Constant Mean Curvature Surfaces in Half Space Models
by
Maria Elisa Esteves Lopes Galvão
IME-USP/ UMC, Brazil
Coauthors: Célia Contin Góes

Poster

We will study surfaces with constant mean curvature c in a half space model for the space form with constant curvature -c². For these surfaces we have a Weierstarss type representation as in the case of curvature one ( [B], [UY-1], [GG]). The main goal is to construct explicitally a sequence of constant mean curvature surfaces that gives a deformation of a constant mean curvature one surface in a constant mean curvature one half space model into a minimal surface in the euclidean space. A general theorem in this direction has been proved by Umehara and Yamada [UY-2].

These sequences will be constructed in a half space model for the space form with constant curvature -c² in order that we can take a sequence of such half spaces with 1 >= c > 0 converging to the euclidean space when c goes to zero.

In the Poincaré half space model given by the half space

{ (y1, y2, y3) : y3 > 1-1/c }

with the metric

ds21-1/c =  1 (c y3+1-c)2 (dy12+dy22+dy32)

we can have the pictures of the deformations for embedded or immersed cathenoids with mean curvature c.

References.

[B] Bryant, R. L. Surfaces of mean curvature one in hyperbolic space, Astérisque, 155 ( 1987 ), ( exposé XVI ), 321-347.

[GG] Galvão, M. E. E. L., Góes, C. C., A Weierstrass Type Representation for Mean Curvature One Surfaces in the Hyperbolic Space , Note di Matematica, 1998

[GG] Galvão, M. E. E. L., Góes, C. C., Constant Mean Curvature Surfaces in Half Space Models, in preparation.

[UY-1] Umehara, M., Yamada, K., Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space, Annals of Math., 137, ( 1993 ), 611-638.

[UY-2] Umehara, M., Yamada, K., A Parametrization of the Weierstrass Formulae and perturbation of Complete Minimal Surfaces in R³ into the Hyperbolic 3- space, J, Reine Angew Math., ( 1992), 93-116.

Date received: June 20, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-31.