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2021 Local rigidity, bifurcation, and stability of $H_f$-hypersurfaces in weighted Killing warped products
Marco A. L. Velásquez, Henrique F. de Lima, André F. A. Ramalho
Publ. Mat. 65(1): 363-388 (2021). DOI: 10.5565/PUBLMAT6512113

Abstract

In a weighted Killing warped product $M^n_f\times_\rho\mathbb R$ with war\-ping me\-tric $\langle\,,\rangle_{\!M}+{\rho}^2\,dt$, where the warping function $\rho$ is a real positive function defined on $M^n$ and the weighted function $f$ does not depend on the parameter $t\in\mathbb R$, 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 $\{\Omega_{\gamma}\}_{\gamma\in I}$ whose boundaries $\partial\Omega_{\gamma}$ are hypersurfaces with constant weighted mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schrödinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface $x\colon \Sigma^n\hookrightarrow M^n_f\times_\rho\mathbb R$ with constant weighted mean curvature in terms of the first eigenvalue of the $f$-Laplacian of $\Sigma^n$.

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Marco A. L. Velásquez. Henrique F. de Lima. André F. A. Ramalho. "Local rigidity, bifurcation, and stability of $H_f$-hypersurfaces in weighted Killing warped products." Publ. Mat. 65 (1) 363 - 388, 2021. https://doi.org/10.5565/PUBLMAT6512113

Information

Received: 12 November 2019; Revised: 2 March 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185838
Digital Object Identifier: 10.5565/PUBLMAT6512113

Subjects:
Primary: 35B32 , 53C42 , 58J55
Secondary: 35P15

Keywords: $f$-minimal hypersurfaces , $f$-stability , $H_f$-hypersurfaces , bifurcation , local rigidity , weighted Killing warped products

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.65 • No. 1 • 2021
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