Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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RIGIDITY AND NONEXISTENCE OF RIEMANNIAN IMMERSIONS IN SEMI-RIEMANNIAN WARPED PRODUCTS VIA PARABOLICITY

  • Railane Antonia (Departamento de Matematica Universidade Federal da Paraiba) ;
  • Henrique F. de Lima (Departamento de Matematica Universidade Federal de Campina Grande) ;
  • Marcio S. Santos (Departamento de Matematica Universidade Federal da Paraiba)
  • Received : 2022.10.25
  • Accepted : 2023.10.19
  • Published : 2024.01.01
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Abstract

In this paper, we study complete Riemannian immersions into a semi-Riemannian warped product obeying suitable curvature constraints. Under appropriate differential inequalities involving higher order mean curvatures, we establish rigidity and nonexistence results concerning these immersions. Applications to the cases that the ambient space is either an Einstein manifold, a steady state type spacetime or a pseudo-hyperbolic space are given, and a particular investigation of entire graphs constructed over the fiber of the ambient space is also made. Our approach is based on a parabolicity criterion related to a linearized differential operator which is a divergence-type operator and can be regarded as a natural extension of the standard Laplacian.

Keywords

Acknowledgement

The authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments which improved the paper.

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