極小曲上面的聯絡估計

Abstract

證明R^3中的2維曲面(x,y,u(x,y))上,存在正交標架使得由Levi-Civita聯絡所定義的一次微分形式的長度平方,可以在局部上有一個上界-K.

In this paper,We prove that if the minimal surface Sigma^2 contains in R^3 is a graph, then there exists an orthonormal frame such that the norm of Levi-Civita connection one form defined on this frame has an upper bound -K which is the Guassian curvature.With this upper bound, we can find another prove on the theorem of Bernstein.