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Topological and rigidity results for four-dimensional hypersurfaces in space forms

Davide Dameno, Aaron J. Tyrrell

Abstract

Exploiting the special features of four-dimensional Riemannian Geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5: we characterize isoparametric hypersurfaces by means of the Weyl tensor, we prove sharp topological bounds on the Weyl functional and, inspired by a famous conjecture by Chern, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under some volume constraints. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form. We also extend some of the results to the case of a locally conformally flat 5-dimensional ambient space.

Topological and rigidity results for four-dimensional hypersurfaces in space forms

Abstract

Exploiting the special features of four-dimensional Riemannian Geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5: we characterize isoparametric hypersurfaces by means of the Weyl tensor, we prove sharp topological bounds on the Weyl functional and, inspired by a famous conjecture by Chern, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under some volume constraints. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form. We also extend some of the results to the case of a locally conformally flat 5-dimensional ambient space.
Paper Structure (5 sections, 27 theorems, 227 equations)

This paper contains 5 sections, 27 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Four-Dimensional hypersurfaces in space forms and the Weyl tensor
  4. Topological bounds for the second fundamental form
  5. Bochner-type formulas and rigidity results

Key Result

Theorem 2.1

Let $(M^4,g)$ be an oriented, isometrically immersed hypersurface in a space form $(N^5(c),g_N)$ with constant sectional curvature $c$. Then, at every $p\in M$,

Theorems & Definitions (65)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Example 2.4
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8: Extrinsic Chern--Gauss--Bonnet formula
  • Proposition 2.9
  • ...and 55 more