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SPACELIKE CAPILLARY SURFACES IN THE LORENTZ–MINKOWSKI SPACE

Part of: Classical differential geometry

Published online by Cambridge University Press:  09 August 2011

JUNCHEOL PYO  and
KEOMKYO SEO
JUNCHEOL PYO
Affiliation:
Department of Mathematics, Pusan National University, Busan 609-735, Korea (email: [email protected])
KEOMKYO SEO*
Affiliation:
Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul 140-742, Korea (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann.323(1) (2002), 143–156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincaré–Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in is part of (spacelike) plane or a hyperbolic plane.

Keywords

capillary surfacesspacelike surfacesconstant mean curvature

MSC classification

Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 362 - 371
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022951 and 2010-0004246).

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