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Rigidity and nonexistence of Riemannian immersions in semi-Riemannian warped products via parabolicity

Henrique F. de Lima|Railane Antonia|Marcio S. Santos
대한수학회 (The Korean Mathematical Society)|2024.01|대한수학회지 (Journal of the KMS)|제61권 제1호|pp. 41-63|DOI 10.4134/JKMS.j220523

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키워드

Einstein manifoldsentire graphshigher order mean curvaturesSemi-Riemannian warped productsRiemannian immersionsparabolicity criterion
주제분류수학

초록

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 pa\-ra\-bo\-li\-ci\-ty 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.

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 pa\-ra\-bo\-li\-ci\-ty 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.

참고문헌 (31)

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