Bernstein定理的短证明：基于Omori-Yau最大值原理
A short proof of Bernstein's theorem via the Omori--Yau maximum principle
浏览论文内容
中文总结 AI 辅助
本文利用Omori-Yau最大值原理，通过构造显式函数并应用逐点微分不等式，简洁地证明了三维空间中完整极小图必为平面的Bernstein定理。
中文摘要 AI 辅助
我们给出Bernstein定理的一个简短且自包含的证明：$R^3$中的每一个完整极小图都是一个平面。唯一的全局工具是Omori-Yau最大值原理，该原理在具有诱导度量的完整极小图上成立。我们恰好将其应用一次，作用于一个由曲面的角度函数和主曲率构造的显式函数$F$。关于$F$的逐点微分不等式，结合Omori-Yau原理，迫使$F$处处达到其最小值，从而得出曲面的平面性，进而得到Bernstein定理。
英文摘要
We give a short, self-contained proof of Bernstein's theorem: every entire minimal graph in $R^3$ is a plane. The only global tool is the Omori-Yau maximum principle, which holds on entire minimal graphs with their induced metric. We apply it exactly once to a single explicit function $F$ built from the surface's angle function and principal curvatures. A pointwise differential inequality for $F$, combined with the Omori-Yau principle, forces it to attain its minimum value everywhere, which yields the planarity of the surface and hence Bernstein's theorem. As a by-product of our approach, we show that the same method, with a different function and with some extra work, also proves in a simple and self-contained way the generalization of Bernstein's theorem given by Osserman that a complete minimal surface in $R^3$ whose normals omit a neighbourhood of some direction must be a plane.
发表机构
- Universidad de Murcia（穆尔西亚大学）
机构由 AI 辅助整理，请以论文原文为准。
