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2 Bulletin of the Polish Academy of Sciences. Mathematics

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A sharp integral inequality for compact Weingarten hypersurfaces under an Okumura type inequality

Lima Eudes L. de, Lima Henrique F. de

Bulletin of the Polish Academy of Sciences. Mathematics

2021

Vol. 69, no. 2

139--148

Our aim in this paper is to provide a sharp integral inequality involving the norm of the traceless second fundamental form of a wide class of compact (without boundary) linear Weingarten hypersurfaces (including those with two distinct principal curvatures) immersed into a Riemannian space form. In particular, we generalize the results of Alías Meléndez (2020) when the ambient space form is the unit Euclidean sphere and give a new estimate when the space form is either the Euclidean space or the hyperbolic space. The sharpness of our integral inequality is realized by the totally umbilical spheres and, when the ambient space is the unit Euclidean sphere, by Clifford tori.

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.