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On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

Alex Moriani, Enrico Trebeschi

Abstract

We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of Ishihara, a sharp bound on the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces of every signature. We show that achieving the bound at a point is equivalent to achieving it identically, and explicitely describe the submanifolds achieving the bound. When the codimension is equal to 1, we deduce a sharp upper bound on the Ricci curvature of complete maximal hypersurfaces in Anti-de Sitter spaces, and characterize the hypersurfaces achieving it. Finally, we discuss the link between scalar curvature and Gromov-hyperbolicity for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.

On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

Abstract

We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of Ishihara, a sharp bound on the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces of every signature. We show that achieving the bound at a point is equivalent to achieving it identically, and explicitely describe the submanifolds achieving the bound. When the codimension is equal to 1, we deduce a sharp upper bound on the Ricci curvature of complete maximal hypersurfaces in Anti-de Sitter spaces, and characterize the hypersurfaces achieving it. Finally, we discuss the link between scalar curvature and Gromov-hyperbolicity for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.

Paper Structure

This paper contains 51 sections, 30 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Historical background
  4. Geometric relevance
  5. Main ingredients
  6. Organization of the paper
  7. Pseudo-Riemannian spaceforms
  8. Explicit models of spaceforms
  9. Spacelike submanifolds
  10. First and second fundamental forms
  11. Curvature
  12. Fundamental equations in a spaceform
  13. Harmonicity of the second fundamental form in a spaceform
  14. Two results in $\mathbb{H}^{p,q}$
  15. Boundary of $\mathbb{H}^{p,q}$
  16. ...and 36 more sections

Key Result

Theorem A.1

Let $M$ be a complete maximal spacelike $p-$submanifold in $\mathbb{H}^{p,q}$, then Moreover, if the bound is achieved at one point, then $M$ is a pseudo-flat.

Theorems & Definitions (66)

  • Theorem A.1
  • Theorem B.1
  • Theorem : ish88
  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 1.2
  • Theorem 3
  • Theorem 3
  • Remark 1.3
  • ...and 56 more