Nonlinear Geo metric Equations of Monge-Ampere Type and Canonical Metrics

Monge-Ampere型非线性几何方程与正则度量

• 批准号: 0808631
• 负责人: Jian Song
• 金额: $ 8.23万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-10 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808631&HistoricalAwards=false
• 关键词: Nonlinear; Geo; metric; Equations; Monge

AbstractAward: DMS-0604805Principal Investigator: Jian SongThis proposal concerns existence and regularity problems of thenonlinear Monge-Ampere type equations from geometry and physics.The problem of finding canonical metrics on a compact Kahlermanifold has been the subject of intense study over the last fewdecades. In his solution to Calabi's conjecture, Yau proved theexistence of a Kahler-Einstein metric on compact Kahler manifoldswith vanishing or negative first Chern class. An alternativeproof of Calabi's conjecture is given by Cao using theKahler-Ricci flow. However, most projective algebraic varietiesdo not have definite or trivial first Chern classes. RecentlyPerelman has made major breakthrough in Hamilton's program as anapproach to the Poincare conjecture and Thurston's geometrizationconjecture. The principal investigator proposes to study thecanonical metrics on the canonical models of projective varietiesof positive Kodaira dimension by applying the Kahler-Ricciflow. Such canonical metrics are constructed by the deformationof the Kaher-Ricci flow on minimal projective surfaces ofpositive Kodaira dimension. These generalized Kahler-Einsteinmetrics can be considered as an analytic version of the abundanceconjecture in algebraic geometry and will also lead to newadvances in the understanding and application of the Ricciflow. In Donaldson's far reaching program, the geometry of theinfinite dimensional symmetric space of Kahler metrics in a fixedclass is related to the existence and uniqueness of constantscalar curvature Kahler metrics. The principal investigator alsointends to study the uniform approximation problem of theMonge-Ampere geodesics in infinite dimensional symmetric space bythose in the finite dimensional Bergman spaces on toricvarieties. The precise understanding of this problem will givenew insight into the conjecture proposed by Yau between therelation of constant scalar curvature Kahler metrics and certainstability in the sense of geometric invariant theory. Theprincipal investigator will also apply the moment map point ofview and study various geometric flows arising from Kahlergeometry as well as symplectic geometry proposed byDonaldson. The first is the J-flow, which is the gradient flow offunctional related to the Mabuchi energy. The second is a momentmap flow in a hyperkahler four manifold. The principalinvestigator intends to study the question of convergence andsingularities os such parabolic flows.Since the discovery of the general relativity, geometric analysishas become crucial to both mathematicians and physicists.Problems in this proposal arise naturally from our attempts tounderstand nonlinear differential equations from geometry andphysics. The solutions to these problems will contribute tovarious fields of sciences such as physics and cosmology in thedeep understanding of our universe. The method of analyzing thesingularities of nonlinear equations will have wide applicationsin engineering and economics.

摘要获奖：DMS-0604805主要研究员：宋健该提案涉及几何和物理学中非线性Monge-Ampere型方程的存在性和正则性问题。在紧凑卡勒流形上寻找规范度量的问题一直是过去几十年来深入研究的主题。在他对卡拉比猜想的解答中，丘证明了在具有消失或负第一陈级的紧卡勒流形上卡勒-爱因斯坦度量的存在性。曹使用卡勒-里奇流给出了卡拉比猜想的另一种证明。然而，大多数射影代数簇没有确定的或平凡的第一陈类。 最近，佩雷尔曼在汉密尔顿方案上取得了重大突破，作为庞加莱猜想和瑟斯顿几何化猜想的一种方法。主要研究者提出应用 Kahler-Ricciflow 来研究正 Kodaira 维数投影变体规范模型的规范度量。这种规范度量是通过 Kaher-Ricci 流在正 Kodaira 维数的最小投影面上的变形来构造的。这些广义的卡勒-爱因斯坦度量可以被认为是代数几何中丰度猜想的解析版本，也将导致对 Ricciflow 的理解和应用的新进展。在唐纳森影响深远的计划中，固定类卡勒度量的无限维对称空间的几何形状与常标量曲率卡勒度量的存在性和唯一性有关。课题负责人还打算研究环面簇上有限维Bergman空间中Monge-Ampere测地线在无限维对称空间中的一致逼近问题。对这个问题的准确理解，将使丘提出的常标量曲率卡勒度量与几何不变量理论意义上的确定稳定性之间的关系的猜想得到新的认识。首席研究员还将应用矩图的观点，研究由卡勒几何学以及唐纳森提出的辛几何学产生的各种几何流。第一个是 J 流，它是与 Mabuchi 能量相关的泛函梯度流。第二个是 hyperkahler 四流形中的矩图流。首席研究员打算研究此类抛物线流的收敛性和奇点问题。自从发现广义相对论以来，几何分析对于数学家和物理学家都变得至关重要。这个提议中的问题自然而然地出现在我们试图从几何和物理学中理解非线性微分方程的过程中。这些问题的解决将有助于物理学和宇宙学等各个科学领域对我们宇宙的深入理解。分析非线性方程奇异性的方法将在工程和经济学中具有广泛的应用。