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标题: "Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces"
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## Mathematics > Differential Geometry

## Title:Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces

Authors:[Gilad Derfner](https://arxiv.org/search/math?searchtype=author&query=Derfner,+G), [Raz Kupferman](https://arxiv.org/search/math?searchtype=author&query=Kupferman,+R), [Cy Maor](https://arxiv.org/search/math?searchtype=author&query=Maor,+C)

[View PDF](https://arxiv.org/pdf/2608.10550) [HTML (experimental)](https://arxiv.org/html/2608.10550v1)

> Abstract:The space of $W^{2,2}$ -isometric immersions of a surface into $\mathbb{R}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy --- the $L^2$ -norm of the second fundamental form --- is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where "too many" asymptotic directions meet --- or, equivalently, where the index of the Gauss map is not $-1$ --- as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$ -isometric immersion.  
> We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$ -convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$ -isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$ -isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$ -isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.

| Subjects: | Differential Geometry (math.DG); Analysis of PDEs (math.AP) |
| --- | --- |
| MSC classes: | 74B20, 74K20, 58D10, 35L71 |
| Cite as: | [arXiv:2608.10550](https://arxiv.org/abs/2608.10550) \[math.DG\] |
|  | (or [arXiv:2608.10550v1](https://arxiv.org/abs/2608.10550v1) \[math.DG\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.10550](https://doi.org/10.48550/arXiv.2608.10550) |

## Submission history

From: Gilad Derfner \[[view email](https://arxiv.org/show-email/4a63d586/2608.10550)\]  
**\[v1\]** Tue, 11 Aug 2026 06:39:54 UTC (745 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.10550) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
