Revista Matemática Iberoamericana


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Volume 36, Issue 1, 2020, pp. 195–206
DOI: 10.4171/rmi/1125

Published online: 2019-09-06

Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^3$ and $\mathbb S^3$

Marcos Petrúcio Cavalcante[1] and Darlan Ferreira de Oliveira[2]

(1) Universidade Federal de Alagoas, Maceió, Brazil
(2) Universidade Estadual de Feira de Santana, Brazil

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 from 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 $M$.

Keywords: Constant mean curvature surfaces, Morse index, spectrum

Cavalcante Marcos Petrúcio, Ferreira de Oliveira Darlan: Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^3$ and $\mathbb S^3$. Rev. Mat. Iberoam. 36 (2020), 195-206. doi: 10.4171/rmi/1125