Surfaces With Constant Mean Curvature
Surfaces With Constant Mean Curvature
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DOI：
10.1090/mmono/221
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发表时间：
2003-10
期刊：
影响因子：
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通讯作者：
K. Kenmotsu
中科院分区：
文献类型：
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作者：
K. Kenmotsu
曲面的平均曲率是衡量曲面在三维空间中如何弯曲的外部参数。每个点的平均曲率为零的曲面是极小曲面，众所周知，这样的曲面是肥皂膜的模型。关于极小曲面，有一个丰富而著名的理论。当我们试图在不改变封闭曲面的体积的情况下最小化闭合曲面的面积时，就得到了平均曲率为常数但不为零的曲面。具有恒定平均曲率的曲面的一个简单例子是球体。常曲率环面提供了一个不平凡的例子，它在1984年的发现给了研究这类曲面的强大动力。后来，人们用各种分析、微分几何和微分方程的方法发现了许多常平均曲率曲面的例子。现在越来越清楚的是，有一个关于常平均曲率曲面的丰富的理论。在这本书中，作者提供了许多常平均曲率曲面的例子和研究它们的技术。许多精细渲染的图形说明了结果，并允许读者可视化和更好地理解这些美丽的对象。这本书适合于对分析和微分几何感兴趣的高级本科生、研究生和研究数学家。
The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. There is a rich and well-known theory of minimal surfaces. A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses. An easy example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in 1984 gave a powerful incentive for studying such surfaces. Later, many examples of constant mean curvature surfaces were discovered using various methods of analysis, differential geometry, and differential equations. It is now becoming clear that there is a rich theory of surfaces of constant mean curvature. In this book, the author presents numerous examples of constant mean curvature surfaces and techniques for studying them. Many finely rendered figures illustrate the results and allow the reader to visualize and better understand these beautiful objects. The book is suitable for advanced undergraduates, graduate students and research mathematicians interested in analysis and differential geometry.