Homotopical and topological rigidity of hypersurfaces of spherical space forms

TL;DR

This paper establishes topological rigidity results for hypersurfaces in spherical space forms, characterizes their moduli spaces via homotopy equivalences, and extends classical theorems with new bounds and constructions.

Contribution

It extends rigidity theorems for hypersurfaces in spherical space forms, providing bounds on fundamental groups and constructing homotopy equivalences for spaces of such hypersurfaces.

Findings

Universal cover of hypersurfaces is diffeomorphic to S^n under certain curvature conditions.

Constructs a weak homotopy equivalence between hypersurface spaces and twisted products involving SO(n+2).

Establishes a homotopy equivalence for hypersurfaces with Gauss maps in convex balls.

Abstract

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the principal curvatures of such a hypersurface $ f \colon N^n \to M^{n+1} $ $( n\ge 2 )$, it asserts that the universal cover of $ N $ must be diffeomorphic to the $ n $-sphere $ {S}^n $, and provides an upper bound for the order of the fundamental group of $ N $ in terms of that of $ M $. In particular, if $ M = {S}^{n+1} $, then $ N $ is diffeomorphic to $ {S}^n $ and either $ f $ or its Gauss map is an embedding. Let $ J \subset (0,\pi) $ be any interval of length less than $ \frac{\pi}{2} $. The second main result constructs a weak homotopy equivalence between the space of all complete immersed hypersurfaces of $ M $ with principal curvatures in $ \cot (J) $ and the twisted product of $ \big( \Gamma\backslash \mathrm{SO}_{n+2} \big) $ and $ \mathrm{Diff}_+({S}^n) $ by $ \mathrm{SO}_{n+1} $, where $ \Gamma $ is the fundamental group of $ M $ regarded as a subgroup of $ \mathrm{SO}_{n+2} $. Relying on another rigidity criterion due to Wang/Xia, the third main result constructs a homotopy equivalence between the space of all complete immersed hypersurfaces of $ S^{n+1} $ whose Gauss maps have image contained in a strictly convex ball and the same twisted product, with $ \Gamma $ the trivial group.