Gap Results for Free Boundary CMC Surfaces in Radially Symmetric Conformally Euclidean Three-Balls

Abstract

In this work, we consider $$M=(\mathbb {B}_r^3,{\bar{g}})$$ 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, where the function $$h=h(x)$$ depends only on the distance of x to the center of $$\mathbb {B}_r^3$$ . We show that if a free boundary CMC surface $$\varSigma $$ in M satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $$\varSigma $$ , the positional conformal vector field $$\mathbf {x}$$ and its potential function $$\sigma ,$$ then either $$\varSigma $$ is a disk or $$\varSigma $$ is an annulus rotationally symmetric. These results extend to the CMC case and to many other different conformally Euclidean spaces the main result obtained by Li and Xiong (J Geom Anal 28(4):3171–3182, 2018).