Differential equations and the geometry of manifolds

微分方程和流形几何

• 批准号: 0503506
• 负责人: Jeff Viaclovsky
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 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: 2005-06-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503506&HistoricalAwards=false
• 关键词: Differential; equations; geometry; manifolds

AbstractAward: DMS-0503506Principal Investigator: Jeff A. ViaclovskyThe first project supported by this award deals with compactnessresults for certain classes of Riemannian metrics in dimensionfour, particularly for anti-self-dual metrics and extremalKaehler metrics. With certain geometric noncollapsingassumptions, the appropriate moduli spaces can be compactified byadding metrics with generalized orbifold singularities. Along-term goal of this project is to use this compactification todefine new smooth invariants of four-manifolds. The secondproject considers conformal deformations of metrics, certainfully nonlinear curvature equations, and relations withRiemannian functions in dimensions three and four. A crucialproblem in this direction is to prescribe a symmetric function ofthe eigenvalues of the Ricci tensor, generalizing the Yamabeproblem.A recurring theme in geometry is the relationship betweentopology, a soft or stretchy version of the shape of a space, andthe possible geometries on that space. In the same way that thefamiliar round geometry on the two-dimensional sphere is moreappealing and geometrically distinctive than any of the nearbygeometries obtained by pushing in or pulling small regions on thesphere, one likes to hope that most manifolds of low dimensionswill carry geometries that are somehow optimal - especiallysymmetric geometries, or minimizers for some sort of total energymeasurement. The projects described above seek to discover suchoptimal or extremal geometries in some important three- andfour-dimensional cases.

摘要奖：DMS-0503506主要研究者：Jeff A. Viaclovsky该奖项支持的第一个项目涉及维数4中某些类黎曼度量的紧致性结果，特别是反自对偶度量和极值Kaehler度量。 在一定的几何非对称性假设下，适当的模空间可以通过增加具有广义轨道奇异性的度量而被紧化。 这个项目的长期目标是利用这种紧化来定义四维流形的新的光滑不变量。第二个项目考虑度量的共形变形，一定的非线性曲率方程，以及与三维和四维黎曼函数的关系。 在这个方向上的一个关键问题是规定Ricci张量的特征值的对称函数，推广Yamabe问题。几何学中一个反复出现的主题是拓扑（空间形状的柔软或拉伸版本）与该空间上可能的几何之间的关系。 同样地，二维球面上熟悉的圆形几何比任何通过推入或拉动球面上的小区域而获得的邻近几何都更吸引人，在几何上更独特，人们希望大多数低维流形将携带某种程度上最优的几何-特别是对称几何，或者某种总能量测量的最小化者。 上述项目试图在一些重要的三维和四维情况下发现这种最优或极值几何。