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

AbstractAward：DMS-0604805主要研究者：宋健本论文主要研究几何和物理中的非线性Monge-Ampere型方程的存在性和正则性问题，在紧致Kahler流形上寻找正则度量的问题是近几十年来研究的热点。在他对卡拉比猜想的解答中，丘证明了在第一陈类为零或为负的紧致Kahler流形上存在Kahler-Einstein度量。Cao利用Kahler-Ricci流给出了Calabi猜想的另一种证明。然而，大多数射影代数簇没有确定的或平凡的第一陈类。 Perelman最近在汉密尔顿方案上取得了重大突破，作为Poincare猜想和Thurston几何化猜想的一种方法。本文主要研究正科代拉维投影变量的典范模型的典范度量.这样的正则度量是由正科代拉维极小射影曲面上的Kaher-Ricci流的变形构造的。这些推广的Kahler-Einstein度量可以看作是代数几何中丰度猜想的一个解析版本，也将导致Ricciflow的理解和应用的新进展。在唐纳森的深远计划中，固定类中Kahler度量的无限维对称空间的几何关系到常数量曲率Kahler度量的存在唯一性.主要研究者还打算研究无限维对称空间中的Monge-Ampere测地线在环面簇上用有限维Bergman空间中的Monge-Ampere测地线的一致逼近问题。对这个问题的准确理解将使我们对Yau提出的关于常数量曲率Kahler度量与几何不变理论意义下的稳定性之间关系的猜想有新的认识。首席研究员还将应用矩图的观点和研究各种几何流动所产生的Kahler几何以及辛几何提出的唐纳森。第一种是J流，它是与Mabuchi能量有关的梯度流函数。第二个是hyperkahler四维流形中的矩映射流。本文主要研究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都是至关重要的，我们试图从几何学和物理学的角度来理解非线性微分方程，这一问题自然就产生了。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。分析非线性方程组奇异性的方法在工程和经济中有着广泛的应用。

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.