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Lower bounds for the index of compact constant mean curvature surfaces in \(\mathbb{R}^3\) and \(\mathbb S^3\). (English) Zbl 1437.53044

Summary: 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 that of Hodge Laplacian of 1-forms on \(M\).

MSC:

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