Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if $\Sigma$ has constant mean curvature, then $\Sigma$ should be a sphere. Whether this is true?
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3$\begingroup$ Have you tried to do the calculation? $\endgroup$– Deane YangCommented Oct 31 at 0:28
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