Einstein Manifolds and Related Geometric Structures

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A Riemannian metric is a mathematical device that allows us to compute lengths and angles between tangent vectors in n-dimensional spaces.  The Riemann curvature tensor is the basic object derived from a metric for local comparisons of its geometry with that of flat space. There is a part of the Riemann curvature tensor that is of the same mathematical type as the metric g--the Ricci tensor Ric(g). Since conditions imposed on the Ricci tensor are in general neither over nor underdetermined, they give the most reasonable constraints one can place on the geometry. An Einstein metric g is one such that Ric(g) = cg, where c is a real constant.  This condition is a nonlinear system of partial differential equations on the underlying space. In General Relativity, the Ricci tensor of a Lorentz metric occurs as a term in Einstein's equation. Einstein metrics with Euclidean signature are important ingredients in supergravity theories such as string and M-theory. Understanding Einstein metrics has for decades been a central endeavour in Differential Geometry with significant impact on Theoretical Physics. My proposal focuses on the existence, moduli, and geometric properties of Einstein metrics, where two metrics are identified if they differ by a diffeomorphism.  Special emphasis will be placed on Einstein metrics whose holonomy algebra is generic because this case is currently the least understood by geometers. For positive Einstein metrics (c > 0) I will investigate their stability properties and develop a variational approach for the existence problem, starting with the cohomogeneity one case.  Einstein metrics are also studied using the Ricci flow by which one evolves a metric in the direction of -2Ric(g). This leads to a flow equation which has many properties similar to those of heat flow. Perelman discovered two useful functionals for studying the Ricci flow. The critical points of these functionals consist of Einstein metrics and their generalizations--the gradient Ricci solitons. I will also investigate the existence, moduli, and geometric properties of these and related structures, particularly in the much less understood non-Kahler case. In studying the Ricci flow, singularities of space tend to form in finite time. To analyse their formation, blow-up models are constructed by a dilation and limiting process, resulting in non-collapsed ancient or eternal solutions of the Ricci flow. I plan to study the construction of such solutions on non-compact spaces, including the important special case of shrinking solitons.  The proposed research may lead to large classes of new Einstein spaces, Ricci solitons, and ancient solutions with a wide range of topological and geometric properties. Some of the examples, especially explicit ones, may be useful to physicists as background geometries for sigma models or models in supergravity theories. The techniques developed may be useful for handling similar equations arising in geometry or physics.