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

摘要奖：DMS-0604805首席研究员：宋健这项建议从几何和物理两个方面讨论了非线性Monge-Ampere型方程的存在性和正则性问题。在过去的几十年里，在紧致的Kahler流形上寻找正则度量的问题一直是人们密切研究的主题。在他对Calabi猜想的解答中，Yau证明了第一类为零或负的紧致Kahler流形上Kahler-Einstein度量的存在性。CaO用Kahler-Ricci流给出了Calabi猜想的另一种证明。然而，大多数射影代数簇都没有确定或平凡的第一类。最近，佩雷尔曼在汉密尔顿的程序中取得了重大突破，作为解决庞加莱猜想和瑟斯顿几何猜想的方法。主要研究者建议应用Kahler-Ricciflow来研究正Kodaira维射影变分的典范模型上的正则度量。这种正则度量是由Kodaira维极小射影曲面上的Kaher-Ricci流的形变构造的。这些广义的Kahler-Einstein度规可以看作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Donaldson意义深远的程序中，固定类Kahler度量的无穷维对称空间的几何关系到常标量曲率Kahler度量的存在唯一性。主要研究了无限维对称空间中的Monge-Ampere测地线与有限维Bergman空间中的一致逼近问题。对这一问题的准确理解将使我们对Yau提出的常标量曲率Kahler度规的关系与几何不变理论意义上的确定性之间的猜想有新的认识。主要研究人员还将应用矩映射的观点，研究由Kahler几何和Donaldson提出的辛几何产生的各种几何流动。第一种是J流，它是与Mabuchi能量有关的泛函的梯度流。第二个是超卡勒4流形中的动量图流。这位首席调查者打算研究这种抛物线流动的收敛和奇点问题。由于广义相对论的发现，几何分析对数学家和物理学家来说都是至关重要的。这一建议中的问题自然产生于我们从几何和物理上理解非线性微分方程的尝试。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刻理解。这种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.