Clustering phenomena in low dimensions for a boundary Yamabe problem

TL;DR

This paper investigates boundary blow-up phenomena in a geometric boundary Yamabe problem, revealing clustering blow-up points in low dimensions, which contrasts with the isolated blow-up points observed in three dimensions.

Contribution

It demonstrates the existence of non-isolated blow-up points in low dimensions for a perturbed boundary Yamabe problem, expanding understanding of blow-up behavior.

Findings

Existence of clustering blow-up boundary points in dimensions 4 to 7.

Non-umbilic blow-up points that are local minimizers of the trace-free second fundamental form.

Contrasts with the isolated blow-up points in three-dimensional cases.

Abstract

We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.