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极小曲面的Bernstein型定理与Gauss映照的值分布
Bernstain's Theorems of a Minimal Surface in R~3 and the Value Distribution of its Gauss Map

忻元龙
Xin Yuanlong

复旦大学数学研究所
(Institute of Mathematics, Fudan Univ.

收稿日期: 1989-12-25
出版日期: 1989-10-15

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摘要  极小曲面的研究已有220多年的历史.一般认为是J.L.Lagrange于1760年开始的.他考虑三维欧氏空间R~3中的光滑函数z=f(x,y)决定的图M.如果M于某区域DR~2中在所有与共边界D上有相同值的曲面中面积最小,那么函数满足下列方程
Abstract:There is a famous theorem due to S.Bernstain which states that the entire solutions to the minimal surface equation in R3 must be linear functions. Since then variant generalizations to Bernstain's theorem have been developed. One of the beautiful viewpoint to this problem is the value distribution of the Gauss map of a minimal surface in Euclidean space.This is an expository paper. Following the historical development, the main theorems in this direction have been described in certain detail,such as Osserman's theorem, Xavier's theorem and Fujimoto's theorem. For completeness the paper begins with the classical Weierstrass representation of a minimal surface in R3.
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