Veronese minimizes normal curvatures

TL;DR

This paper establishes an optimal bound on the normal curvatures of closed submanifolds in high-dimensional Euclidean balls, showing that such bounds imply the submanifold is a sphere, with Veronese embeddings as extremal cases.

Contribution

It provides the first optimal curvature bounds that characterize spheres among submanifolds, with Veronese embeddings serving as the boundary cases.

Findings

Optimal normal curvature bounds guarantee the submanifold is a sphere.

Veronese embeddings of projective planes are the extremal cases.

The results extend classical sphere theorems to higher codimension.

Abstract

Suppose M is a closed submanifold in a Euclidean ball of sufficiently large dimension. We give an optimal bound on the normal curvatures, guaranteeing that M is a sphere. The border cases consist of Veronese embeddings of the four projective planes.