Index and total curvature of surfaces with constant mean curvature

do Carmo, Manfredo P.; Da Silveira, Alexandre M.

doi:10.1090/S0002-9939-1990-1039255-5

Index and total curvature of surfaces with constant mean curvature
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by Manfredo P. do Carmo and Alexandre M. Da Silveira PDF

Proc. Amer. Math. Soc. 110 (1990), 1009-1015 Request permission

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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