A Nash-Kuiper theorem for $C^{1,\frac{1}{5}-\delta}$ immersions of surfaces in $3$ dimensions

TL;DR

This paper proves that any short $C^1$ immersion of a 2D disk with a given Riemannian metric into 3D space can be approximated by $C^{1,\alpha}$ isometric immersions for any \( \alpha < \frac{1}{5} \), advancing the regularity of such approximations.

Contribution

It extends the Nash-Kuiper theorem by establishing $C^{1,\alpha}$ isometric immersions with \( \alpha < \frac{1}{5} \) for surfaces in 3D, improving previous regularity results.

Findings

Achieved $C^{1,\alpha}$ isometric immersions for \( \alpha < \frac{1}{5} \).

Improved regularity bounds over previous work by Borisov and others.

Provided a constructive approximation method for short immersions.

Abstract

We prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensional disk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb R^3$ can be uniformly approximated with $C^{1,\alpha}$ isometric immersions for any $\alpha < \frac{1}{5}$. This statement improves previous results by Yu.F. Borisov and of a joint paper of the first and third author with S. Conti.