Isometric Immersions with Controlled Curvatures

TL;DR

This paper demonstrates how to approximate strictly short maps between Riemannian manifolds with smooth isometric immersions that have controlled curvature, as the approximation parameter approaches zero.

Contribution

It introduces a method to approximate short maps by smooth isometric immersions with bounded curvature in high codimension.

Findings

Approximate strictly short maps with smooth isometric immersions.

Curvature of approximations is controlled and tends to infinity as delta approaches zero.

Method applicable for high codimension embeddings.

Abstract

We $\delta$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_\delta:X^m\to Y^N$ with curvatures $curv(f_\delta) < \frac {\sqrt 3}{\delta}$, for $\delta\to 0$