Differential Geometry and Partial Differential Equations

微分几何和偏微分方程

• 批准号: 1105323
• 负责人: Richard Schoen
• 金额: $ 52.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2016-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105323&HistoricalAwards=false
• 关键词: Differential; Geometry; Partial; Equations

This project lies at the interface between Differential Geometry, General Relativity, and Partial Differential Equations. One theme will be the study of manifolds of positive curvature. Specifically the proposer plans to further his study of manifolds of positive isotropic curvature (PIC) using Ricci flow and minimal surface techniques. He also hopes to prove sphere theorems under pointwise pinching conditions with pinching slightly below 1/4. In relativity he plans to study the geometry of static matter solutions hoping to show that matter bodies cannot be separated by convex sets in such solutions. He also plans to study the general Penrose inequality; that is, the conjectured inequality for black hole initial sets with nonzero second fundamental form. He will also pursue a range of questions concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. Finally he plans to continue his study of minimal lagrangian and special lagrangian submanifolds of Kahler-Einstein manifolds. He will attempt to prove a conjecture concerning the invariance of the subgroup of the integral homology of a Calabi-Yau manifold which is generated by minimal lagrangian cycles when one deforms the ambient Calabi-Yau structure.Understanding spaces in terms of their curvature properties is a fundamental idea in mathematics and science. The laws of nature often provide information about curvature properties, and from those we must deduce information about the spaces. A prime example of this is General Relativity where the equations of motion are described by curvature, and from those we must determine concrete properties of the universe. This project deals with a range of questions of this type such as the way in which black holes form from smooth initial data. Furthermore, geometry is important in application areas such as imaging and computer graphics. The research of this project will advance the core geometric ideas which form the basis of such applications.

该项目位于微分几何，一般相对论和部分微分方程之间的界面。一个主题将是研究正曲率的流形。具体而言，提议者计划使用RICCI流动和最小的表面技术进一步研究他对阳性曲率歧管（PIC）的歧管的研究。他还希望在尖锐的捏合条件下证明球体定理，夹紧略低于1/4。 在相对论中，他计划研究静态物质解决方案的几何形状，希望证明物体不能通过此类溶液中的凸集分开。他还计划研究一般的彭罗斯不平等。也就是说，具有非零第二基本形式的黑洞初始集的猜想不等式。他还将提出一系列有关满足自由边界条件以及与特征值问题联系的最小子曼群的问题。最后，他计划继续研究卡勒因斯坦歧管的最低限度拉格朗日和特殊的拉格朗日submanifolds。他将试图证明关于卡拉比野流歧管的整体同源性不变性的猜想，当一个人的曲率属性在弯曲属性方面，它是在数学和科学方面的基本思想，这是由Lagrangian Cycles产生的。自然定律通常提供有关曲率属性的信息，以及我们必须从这些属性的信息中推断出有关空间的信息。一个主要的例子是一般相对论，其中运动方程是通过曲率描述的，我们必须从我们必须确定宇宙的具体特性的方程。该项目处理了一系列此类问题，例如从平滑的初始数据中形成黑洞的方式。 此外，几何形状在成像和计算机图形等应用领域很重要。该项目的研究将推进构成此类应用程序基础的核心几何思想。