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

摘要/ abstract摘要：项目负责人：宋健，从几何和物理的角度研究非线性Monge-Ampere型方程的存在性和正则性问题。在紧化卡勒流形上寻找正则度量的问题在过去几十年中一直是研究的热点。在他对Calabi猜想的解中，Yau证明了具有消失或负第一陈氏类的卡勒流形上的卡勒-爱因斯坦度规的存在性。Cao利用kahler - ricci流给出了Calabi猜想的另一种证明。然而，大多数射影代数变种没有确定或平凡的第一陈氏类。最近，佩雷尔曼在汉密尔顿的程序中取得了重大突破，作为庞加莱猜想和瑟斯顿几何猜想的一种方法。主要研究者提出应用Kahler-Ricciflow研究正Kodaira维的射影变异的典型模型的规范度量。这种规范度量是由Kaher-Ricci流在正Kodaira维的最小射影表面上的变形构造的。这些广义的kahler - einstein度量可以看作是代数几何中丰度猜想的解析版本，也将在理解和应用里奇流方面取得新的进展。在Donaldson的深远规划中，固定类中无限维Kahler度量对称空间的几何关系到常标量曲率Kahler度量的存在唯一性。研究了无限维对称空间中monge - ampere测地线在环型上有限维Bergman空间中的一致逼近问题。对这一问题的准确理解将对丘德威在几何不变理论意义上提出的常数标量曲率Kahler度规与确定性稳定性之间的关系的猜想有新的认识。首席研究员还将应用矩图的观点，研究由Kahlergeometry以及由donaldson提出的辛几何引起的各种几何流动。第一个是j流，它是与Mabuchi能量相关的梯度流函数。第二种是超kahler四流形中的动量映射流。主要研究者打算研究这类抛物流的收敛性和奇异性问题。自从广义相对论被发现以来，几何分析对数学家和物理学家来说都变得至关重要。由于我们试图从几何和物理上理解非线性微分方程，这个建议中的问题自然产生了。这些问题的解决方案将有助于科学的各个领域，如物理学和宇宙学，在我们对宇宙的深刻理解。分析非线性方程奇异性的方法将在工程和经济上有广泛的应用。

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.