Fully Nonlinear and Higher Order Equations in Geometry

几何中的完全非线性和高阶方程

• 批准号: 0500538
• 负责人: Matthew Gursky
• 金额: $ 9.6万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500538&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Higher; Order; Equations

Fully Nonlinear and Higher Order Equations in GeometryMatthew J. GurskyUniversity of Notre DameAbstractThe interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world, and differential geometry was developed to understand the geometry of curved spaces--for example, the curvature of the surface of the earth, or the curvature of space by matter predicted by general relativity. In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this proposal involves several such problems from Riemmanian and conformal geometry, whose analysis requires techniques from various fields within mathematical analysis.The specific problems we pose are at the intersection of three fields: higher order elliptic partial differential equations, fully nonlinear equations, and differential geometry. The equations we study are geometric in origin, and given by symmetric functions of the eigenvalues of the Ricci tensor, specifically under conformal deformations of the metric. There is a strong structural analogy between this problem and the more classical problem of prescribing the curvature(s) of a surface in three-dimensional space. To analyze our equations we use techniques from both the theory of elliptic equations and from comparison geometry. For example, to understand on a microscopic level the blow-up behavior of a sequence of solutions, we are confronted with the problem of understanding the tangent cone at infinity of certain asymptotically flat spaces. The geometric consequences of these results are most interesting in low dimensions: for example, we have developed a technique for constructing large families of conformal manifolds which admit metrics with positive Ricci curvature.

几何中的完全非线性和高阶方程Matthew J. Gursky圣母大学摘要几何与分析的相互作用至少可以追溯到十八世纪，但仍然是数学研究的一个重要且高度活跃的领域。经典的几何学是从我们对物理世界的某些性质的渴望中产生的，微分几何是为了理解弯曲空间的几何而发展起来的，例如，地球表面的曲率，或者广义相对论预测的物质空间的曲率。在同样的方式，笛卡尔意识到，平面几何可以研究使用代数，所以微分几何可以研究使用技术分析，特别是微分方程。本文的研究涉及黎曼几何和共形几何中的几个问题，这些问题的分析需要数学分析中各个领域的技术，具体问题涉及高阶椭圆型偏微分方程、完全非线性方程和微分几何三个领域的交叉点。我们研究的方程是几何的起源，并给出了对称函数的特征值的里奇张量，特别是在共形变形的度量。在这个问题和在三维空间中规定曲面曲率的更经典的问题之间有很强的结构相似性。为了分析我们的方程，我们使用的技术从椭圆方程理论和比较几何。 例如，为了在微观水平上理解一系列解的爆破行为，我们面临着理解某些渐近平坦空间在无穷远处的切锥的问题。这些结果的几何后果是最有趣的低维：例如，我们已经开发出一种技术，用于构建大家庭的共形流形承认度量与积极的里奇曲率。