This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The theme of this project is to generalize the methods of global Riemannian geometry to better understand metric measure spaces, which occur naturally as singularity models for the Ricci flow. This proposal builds on joint work by the PI and Petersen, who have proven a number of classification theorems for gradient Ricci solitons, or self-similar solutions to the flow. An important tool in this work is the theory of curvature dimension inequalities developed by Bakry and Emery and their collaborators, which was also researched by the PI in joint work with Wei. One aspect of this project is to investigate new theorems relating Ricci solitons and curvature dimension inequalities. Another component is to investigate new ideas about classical Riemannian geometry that arise in the novel setting of metric measure spaces. As an example, the PI will investigate the interesting relationship between curvature dimension inequalities and warped product Einstein manifolds.
Global Riemannian geometry studies the relationship between the infinitesimal warping (or curvature) of a space, and the global shape of that space. These spaces, or manifolds, appear frequently in science and engineering problems because physical laws can generally be modeled by partial differential equations on a manifold. Partial differential equations are also important in geometry because they are used to draw global conclusions from curvature, which is a type of local information. This structure on the overall shape in turn allows new insights into the behavior of differential equations on the manifold. The Ricci flow, which is one of the most important partial differential equations in mathematics, has been used to classify the shape of all three dimensional manifolds. A better understanding of singularities in the Ricci flow promises deep new discoveries applicable to many areas of mathematics and science. Because this project advances our understanding of the geometry of manifolds, it will also contribute to a better understanding of many physical models.
