Symmetry of properly embedded special Weingarten surfaces in 𝐇³

Sa Earp, Ricardo; Toubiana, Eric

doi:10.1090/S0002-9947-99-02511-8

Symmetry of properly embedded special Weingarten surfaces in $\mathbf {H}^3$
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by Ricardo Sa Earp and Eric Toubiana

Trans. Amer. Math. Soc. 351 (1999), 4693-4711

DOI: https://doi.org/10.1090/S0002-9947-99-02511-8

Published electronically: August 25, 1999

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Abstract:

In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.

References

J. L. Barbosa, R. Sa Earp. New results on prescribed mean curvature hypersurfaces in space forms. An Acad. Bras. Ci., 67, 1,1-5, (1995).
J. Lucas M. Barbosa and Ricardo Sa Earp, Prescribed mean curvature hypersurfaces in $H^{n+1}(-1)$ with convex planar boundary. I, Geom. Dedicata 71 (1998), no. 1, 61–74. MR 1624726, DOI 10.1023/A:1005046021921
Fabiano Gustavo Braga Brito and Ricardo Sa Earp, On the structure of certain Weingarten surfaces with boundary a circle, Ann. Fac. Sci. Toulouse Math. (6) 6 (1997), no. 2, 243–255 (English, with English and French summaries). MR 1611824
R. Bryant. Complex analysis and a class of Weingarten surfaces. Preprint.
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Manfredo P. do Carmo and H. Blaine Lawson Jr., On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J. 50 (1983), no. 4, 995–1003. MR 726314, DOI 10.1215/S0012-7094-83-05041-X
M. P. do Carmo, J. de M. Gomes, and G. Thorbergsson, The influence of the boundary behaviour on hypersurfaces with constant mean curvature in $H^{n+1}$, Comment. Math. Helv. 61 (1986), no. 3, 429–441. MR 860133, DOI 10.1007/bf02621926
J.M. Gomes. Sobre hirpersuperfícies com curvatura média constante no espaço hiperbólico. Tese de doutorado. IMPA, (1985).
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR 707850, DOI 10.1007/978-3-662-21563-0
Wu-yi Hsiang, On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature, Duke Math. J. 49 (1982), no. 3, 485–496. MR 672494
Nicholas J. Korevaar, Rob Kusner, William H. Meeks III, and Bruce Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), no. 1, 1–43. MR 1147718, DOI 10.2307/2374738
Gilbert Levitt and Harold Rosenberg, Symmetry of constant mean curvature hypersurfaces in hyperbolic space, Duke Math. J. 52 (1985), no. 1, 53–59. MR 791291, DOI 10.1215/S0012-7094-85-05204-4
Barbara Nelli and Harold Rosenberg, Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary, Ann. Global Anal. Geom. 13 (1995), no. 1, 23–30. MR 1327108, DOI 10.1007/BF00774564
Harold Rosenberg and Ricardo Sa Earp, The geometry of properly embedded special surfaces in $\textbf {R}^3$, e.g., surfaces satisfying $aH+bK=1$, where $a$ and $b$ are positive, Duke Math. J. 73 (1994), no. 2, 291–306. MR 1262209, DOI 10.1215/S0012-7094-94-07314-6
Ricardo Sá Earp and Éric Toubiana, A note on special surfaces in $\textbf {R}^3$, Mat. Contemp. 4 (1993), 109–118. VIII School on Differential Geometry (Portuguese) (Campinas, 1992). MR 1302497
Ricardo Sa Earp and Eric Toubiana, Sur les surfaces de Weingarten spéciales de type minimal, Bol. Soc. Brasil. Mat. (N.S.) 26 (1995), no. 2, 129–148 (French, with English summary). MR 1364263, DOI 10.1007/BF01236989
R. Sa Earp et E. Toubiana. Classification des surfaces de type Delaunay et applications. Amer. J. Math. 221, 671–700, (1999).