Isothermal Coordinates of W2,2 Immersions: A Counterexample Full article
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Proceedings of the Steklov Institute of Mathematics
ISSN: 0081-5438 |
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| Output data | Year: 2024, Volume: 327, Number: 1, Pages: 251-267 Pages count : 17 DOI: 10.1134/s008154382406018x | ||
| Tags | isothermal coordinates, conformal factor, immersions with square integrable second fundamental form | ||
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Abstract:
We study isothermal coordinates for the immersions of two-dimensional manifolds into Euclidean space and consider a class of immersions with square integrable second fundamental form, which are also called immersions. It is a widespread statement in the literature that such immersions have isothermal coordinates with uniformly bounded logarithm of the conformal factor. We show that this is not the case: We give an example of an immersion of the two-dimensional sphere into three-dimensional Euclidean space for which the logarithm of the conformal factor is unbounded. The reason is that immersions with square integrable second fundamental form do not admit a smooth approximation. In other words, they do not satisfy the hypotheses of the Toro theorem on bi-Lipschitz conformal coordinates.
Cite:
Plotnikov P.I.
Isothermal Coordinates of W2,2 Immersions: A Counterexample
Proceedings of the Steklov Institute of Mathematics. 2024. V.327. N1. P.251-267. DOI: 10.1134/s008154382406018x WOS Scopus OpenAlex
Isothermal Coordinates of W2,2 Immersions: A Counterexample
Proceedings of the Steklov Institute of Mathematics. 2024. V.327. N1. P.251-267. DOI: 10.1134/s008154382406018x WOS Scopus OpenAlex
Dates:
| Submitted: | May 3, 2024 |
| Accepted: | Sep 12, 2024 |
| Published print: | Apr 1, 2025 |
Identifiers:
| Web of science: | WOS:001457341300005 |
| Scopus: | 2-s2.0-105001519319 |
| OpenAlex: | W4409078339 |
Citing:
| DB | Citing |
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| OpenAlex | Нет цитирований |
| Scopus | Нет цитирований |