A Nash-Kuiper theorem for isometric immersions in a high codimension

TL;DR

This paper advances the regularity of isometric immersions of Riemannian manifolds into Euclidean spaces with high codimension, improving previous results and providing explicit estimates relating initial metric error to immersion steepness.

Contribution

It improves the Hölder regularity of isometric immersions in high codimension and provides explicit $C^{1}$ estimates linking initial metric error to the steepness of the immersion.

Findings

Achieves $C^{1, heta}$ regularity for all $ heta ext{ in }(0,1/n)$ in odd dimensions.

Achieves $C^{1, heta}$ regularity for all $ heta ext{ in }(0,1/(n+1))$ in even dimensions.

Provides explicit $C^{1}$ estimates showing larger initial metric error leads to steeper immersions.

Abstract

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into $2n$-dimensional Euclidean space can be uniformly approximated by $C^{1,\theta}$ isometric immersions with any $\theta\in(0,1/(n+2))$ in dimensions $n\geq3$. In this paper, we improve the H\"{o}lder regularity of the constructed isometric immersions in the local setting, achieving $C^{1,\theta}$ for all $\theta\in(0,1/n)$ in odd dimensions and all $\theta\in(0,1/(n+1))$ in even dimensions. Moreover, we also establish explicit $C^{1}$ estimates for the isometric immersions, which indicate that the larger the initial metric error is, the greater the $C^{1}$ norms of the resulting isometric maps become, meaning that their slope become steeper.