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

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.