On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910003805290496 |
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| author | Li, Siran Su, Xiangxiang |
| author_facet | Li, Siran Su, Xiangxiang |
| contents | A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric immersion $ι$ of $(\mathcal{M}^n,g)$ into the Euclidean space $\mathbb{R}^{n+k}$ whose extrinsic geometry coincides with $\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\mathfrak{S}$ and $ι$. The best result up to date is $\mathfrak{S} \in L^p$ and $ι\in W^{2,p}$ for $p>n \geq 3$ or $p=n=2$.
In this paper, we extend the above result to $ι\in \mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n \geq 3$. Indeed, $\mathcal{X}$ is the weak Morrey space $L^{p, n-p}_{2,w}$ with arbitrary $p \in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss--Codazzi--Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges -- in particular, Rivière--Struwe's work [Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008)] on harmonic maps in arbitrary dimensions and codimensions -- and compensated compactness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity Li, Siran Su, Xiangxiang Differential Geometry Analysis of PDEs A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric immersion $ι$ of $(\mathcal{M}^n,g)$ into the Euclidean space $\mathbb{R}^{n+k}$ whose extrinsic geometry coincides with $\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\mathfrak{S}$ and $ι$. The best result up to date is $\mathfrak{S} \in L^p$ and $ι\in W^{2,p}$ for $p>n \geq 3$ or $p=n=2$. In this paper, we extend the above result to $ι\in \mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n \geq 3$. Indeed, $\mathcal{X}$ is the weak Morrey space $L^{p, n-p}_{2,w}$ with arbitrary $p \in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss--Codazzi--Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges -- in particular, Rivière--Struwe's work [Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008)] on harmonic maps in arbitrary dimensions and codimensions -- and compensated compactness. |
| title | On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2405.16249 |