Principal Investigator: Guofang Wei
This proposal studies the geometry and topology of smooth metric measure spaces with Bakry-Emery Ricci curvature bounded from below and quasi-Einstein metrics. The Bakry-Emery Ricci curvature is an important generalization of Ricci curvature for smooth metric measure space, which occurs naturally as the collapsed measured Gromov-Haudorff limit. Furthermore, the Bakry-Emery Ricci curvature plays important roles in a variety of topics such as the study of Ricci soliton, warped Einstein metric, diffusion process, logrithmic Sobolev inequality, as well as in the extension of Ricci curvature lower bound for metric measure space. The principal investigator will study the geometry of Ricci solitons; relation between comparison geometry and Ricci flow and obtaining a combined estimate for the first eigenvalue for Ricci positive and nonnegative lower bound. The PI will also study the structures of the fundamental groups for manifolds with lower integral Ricci curvature bound as well as minimal volume of hyperbolic orbifolds.
Bakry-Emery Ricci curvature and Ricci soliton play a very important role in Ricci flow, which led to the solution of Poincare conjecture. Quasi-Einstein metrics is are important both in mathematics and physics. Smooth metric measure space is also related to optimal transport, information geometry, discrete geometry. The proposed activities would have impact on all these directions. The fundamental group is the most fundamental (as the name implies) topological information. Its understanding will greatly advance the study of the effect of curvature bounds on the global topology of Riemannian manifolds, therefore will be helpful in answering questions about the shape of the universe.