Lower bounds for the index of compact constant mean curvature surfaces in \(\mathbb{R}^3\) and \(\mathbb S^3\)

DOI10.4171/RMI/1125zbMATH Open1437.53044arXiv1711.07233OpenAlexW2971515122MaRDI QIDQ2177516

Darlan Ferreira de Oliveira, Marcos P. Cavalcante

Publication date: 6 May 2020

Published in: Revista Matemática Iberoamericana (Search for Journal in Brave)

Abstract: Let $M$ be a compact constant mean curvature surface either in $mathbb{S}^3$ or $mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.

Full work available at URL: https://arxiv.org/abs/1711.07233

zbMATH Keywords

spectrumMorse indexconstant mean curvature surfaces

Mathematics Subject Classification ID

Estimates of eigenvalues in context of PDEs (35P15) Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42) Optimization of shapes other than minimal surfaces (49Q10)

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Related Items (6)

Index estimates for closed minimal submanifolds of the sphere ⋮ Stability for a second type partitioning problem ⋮ On the genus and area of constant mean curvature surfaces with bounded index ⋮ On some inequalities for low eigenvalues of closed surfaces in ⋮ Morse index bounds for minimal submanifolds ⋮ Index estimates for surfaces with constant mean curvature in 3-dimensional manifolds

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