Local rigidity, bifurcation, and stability of Hf-hypersurfaces in weighted Killing warped products.

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Marco Antonio Velásquez
niversidade Federal de Campina Grande. Departamento de Matemática
Henrique F de Lima
Universidade Federal de Campina Grande. Departamento de Matemática
Andre F. A. Ramalho
Universidade Federal de Campina Grande. Departamento de Matem´atica

In a weighted Killing warped product Mn f ×ρR with warping metric h , iM+ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weighted
mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schr¨odinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x: Σn ,→ Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f-Laplacian of Σn.

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Velásquez, Marco Antonio et al. «Local rigidity, bifurcation, and stability of Hf-hypersurfaces in weighted Killing warped products». Publicacions Matemàtiques, 2021, vol.VOL 65, núm. 1, p. 363–388, http://raco.cat/index.php/PublicacionsMatematiques/article/view/384462.
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