Calculus of Variations and Geometric Measure Theory

G. Catino - A. Roncoroni - L. Vezzoni

On the umbilic set of immersed surfaces in three-dimensional space forms

created by catino on 17 Apr 2019
modified by roncoroni on 05 Nov 2020


Accepted Paper

Inserted: 17 apr 2019
Last Updated: 5 nov 2020

Journal: Bull. Sci. Math.
Year: 2020

ArXiv: 1904.08145 PDF


We prove that under some assumptions on the mean curvature 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.