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), 10091015
MSC:
Primary 53C42; Secondary 53A10, 58G30
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 FischerColbrie 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.
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 R. Bryant, Surfaces of mean curvature one in hyperbolic spaces, Astérisque (to appear). MR 955072
 [2]
 D. FischerColbrie, On complete minimal surfaces with finite Morse index in threemanifolds, Invent. Math. 82 (1985), 121132. MR 808112 (87b:53090)
 [3]
 R. Gulliver, Index and total curvature of complete minimal surfaces, Proc. Sympos. Pure Math. 44 (1986), 207211. 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), 213237. MR 840275 (87g:53091)
 [5]
 A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 1372. MR 0094452 (20:970)
 [6]
 A. M. da Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math. Ann. 277 (1987), 629638. MR 901709 (88h:53053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010392555
PII:
S 00029939(1990)10392555
Article copyright:
© Copyright 1990
American Mathematical Society
