An analogue of minimal surface theory in

Authors:
M. Kokubu, M. Takahashi, M. Umehara and K. Yamada

Journal:
Trans. Amer. Math. Soc. **354** (2002), 1299-1325

MSC (2000):
Primary 53A10; Secondary 53A35, 53A07

DOI:
https://doi.org/10.1090/S0002-9947-01-02935-X

Published electronically:
November 19, 2001

MathSciNet review:
1873007

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Abstract | References | Similar Articles | Additional Information

Abstract: We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. ), which contains minimal surfaces in and constant mean curvature surfaces in . A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.

**[B]**R. Bryant,*Surfaces of constant mean curvature one in hyperbolic space*, Astérisque Vol. 154-155, (1987), 321-347. CMP**20:16****[CL]**E. A. Coddington, N. Levinson,*Theory of Ordinary Differential Equations*, McGraw-Hill, 1955. MR**16:1022b****[CO]**S. Chern and R. Osserman,*Complete minimal surface in euclidean**-space*, J. Analyse Math.**19**(1967) 15-34. MR**37:2103****[F]**R. Finn,*On a class of conformal metrics with application to geometry in large*, Comm. Math. Helv.**40**(1965) 1-30. MR**34:3467****[H]**S. Helgason,*Differential Geometry, Lie groups, and Symmetric Spaces*, Academic Press, New York-San Francisco-London, 1978. MR**80k:53081****[KUY]**M. Kokubu, M. Umehara and K. Yamada,*Minimal surfaces that attain equality in the Chern-Osserman inequality*, preprint, math.DG/0102037.**[L]**H. B. Lawson,*Lectures on minimal submanifolds*(*Volume 1*), Publish or Perish Inc., 1980. MR**82d:53035b****[RUY]**W. Rossman, M. Umehara and K. Yamada,*Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus*, Tôhoku Math. J. 49 (1997), 449-484 MR**99a:53025****[UY1]**M. Umehara and K. Yamada,*Complete surfaces of constant mean curvature-**in the hyperbolic**-space*, Ann. of Math.**137**(1993), 611-638. MR**94c:53015****[UY2]**M. Umehara and K. Yamada,*A parametrization of Weierstrass formulae and perturbation of some complete minimal surfaces of**into the hyperbolic**-space*, J. Reine Angew. Math.**432**(1992), 93-116. MR**94e:54004****[UY3]**M. Umehara and K. Yamada,*Surfaces of constant mean curvature**in**with prescribed hyperbolic Gauss map*, Math. Ann.**304**(1996), 203-224. MR**97b:53017****[UY4]**M. Umehara and K. Yamada,*A duality on CMC-1 surface in the hyperbolic**-space and a hyperbolic analogue of the Osserman Inequality*, Tsukuba J. Math.**21**(1997), 229-237. MR**99e:53012****[Y]**Z. Yu,*The value distribution of the hyperbolic Gauss map*, Proc. Amer. Math. Soc.**125**(1997) 2997-3001 MR**97m:53016****[Y2]**Z. Yu,*The inverse surface and the Osserman inequality*, Tsukuba J. Math.**22**(1998) 57-588. MR**2000c:53008**

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Additional Information

**M. Kokubu**

Affiliation:
Department of Natural Science, School of Engineering, Tokyo Denki University, 2-2, Kanda-Nishiki-Cho, Chiyoda-Ku, Tokyo, 101-8457 Japan

Email:
kokubu@cck.dendai.ac.jp

**M. Takahashi**

Affiliation:
Department of General Education, Kurume National College of Technology, Kurume, Fukuoka 830-8555, Japan

Email:
taka@GES.kurume-nct.ac.jp

**M. Umehara**

Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan

Email:
umehara@math.sci.hiroshima-u.ac.jp

**K. Yamada**

Affiliation:
Faculty of Mathematics, Kyushu University 36, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan

Email:
kotaro@math.kyushu-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-01-02935-X

Received by editor(s):
March 8, 2001

Published electronically:
November 19, 2001

Article copyright:
© Copyright 2001
American Mathematical Society