Variational Problems in Riemannian Geometry

黎曼几何中的变分问题

• 批准号: EP/G039593/1
• 负责人: Kevin Houston
• 金额: $ 1.26万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG039593%2F1
• 关键词: Variational; Problems; Riemannian; Geometry

In differential geometry, many of the particularly interesting maps and geometric structures arise in the study of extrema of variational problems. Solutions of such a problem, in general, satisfy an elliptic partial differential equation and can often be obtained by deformation under a geometric flow. Particular examples include harmonic maps, Willmore surfaces and metrics with constant scalar curvature (in a given conformal class). These are critical points of, respectively, the energy, the Willmore and the Yamabe functional. Many other geometric problems, defined originally in some other way, turn out to admit a variational formulation, which then sheds new light on the question. This occurred, for example, with such highly relevant topics as manifolds with $G_2$ holonomy and with special Lagrangian submanifolds. Hence, geometric variational problems interact with many other areas of mathematics, and have strong relevance to integrable systems, mathematical physics and PDE.

在微分几何中，许多特别有趣的映射和几何结构出现在变分问题极值的研究中。这样的问题的解决方案，在一般情况下，满足椭圆型偏微分方程，往往可以通过变形下的几何流。特别的例子包括调和映射、Willmore曲面和具有常数标量曲率的度量（在给定的共形类中）。这些分别是能量泛函、Willmore泛函和Yamabe泛函的临界点。许多其他的几何问题，原来定义在一些其他的方式，原来承认一个变分制定，然后揭示了新的光的问题。这种情况发生，例如，与这些高度相关的主题作为流形与$G_2$ holonomy和特殊的拉格朗日子流形。因此，几何变分问题与许多其他数学领域相互作用，并与可积系统，数学物理和偏微分方程有很强的相关性。