Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension
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| Format: | Preprint |
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2025
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| _version_ | 1866911381418147840 |
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| author | Lewicka, Marta |
| author_facet | Lewicka, Marta |
| contents | A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $ω\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in\mathcal{C}^{r,β}$, an isometric immersion of regularity $\mathcal{C}^{1,α}(\barω,\mathbb{R}^4)$ for any $α<\min\{\frac{r+β}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained byt Källen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Ampère system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension Lewicka, Marta Differential Geometry Analysis of PDEs A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $ω\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in\mathcal{C}^{r,β}$, an isometric immersion of regularity $\mathcal{C}^{1,α}(\barω,\mathbb{R}^4)$ for any $α<\min\{\frac{r+β}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained byt Källen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Ampère system. |
| title | Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2511.16305 |