Flexibility of Codimension One $C^{1,\theta}$ Isometric Immersions

TL;DR

This paper advances the understanding of $C^{1, heta}$ isometric immersions by establishing new approximation results with improved Hölder exponents for dimensions three and higher, using a refined convex integration method.

Contribution

It demonstrates that any short immersion can be approximated by $C^{1, heta}$ isometric immersions with a higher exponent than previously known, for $n eq 2$, through a novel convex integration scheme.

Findings

Improved Hölder exponent for $n eq 2$ in $C^{1, heta}$ isometric immersions.

Any short immersion can be approximated by $C^{1, heta}$ isometries for $ heta< 1/(1+2(n-1))$.

Enhanced convex integration technique with refined iterative integration by parts.

Abstract

We study the problem of constructing $C^{1,\theta}$ isometric immersions of Riemannian metrics on $n$-dimensional domains into $\mathbb{R}^{n+1}$. While the classical Nash--Kuiper theorem established the flexibility of $C^1$ isometries, subsequent work has extended this to $C^{1,\theta}$ isometries for certain $\theta$, though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by $C^{1,\theta}$ isometric immersions for $\theta< 1/(1+2(n-1))$, improving upon the previously known exponent for $n\geq 3$. The improvement is obtained via a convex integration scheme incorporating a refined iterative integration by parts procedure resting on a detailed structural analysis of error terms and the interaction of multiple frequency scales.