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1990•Proceedings of the American Mathematical SocietyOpen access

Index and Total Curvature of Surfaces with Constant Mean Curvature

Manfredo P. do Carmo, Alexandre M. da Silveira

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Abstract

We prove an analogue, for surfaces with constant mean curvature in hyperbolic space, of a theorem of Fischer-Colbrie and Gulliver about minimal surfaces in Euclidean space. That is, for a complete surface ${M^2}$ in hyperbolic $3$-space with constant mean curvature 1, the (Morse) index of the operator $L = \Delta - 2K$ is finite if and only if the total Gaussian curvature is finite.

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We prove an analogue, for surfaces with constant mean curvature in hyperbolic space, of a theorem of Fischer-Colbrie and Gulliver about minimal surfaces in Euclidean space. That is, for a complete surface ${M^2}$ in hyperbolic $3$-space with constant mean curvature 1, the (Morse) index of the operator $L = \Delta - 2K$ is finite if and only if the total Gaussian curvature is finite.

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Available abstract

We prove an analogue, for surfaces with constant mean curvature in hyperbolic space, of a theorem of Fischer-Colbrie and Gulliver about minimal surfaces in Euclidean space. That is, for a complete surface ${M^2}$ in hyperbolic $3$-space with constant mean curvature 1, the (Morse) index of the operator $L = \Delta - 2K$ is finite if and only if the total Gaussian curvature is finite.

Key concepts: Mean curvature, Gaussian curvature, Constant-mean-curvature surface, Total curvature, Mathematics, Hyperbolic space, Mathematical analysis, Curvature

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