Weingarten重图的拟共形Gauss映射和Bernstein问题
Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs
摘要：
证明了欧氏$3$ -空间中任何完备的、一致椭圆的Weingarten曲面，如果其高斯映射像省略了一个开半球，则该曲面是柱面或平面.这推广了Hoffman，Osserman和Schoen关于常中曲率曲面的经典定理。特别地，这证明了平面是唯一完全的、一致椭圆的Weingarten重图。我们还表明，这一结果适用于一个大类的非一致椭圆型Weingarten方程。特别是，这解决了肯定的伯恩斯坦问题，整个图形的一类椭圆型方程。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型的完全重图。
英文摘要：
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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