Differentiable sphere theorems for compact submanifolds

Juan Li, Hongwei Xu and Entao Zhao

Vol. 343 (2026), No. 2, 427–452

DOI: 10.2140/pjm.2026.343.427

Abstract

We investigate the differentiable structure on compact simply connected submanifolds in Riemannian manifolds under curvature pinching conditions. We prove a sharp differentiable sphere theorem that an $n$-dimensional compact simply connected submanifold $M^n$ $(n\ge 5,n\ne 7,8)$ in the sphere ${\mathbb{𝕊}}^N(1∕\sqrt{c})$ $(c>0)$ with the second fundamental form $A$ and the mean curvature vector $H$ satisfying $|A{|}^2\le 4c+\frac{|H{|}^2}{n-2}$ is diffeomorphic to the standard sphere. The similar differentiable sphere theorem also holds for compact simply connected submanifolds in the space form ${\mathbb{𝔽}}^N(c)$ with $c\le 0$.

Keywords

differentiable sphere theorem, compact submanifold, curvature pinching, mean curvature flow, homology vanishing theorem

Mathematical Subject Classification

Primary: 53C40

Milestones

Received: 5 September 2025

Revised: 31 January 2026

Accepted: 23 March 2026

Published: 22 May 2026