Mean Curvature, the Laplacian, and Soap Bubbles
Mean Curvature, the Laplacian, and Soap Bubbles
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平均曲率、拉普拉斯算子和肥皂泡
DOI：
10.1080/00029890.1982.11995407
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
R. Reilly
中科院分区：
文献类型：
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作者：
R. Reilly
1. 介绍。曲面平均曲率的概念是经典微分几何中最有成果的概念之一，也是许多研究论文的主题。在所有标准教科书中（例如，参见[2]，[6]），它都被处理，但通常是作为一个特殊的主题嵌入。主要讨论基本形式、科达齐方程和高斯映射。（希尔伯特（Hilbert）和科恩-沃森（Cohn-Vossen）的经典论述是个例外。）在本文中，我以一种简单的方式发展了平均曲率的概念，只使用微积分中众所周知的事实，避免了微分几何中复杂的术语。我指出了这个概念与拉普拉斯算子~]= 1 （a 2; axj）的关系，并利用这种关系证明了AD Aleksandrov著名的“肥皂泡定理”。
1. Introduction. The notion of the mean curvature of a surface is one of the most fruitful in classical differential geometry and is the subject of many research papers. It is treated in all the standard textbooks (see, for example,[2],[6]), but usually as a special topic embedd. ed in a general discussion of fundamental forms, Codazzi equations, and Gauss maps.(An exception is the classic exposition by Hilbert and Cohn-Vossen [4].) In this paper I develop the concept of mean curvature in a simple manner, using only well-known facts from calculus and avoiding the complicated terminology of differential geometry. I point out how this concept is related to the Laplace operator~]= 1 (a 2; axj) and I exploit this relation to prove the well-known" Soap-Bubble Theorem" of AD Aleksandrov.