Singularities of the Kahler-Ricci Flow, Einstein 4-Manifolds and Seiberg-Witten Theory

摘要建议：DMS-9803549主要研究者：曹怀东，周健我们提出研究几何分析中的某些问题，包括Kaehler-Ricci流的渐近行为，某些Ricci平坦四维流形的研究，封闭Einstein四维流形上的Miyaoka-Yau不等式。对于Kaehler-Ricci流，我们希望了解某些梯度Kaehler-Ricci孤立子的结构，这些孤立子是Ricci流奇点扩张的极限。这在Kaehler几何中有重要的应用，并可能导致几何演化方程领域的新知识和新见解。我们的研究表明，梯度Ricci孤子的研究与某些特殊Hamilton向量场闭轨存在的辛几何有着密切的联系。对于第二个问题，我们的目标是通过Seiberg-Witten理论对局部欧氏Ricci平坦四维流形进行渐近分类。这与霍金和波普的广义正作用猜想有关。 对于第三个问题，注意Kaehler曲面上Kaehler-Einstein度量存在的一个显著结果是Kaehler曲面的Euler特征数与基础四维流形的签名之间的Miyaoka-Yau不等式。在这个方案中，我们还提出研究一个有趣的问题：是否每个闭的Einstein四维流形都满足Miyaoka-Yau不等式。一般来说，热方程描述的是材料温度变化到稳定状态的过程. Ricci流描述了度量的变化。它的“稳态”是一种爱因斯坦度规，它的存在在几何学、拓扑学和广义相对论中具有重要意义。我们的建议涉及非线性偏微分方程，数学中的微分和复几何，物理学中的引力和广义相对论。对所提出的问题的透彻理解应该会促进我们在这些方面的知识，并给我们新的见解。

AbstractProposal: DMS-9803549Principal Investigator: Huai-Dong Cao and Jian ZhouWe propose to study certain problems in geometric analysis, includingthe asymptotic behavior of the Kaehler-Ricci flow, the study ofcertain Ricci flat four-manifolds, and Miyaoka-Yau Inequalities onclosed Einstein four-manifolds. For the Kaehler-Ricci flow, we wouldlike to understand the structure of certain gradient Kaehler-Riccisolitons, which arise as limits of dilations of singularities of theRicci flow. This has important applications in Kaehler geometry andmay lead to new knowledge and new insights in the field of geometricevolution equations. It turns out, as our research indicates, that thestudy of gradient Ricci solitons has a close link to the symplecticgeometry of existence of closed orbits for certain special Hamiltonianvector field. For the second problem, our goal is to classifyasymptotically locally Euclidean Ricci-flat four-manifolds viaSeiberg-Witten theory. This is related to the generalized positiveaction conjecture of Hawking and Pope. For the third problem, notethat a remarkable consequence of the existence of a Kaehler-Einsteinmetric on a Kaehler surface is the Miyaoka-Yau inequality between theEuler characteristic number and signature of the underlyingfour-manifold of the Kaehler surface. In this proposal we also proposeto study the interesting question whether every closed orientedEinstein four-manifold satisfies the Miyaoka-Yau Inequality.The Ricci flow is an important geometric "heat" equation. In general,heat equations describe the process of changes of tempreature ofmaterial to a steady state. The Ricci flow describes changes ofmetrics. Its "steady state" is an Einstein metric, whose existence isof fundemental importance in geometry, topology, and generalrelativity. Our proposal relates nonlinear partial differentialequations, differential and complex geometry in mathematics, andgravity, general relativity in physics. A thorough understanding ofthe problems proposed should advance our knowledge in these aspectsand give us new insight.