Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914379993186304 |
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| author | Inauen, Dominik |
| author_facet | Inauen, Dominik |
| contents | We study the problem of constructing $C^{1,θ}$ 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 $C^{1,θ}$ isometries for certain $θ$, though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by $C^{1,θ}$ isometric immersions for $θ< 1/(1+2(n-1))$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08382 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions Inauen, Dominik Analysis of PDEs Differential Geometry 58D10, 53C21, 57N35 We study the problem of constructing $C^{1,θ}$ 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 $C^{1,θ}$ isometries for certain $θ$, though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by $C^{1,θ}$ isometric immersions for $θ< 1/(1+2(n-1))$, 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. |
| title | Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions |
| topic | Analysis of PDEs Differential Geometry 58D10, 53C21, 57N35 |
| url | https://arxiv.org/abs/2603.08382 |