Rigidity of riemannian manifolds containing an equator

TL;DR

This paper proves a rigidity theorem for Riemannian manifolds with curvature bounds containing specific minimal spheres, showing they must have constant sectional curvature 1, using ancient mean curvature flows.

Contribution

It establishes a new rigidity result for manifolds with minimal spheres of a given area and index, employing ancient mean curvature flow techniques.

Findings

Manifolds with a minimal 2-sphere of area 4π and index ≥ n-2 have constant sectional curvature 1.

The proof involves constructing ancient mean curvature flows from minimal submanifolds.

Results extend to rigidity of Simon-Smith minimal spheres.

Abstract

In this paper, we prove that a Riemannian $n$-manifold $M$ with sectional curvature bounded above by $1$ that contains a minimal $2$-sphere of area $4\pi$ which has index at least $n-2$ has constant sectional curvature $1$. The proof uses the construction of ancient mean curvature flows that flow out of a minimal submanifold. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres.