Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-29 DOI:10.1353/ajm.2023.a913297

Isabel Fernández, José A. Gálvez, Pablo Mira

引用次数: 4

Abstract

Abstract:

We prove that any complete, uniformly elliptic Weingarten surface in Euclidean $3$-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. To obtain these results, we prove that planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.

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Weingarten多图的拟共形高斯映射和Bernstein问题

摘要:我们证明了欧几里得$3$-空间中任何完备的均匀椭圆Weingarten曲面，其高斯映射图像省略了一个开半球，即为圆柱体或平面。这推广了Hoffman, Osserman和Schoen关于常平均曲率曲面的经典定理。特别地，这证明了平面是唯一完备的一致椭圆Weingarten多图。我们还证明了这一结果适用于一大类非一致椭圆Weingarten方程。特别地，这肯定地解决了该类椭圆方程的整个图的Bernstein问题。为了得到这些结果，我们证明了平面是唯一具有拟共形高斯映射和有界第二基本形式的完备多图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.