Weingarten多图的拟共形高斯映射和Bernstein问题

IF 1.7 1区 数学 Q1 MATHEMATICS
Isabel Fernández, José A. Gálvez, Pablo Mira
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引用次数: 4

摘要

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。
Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs

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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来源期刊
CiteScore
3.20
自引率
0.00%
发文量
35
审稿时长
24 months
期刊介绍: The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.
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