Isothermal Coordinates of W2,2 Immersions: A Counterexample | Sciact

Isothermal Coordinates of W2,2 Immersions: A Counterexample Научная публикация

Журнал

Proceedings of the Steklov Institute of Mathematics
ISSN: 0081-5438

Вых. Данные

Год: 2024, Том: 327, Номер: 1, Страницы: 251-267 Страниц : 17 DOI: 10.1134/s008154382406018x

Ключевые слова

isothermal coordinates, conformal factor, immersions with square integrable second fundamental form

Авторы

Plotnikov P.I. ¹

Организации

1	Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

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.

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

Поступила в редакцию:	3 мая 2024 г.
Принята к публикации:	12 сент. 2024 г.
Опубликована в печати:	1 апр. 2025 г.

Web of science:	WOS:001457341300005
Scopus:	2-s2.0-105001519319
OpenAlex:	W4409078339

БД	Цитирований
OpenAlex	Нет цитирований
Scopus	Нет цитирований