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2 Bulletin of the Polish Academy of Sciences. Mathematics

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A short note on rigidity of compact surfaces in space forms

Costa-Filho Wagner Oliveira

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

159--164

A rigidity result for a compact (without boundary) surface in a space form is obtained under a certain condition involving the Gaussian curvature and mean curvature of the surface. Moreover, as a byproduct of our approach, we shall present an alternative proof of an extension of Li’s theorem (2001) due to Chang (2013), on an integral inequality for Willmore surfaces in space forms.

Revisiting Liebmann’s theorem in higher codimension

Araújo Jogli G., Lima Henrique F. de

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

179--185

We deal with compact surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in a space form Qc2+p of constant sectional curvature c ϵ {−1, 0, 1}. Such a surface is called an LW-surface when it satisfies a linear Weingarten condition of the type K = aH + b for some real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively. In this setting, we extend the classical rigidity theorem of Liebmann (1899) showing that a non-flat LW-surface with non-negative Gaussian curvature must be isometric to a totally umbilical round sphere.