Proposal: DMS-9803549 Principal Investigator: Huai-Dong Cao and Jian Zhou
We propose to study certain problems in geometric analysis, including the asymptotic behavior of the Kaehler-Ricci flow, the study of certain Ricci flat four-manifolds, and Miyaoka-Yau Inequalities on closed Einstein four-manifolds. For the Kaehler-Ricci flow, we would like to understand the structure of certain gradient Kaehler-Ricci solitons, which arise as limits of dilations of singularities of the Ricci flow. This has important applications in Kaehler geometry and may lead to new knowledge and new insights in the field of geometric evolution equations. It turns out, as our research indicates, that the study of gradient Ricci solitons has a close link to the symplectic geometry of existence of closed orbits for certain special Hamiltonian vector field. For the second problem, our goal is to classify asymptotically locally Euclidean Ricci-flat four-manifolds via Seiberg-Witten theory. This is related to the generalized positive action conjecture of Hawking and Pope. For the third problem, note that a remarkable consequence of the existence of a Kaehler-Einstein metric on a Kaehler surface is the Miyaoka-Yau inequality between the Euler characteristic number and signature of the underlying four-manifold of the Kaehler surface. In this proposal we also propose to study the interesting question whether every closed oriented Einstein 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 of material to a steady state. The Ricci flow describes changes of metrics. Its "steady state" is an Einstein metric, whose existence is of fundemental importance in geometry, topology, and general relativity. Our proposal relates nonlinear partial differential equations, differential and complex geometry in mathematics, and gravity, general relativity in physics. A thorough understanding of the problems proposed should advance our knowledge in these aspects and give us new insight.
