We prove that, under some integral inequality regarding the mean curvature and the traceless second fundamental form, the set of umbilical points of an immersed surface in a 3-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
On the umbilic set of immersed surfaces in three-dimensional space forms
Vezzoni L.
2020-01-01
Abstract
We prove that, under some integral inequality regarding the mean curvature and the traceless second fundamental form, the set of umbilical points of an immersed surface in a 3-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
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