A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913836227887104 |
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| author | Cao, Wentao Hirsch, Jonas Inauen, Dominik |
| author_facet | Cao, Wentao Hirsch, Jonas Inauen, Dominik |
| contents | Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any Hölder exponent $α<(1+n^2-n)^{-1}$, we construct a $C^{1, α}$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of $C^{1, α}$ isometric immersions beyond Borisov's exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1/3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent Cao, Wentao Hirsch, Jonas Inauen, Dominik Analysis of PDEs Differential Geometry 58D10, 53C21, 57N35 Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any Hölder exponent $α<(1+n^2-n)^{-1}$, we construct a $C^{1, α}$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of $C^{1, α}$ isometric immersions beyond Borisov's exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1/3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors. |
| title | A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent |
| topic | Analysis of PDEs Differential Geometry 58D10, 53C21, 57N35 |
| url | https://arxiv.org/abs/2503.13867 |