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

我们建议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平坦四流形的研究，以及闭爱因斯坦四流形上的Miyaoka-Yau不等式。对于Kaehler-Ricci流，我们希望了解某些梯度Kaehler-Ricci孤子的结构，它是由Ricci流的奇点膨胀极限引起的。这在Kaehler几何中有重要的应用，并可能在几何演化方程领域带来新的知识和新的见解。结果表明，对于某些特殊的哈密顿向量场，梯度Ricci孤子的研究与闭轨存在性的辛几何有着密切的联系。对于第二个问题，我们的目标是利用Seiberg-Witten理论对局部欧几里奇平坦四流形进行渐近分类。这与霍金和波普的广义正作用猜想有关。对于第三个问题，请注意，Kaehler曲面上存在Kaehler-Estein度量的一个显著结果是Kaehler曲面下面的四个流形的欧拉特征数和符号之间的Miyaoka-Yau不等式。在这个方案中，我们还提出了一个有趣的问题，即是否每个封闭定向爱因斯坦四流形都满足Miyaoka-Yau不等式。Ricci流是重要的几何“热”方程。一般来说，热方程描述了材料温度变化到稳定状态的过程。RICCI流描述了指标的变化。它的“稳态”是爱因斯坦度规，它的存在在几何学、拓扑学和广义相对论中具有基本的重要性。我们的建议将非线性偏微分方程式、数学中的微分和复杂几何，以及物理学中的重力和广义相对论联系起来。对提出的问题有一个透彻的认识，应该会促进我们在这些方面的认识，给我们新的洞察力。

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.