论文标题
Quasiconformal Gauss地图和Weingarten Multigraphs的Bernstein问题
Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs
论文作者
论文摘要
我们证明，欧几里得的任何完整的，均匀的椭圆形的温达尔滕表面$ 3 $ - 空间的高斯图像省略了一个敞开的半球，都是圆柱体或平面。这概括了霍夫曼，奥塞尔曼和Schoen的经典定理，以实现恒定的平均曲率表面。特别是，这证明了平面是唯一完整的，椭圆形的Weingarten Multigraphs。我们还表明，这一结果适用于一大批非均匀椭圆形的Weingarten方程。特别是，这解决了该类别椭圆方程的整个图表的肯定性问题。为了获得这些结果，我们证明了平面是唯一具有准正面高斯映射和有界第二基本形式的完整多编码。
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.
