

Blow-up phenomena are characterized under the condition of non-zero Weyl tensor.

The results apply to manifolds with umbilic boundary and negative scalar curvature.

Abstract

We consider a linear perturbation of the classical geometric problem of prescribing the scalar and the boundary mean curvature problem in a Riemannian manifold with umbilic boundary provided the Weyl tensor is non-zero everywhere. We will deal with the case of negative scalar curvature showing the existence of a positive solutions when $n\geq 8$.