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On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

Published online by Cambridge University Press:  20 October 2017

Oscar Palmas
Affiliation:
Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 Mexico City, México ([email protected])
Francisco J. Palomo
Affiliation:
Departamento de Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain ([email protected])
Alfonso Romero
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain ([email protected])

Abstract

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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