Index and total curvature of surfaces with constant mean curvature

Authors:
Manfredo P. do Carmo and Alexandre M. Da Silveira

Journal:
Proc. Amer. Math. Soc. **110** (1990), 1009-1015

MSC:
Primary 53C42; Secondary 53A10, 58G30

DOI:
https://doi.org/10.1090/S0002-9939-1990-1039255-5

MathSciNet review:
1039255

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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 in hyperbolic -space with constant mean curvature 1, the (Morse) index of the operator is finite if and only if the total Gaussian curvature is finite.

**[1]**R. Bryant,*Surfaces of mean curvature one in hyperbolic spaces*, Astérisque (to appear). MR**955072****[2]**D. Fischer-Colbrie,*On complete minimal surfaces with finite Morse index in three-manifolds*, Invent. Math.**82**(1985), 121-132. MR**808112 (87b:53090)****[3]**R. Gulliver,*Index and total curvature of complete minimal surfaces*, Proc. Sympos. Pure Math.**44**(1986), 207-211. MR**840274 (87f:53005)****[4]**R. Gulliver and B. Lawson,*The structure of stable minimal hypersurface near a singularity*, Proc. Sympos. Pure Math.**44**(1986), 213-237. MR**840275 (87g:53091)****[5]**A. Huber,*On subharmonic functions and differential geometry in the large*, Comment. Math. Helv.**32**(1957), 13-72. MR**0094452 (20:970)****[6]**A. M. da Silveira,*Stability of complete noncompact surfaces with constant mean curvature*, Math. Ann.**277**(1987), 629-638. MR**901709 (88h:53053)**

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1039255-5

Article copyright:
© Copyright 1990
American Mathematical Society