New existence results for prescribed mean curvature problem on balls under pinching conditions

TL;DR

This paper establishes new existence results for a Yamabe-type problem involving prescribed boundary mean curvature on unit balls in higher dimensions, using advanced variational and topological methods under specific curvature conditions.

Contribution

It introduces novel existence results for the prescribed mean curvature problem on balls in dimensions n≥5, employing a combination of critical points at infinity and Morse theory.

Findings

Existence of solutions under pinching conditions on mean curvature

Application of critical points at infinity approach in higher dimensions

Use of Morse theory to establish solution existence

Abstract

We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball $\mathbb{B}^n$ and on the boundary $\mathbb{S}^{n-1}$ the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case $n\geq 5$, under suitable pinching conditions on the mean curvature function.