Almost conformally flat hypersurfaces

TL;DR

This paper establishes a lower bound for the Weyl tensor norm in conformally immersed hypersurfaces, determines their homology, and provides conditions for minimal immersions in spheres, extending previous results.

Contribution

It introduces a universal lower bound for the Weyl tensor norm based on Betti numbers and characterizes the homology of almost conformally flat hypersurfaces, extending known immersion results.

Findings

Lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of Betti numbers.

Homology classification of almost conformally flat hypersurfaces.

Necessary conditions for minimal immersions in spheres.

Abstract

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the sphere and extend a result due to Shiohama and Xu \cite{SX} for compact hypersurfaces in any space form.