Prescribing sign-changing mean curvature candidates on the n+1-dimensional unit ball

TL;DR

This paper addresses the problem of prescribing mean curvature on the unit ball with sign-changing functions, using a gradient flow method to realize such functions as boundary mean curvatures of conformal metrics.

Contribution

It introduces a new approach employing negative gradient flow to prescribe sign-changing mean curvature functions on the unit ball.

Findings

Successfully prescribes sign-changing mean curvature functions.

Establishes existence results under Morse index or symmetry conditions.

Utilizes gradient flow method for boundary mean curvature realization.

Abstract

This paper focuses on the problem of prescribing mean curvature on the unit ball. Assume that $f$, which is allowed to change sign, satisfies Morse index counting or certain kind of symmetry condition. By using a negative gradient flow method, we then prove that $f$ can be realized as the boundary mean curvature of some conformal metric.