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On spacelike surfaces in four-dimensional Lorentz–Minkowski spacetime through a light cone

Published online by Cambridge University Press:  17 July 2013

Francisco J. Palomo
Affiliation:
Departamento de Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain ([email protected])
Alfonso Romero
Affiliation:
Departamento de Geometráa y Topologáa, Universidad de Granada, 18071 Granada, Spain ([email protected])

Abstract

On any spacelike surface in a light cone of four-dimensional Lorentz–Minkowski space, a distinguished smooth function is considered. We show how both extrinsic and intrinsic geometry of such a surface are codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann-type result. Two remarkable families of examples of spacelike surfaces in a light cone are explicitly constructed. Finally, several results that involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a light cone are obtained.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013 

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