A Condition in Mean Curvature Prescriptions for Conformal Metrics on the Ball

TL;DR

This paper proves the existence of conformal metrics with zero scalar curvature and prescribed boundary mean curvature on the Euclidean ball, under symmetry and flatness conditions, extending understanding of geometric boundary value problems.

Contribution

It establishes existence results for conformal metrics with prescribed boundary mean curvature on the Euclidean ball under specific symmetry and flatness conditions.

Findings

Existence of conformal metrics with zero scalar curvature and prescribed boundary mean curvature.

Conditions involving sign changes of H' and flatness are sufficient for existence.

Results apply to n-dimensional Euclidean balls with n ≥ 3.

Abstract

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for $n \geq 3$. Given a rotationally symmetric function $H:\partial B^{n}\rightarrow R$, in this work, we will prove that if $H'(r)$ changes signs where $H>0$ and $H(r)$ also satisfies a flatness condition then there exists a metric $g$ conformal to the Euclidean metric, with zero scalar curvature in the ball and mean curvature $H$ on its boundary.