Differential Geometry in the Large

大微分几何

• 批准号: 0503735
• 负责人: Peter Li
• 金额: --
• 依托单位: University of California-Irvine
• 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: 2005-07-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503735&HistoricalAwards=false
• 关键词: Differential; Geometry; Large

AbstractAward: DMS-0503735Principal Investigator: Peter LiThe PI will continue the joint investigation with his coauthor,Jiaping Wang, on rigidity and finiteness properties of certain classesof complete manifolds. Motivated by the theorem of Witten and Yau,the PI and Wang proved some rigidity and finiteness results forn-dimensional complete manifolds with Ricci curvature bounded frombelow and has the spectrum of the Laplacian bounded from below by apositive constant. One of the main projects is to consider a morerelaxed hypothesis than the above mentioned. Instead of assuming thatthe spectrum has a positive lower bound, Li and Wang will considermanifolds on which a weighted Poincare inequality is valid. The moregeneral hypothesis will enlarge the class of manifolds underconsideration to include even Euclidean space of dimension at least3. Other projects related to the wieghted Poincare inequality willalso be considered. In particular, its relationship to Agmon'sdistance in the elliptic setting, the work of Li-Yau in the linearparabolic Schroedinger equation setting, and the Perelman's L-lengthin the non-linear parabolic (Ricci flow) setting.The work of Witten-Yau drew conclusion on a certain physical model ofthe universe and showed that there is no worm-hole. A broader purposeof the project is to seek a substantial generalization of their workin a geometric setting. In particular, this research project willgive further understanding of unbounded (infinite) geometric objects.In this sense, a possible long-term effect is the understanding ofother physical models of the universe. The understanding of geometricstructure will also have broader impact on material science. This lineof investigation will yield direct implications to the theory ofpartial differential equations governing the behavior of many physicalmodels and biological models. It is also related to many engineeringproblems, such as, liquid crystals, heat transfer, and imaging.

摘要奖项：DMS-0503735 首席研究员：Peter Li PI将继续与他的合著者Jiaping Wang联合研究某些类完全流形的刚性和有限性性质。 在Witten和Yau定理的推动下，PI和Wang证明了具有下界里奇曲率的维完备流形的一些刚性和有限性结果，并具有下界为正常数的拉普拉斯谱。 主要项目之一是考虑比上述更宽松的假设。 李和王将考虑加权庞加莱不等式有效的流形，而不是假设谱具有正下界。 更一般的假设将扩大所考虑的流形类别，甚至包括至少 3 维的欧几里得空间。与加权庞加莱不等式相关的其他项目也将被考虑。特别是，它与椭圆设置中的 Agmon 距离、线性抛物线薛定谔方程设置中的 Li-Yau 的工作以及非线性抛物线（里奇流）设置中的佩雷尔曼 L 长度的关系。Witten-Yau 的工作根据宇宙的某种物理模型得出结论，表明不存在虫洞。 该项目更广泛的目的是寻求在几何环境中对他们的工作进行实质性概括。 特别是，该研究项目将进一步了解无界（无限）几何物体。从这个意义上说，可能的长期影响是对宇宙其他物理模型的理解。 对几何结构的理解也将对材料科学产生更广泛的影响。这一研究方向将对控制许多物理模型和生物模型行为的偏微分方程理论产生直接影响。 它还与许多工程问题有关，例如液晶、传热和成像。