Geometric Inequalities and Fully Nonlinear Elliptic Equations

几何不等式和完全非线性椭圆方程

• 批准号: 1205350
• 负责人: Yi Wang
• 金额: $ 12.02万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205350&HistoricalAwards=false
• 关键词: Geometric; Inequalities; Fully; Nonlinear; Elliptic

The Principal Investigator (PI) will study various geometric inequalities and their corresponding nonlinear partial differential equations in the context of conformal geometry and the geometry of submanifolds. One theme plans to investigate the effect of the higher order curvatures on the validity of the isoperimetric inequality. In particular, she intends to quantitatively analyze the interaction between the Q-curvature and the isoperimetric constant. The PI also proposes to study curvature inequalities of different orders for embedded submanifolds. These inequalities, originally considered in the context of convex geometry, have recently known to be valid on a very large class of non-convex domains. The PI's investigation strives to look for the full generality of such inequalities. In the meanwhile, she also aims to develop new skills to understand the corresponding fully nonlinear elliptic partial differential equations that arise naturally in the problem.In the proposed study, the PI's research interest lies at the intersection of conformal geometry, the geometry of submanifolds and partial differential equations (PDEs). The study of geometric inequalities and geometric PDEs focuses on conformal invariants, which form an important machinery from physicists' point of view and have found applications to fundamental principles in mathematical physics. The research project to generalize classical results of convex geometry will improve our understanding on the rigidity of the established theory and will shed light on a greatly larger scope of its application.

首席研究员（PI）将在共形几何和子流形几何的背景下研究各种几何不等式及其相应的非线性偏微分方程。其中一个主题计划研究高阶曲率对等周不等式有效性的影响。特别是，她打算定量分析 Q 曲率和等周常数之间的相互作用。 PI 还建议研究嵌入式子流形的不同阶曲率不等式。这些不等式最初是在凸几何的背景下考虑的，最近发现它们在很大一类非凸域上是有效的。 PI 的调查力求找出此类不平等现象的全部普遍性。与此同时，她还致力于开发新技能来理解问题中自然产生的相应的完全非线性椭圆偏微分方程。在拟议的研究中，PI的研究兴趣在于共形几何、子流形几何和偏微分方程（PDE）的交叉点。几何不等式和几何偏微分方程的研究重点是共形不变量，从物理学家的角度来看，它构成了一种重要的机制，并已在数学物理的基本原理中得到应用。推广凸几何经典结果的研究项目将提高我们对已建立理论的刚性的理解，并将揭示其更大范围的应用。