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).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.