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

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 $n\ge 3$. 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.