Manifolds with positive curvature operator and strictly convex bounday

TL;DR

This paper generalizes a previous result by proving that compact manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to Euclidean disks in any dimension.

Contribution

It extends the classification of manifolds with positive curvature and convex boundary from 3D to higher dimensions.

Findings

Manifolds with positive curvature operator and convex boundary are diffeomorphic to Euclidean disks.

The higher-dimensional generalization of the 3D result is established.

Provides a broader understanding of the topology of positively curved manifolds.

Abstract

In [AMW], it is proved that if a compact $3$-manifold has positive Ricci curvature and strictly convex boundary, then this manifold is diffeomorphic to the standard $3$-dimensional Euclidean disk. In this paper, we prove its higher-dimensional generalization: if a compact $n$-manifold has positive curvature operator and strictly convex boundary, then this manifold is diffeomorphic to the standard $n$-dimensional Euclidean disk.