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

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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.