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Conformal Killing graphs in foliated Riemannian spaces with density: rigidity and stability

Marco L. A. Velásquez;  André F. A. Ramalho;  Henrique F. de Lima;  Márcio S. Santos;  Arlandson M. S. Oliveira — 2021

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold M ¯ f n + 1 endowed with a weight function f and having a closed conformal Killing vector field V with conformal factor ψ V , that is, graphs constructed through the flow generated by V and which are defined over an integral leaf of the foliation V ⊥ orthogonal to V . For such graphs, we establish some rigidity results under appropriate constraints on the f -mean curvature. Afterwards, we obtain some stability results...

Sharp eigenvalue estimates of closed H -hypersurfaces in locally symmetric spaces

Eudes L. de Lima;  Henrique F. de Lima;  Fábio R. dos Santos;  Marco A. L. Velásquez — 2019

Czechoslovak Mathematical Journal

The purpose of this article is to obtain sharp estimates for the first eigenvalue of the stability operator of constant mean curvature closed hypersurfaces immersed into locally symmetric Riemannian spaces satisfying suitable curvature conditions (which includes, in particular, a Riemannian space with constant sectional curvature). As an application, we derive a nonexistence result concerning strongly stable hypersurfaces in these ambient spaces.

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