Gap results for free boundary CMC surfaces in conformally Euclidean three-balls

TL;DR

This paper classifies free boundary constant mean curvature surfaces in conformally Euclidean three-balls, showing they are either disks or rotationally symmetric annuli under certain conditions, and provides explicit examples in Gaussian space.

Contribution

It extends classification results for free boundary CMC surfaces to conformally Euclidean spaces, including Gaussian space, under new pinching conditions.

Findings

Surfaces are either disks or rotationally symmetric annuli.

Constructed explicit minimal surface example in Gaussian space.

Extended previous classification results to broader conformally Euclidean settings.

Abstract

In this work, we consider $M=(\mathbb{B}^3_r,\bar{g})$ as the Euclidean three-ball with radius $r$ equipped with the metric $\bar{g}=e^{2h}\left\langle , \right\rangle$ conformal to the Euclidean metric. We show that if a free boundary CMC surface $\Sigma$ in $M$ satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $\Sigma$, the positional conformal vector field $\vec{x}$ and its potential function $\sigma,$ then either $\Sigma$ is a disk or $\Sigma$ is an annulus rotationally symmetric. In a particular case, we construct an example of minimal surface with strictly convex boundary in $M$, when $M$ is the Gaussian space, that illustrate our results. These results extend to the CMC case and to many others different conformally Euclidean spaces the main result obtained by Haizhong Li and Changwei Xiong.