On $q$-convex hypersurfaces in Riemannian manifolds

TL;DR

This paper proves topological restrictions on convex hypersurfaces in Riemannian manifolds with positive curvature, showing they are rational homology spheres with finite fundamental groups under certain curvature conditions.

Contribution

It establishes new vanishing theorems for Betti numbers of $q$-convex hypersurfaces under curvature bounds, extending classical results to broader geometric contexts.

Findings

Closed convex hypersurfaces are rational homology spheres.

Finite fundamental groups for hypersurfaces under curvature conditions.

Betti number vanishing theorems for $q$-convex hypersurfaces.

Abstract

We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator.