Hopf-type theorem for self-shrinkers (Q2065875)

This paper studies self-sinkers, i.e., self-similar solutions of the mean curvature flow. More precisely the authors prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in \(\mathbb{R}^3\) and having its mean curvature and the norm of the position vector bounded from above in terms of the norm of its traceless second fundamental form, is a round sphere. Furthermore, the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in \(\mathbb{R}^n\) with radial weight. These results represent consequences of a new generalization of the classical Cauchy's Theorem in complex analysis stating that a complex function is identically zero or its zeroes are isolated if it satisfies a certain weak holomorphy.