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: 1547878
• 负责人: Yi Wang
• 金额: $ 1.12万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1547878&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.

主要研究者（PI）将研究各种几何不等式及其相应的非线性偏微分方程的背景下，共形几何和几何的子流形。一个主题是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还建议研究嵌入子流形的不同阶的曲率不等式。这些不等式，最初被认为是在凸几何的背景下，最近被称为是有效的一个非常大的一类非凸域。PI的调查努力寻找这种不平等的全部普遍性。同时，她也致力于发展新的技能，以理解相应的完全非线性椭圆偏微分方程，自然出现在问题中。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集中在共形不变量上，从物理学家的角度来看，共形不变量是一个重要的机制，并在数学物理的基本原理中找到了应用。推广凸几何的经典结果的研究计划将提高我们对已建立理论的刚性的理解，并将揭示其更大的应用范围。