Immersions with small normal curvature

TL;DR

This paper investigates minimal normal curvature immersions within the unit ball, identifying the lowest possible normal curvature for specific manifolds, and establishing related sphere theorems and existence results.

Contribution

It determines the minimal normal curvature for $S^n imes S^1$ and proves a differentiable sphere theorem along with existence of minimizers in this setting.

Findings

Minimal normal curvature for $S^n imes S^1$ identified.

Proved a differentiable sphere theorem.

Established existence of minimizers.

Abstract

We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem and an existence result for minimizers in this context.