Ricci curvature and the structure of manifolds

里奇曲率和流形结构

• 批准号: 1005484
• 负责人: Ovidiu Munteanu
• 金额: $ 12.56万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2012-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005484&HistoricalAwards=false
• 关键词: Ricci; curvature; structure; manifolds

The goal of this project is to obtain further understanding of the structure of complete noncompact manifolds. The general theme is that understanding the solutions of some partial differential equations in terms of curvature can be used to obtain more information about the geometry and topology of the manifold. In this context, the PI will study bounds on the bottom of spectrum of the Laplacian in terms of the Ricci curvature and use the spectral information to study the structure of the manifold. The PI will also study geometric and analytic properties of complete gradient Ricci solitons, aiming at a better understanding of the singularities of the Ricci flow.The proposed project will lead to a better understanding of the shape and structure of infinite geometric objects. These objects, called manifolds, appear frequently in science because physical laws can be modeled by certain partial differential equations on manifolds. Understanding the structure of manifolds is important in many fields, such as black holes and worm holes in physics, the structure of molecules in chemistry or the motion of liquid crystal in engineering. Partial differential equations have been a fundamental tool in geometry, for example the Ricci flow has been used to classify the shape of all three dimensional manifolds. Further advances in the study of geometric partial differential equations on manifolds would lead to new discoveries applicable to many areas of mathematics and science.

该项目的目的是进一步了解完整的非伴随歧管的结构。一般的主题是，可以使用曲率方面了解某些部分微分方程的解决方案，以获取有关歧管的几何和拓扑的更多信息。在这种情况下，PI将根据RICCI曲率研究Laplacian频谱底部的界限，并使用光谱信息研究歧管的结构。 PI还将研究完整梯度RICCI孤子的几何和分析特性，以更好地理解RICCI流动的奇异性。拟议的项目将更好地理解无限几何对象的形状和结构。这些对象（称为歧管）经常出现在科学中，因为物理定律可以通过歧管上的某些部分微分方程来建模。在许多领域中了解歧管的结构很重要，例如物理学中的黑洞和蠕虫孔，化学分子的结构或工程中液晶的运动。部分微分方程一直是几何学的基本工具，例如，RICCI流已用于对所有三个维歧管的形状进行分类。在流形的几何部分微分方程的研究中，进一步的进步将导致适用于许多数学和科学领域的新发现。