Analysis of noncompact and singular Ricci flows

About This Grant

The Ricci flow is the prototypical example of a geometric heat flow -- a natural process which evolves the geometric structure of a space by a heat-type differential equation, smoothing out bumps and other irregularities much in the same way that the laws governing the diffusion of heat drive the temperatures of all objects in a room, hot or cold, toward the same value over time. Geometric flows arise as models for physical phenomena as diverse as the evolution of grain boundaries in annealing metal and the weathering of stones at the ocean's edge, and through their tendency to "improve" a given space into something more symmetric and homogeneous, they have proven to be remarkably effective tools in efforts to resolve fundamental mathematical questions at the intersection of geometry and topology. This project belongs to these efforts, seeking to better understand the extreme situations where the analogy between the (linear) heat equation and the (nonlinear) Ricci flow begins to break down. The main aims are to study the nature of solutions in singular regions (where the space is becoming irrecoverably curved or pinched), and to extend the analytic theory of the equation to solutions which may become arbitrarily highly curved near spatial infinity. The project also includes an educational component, naturally incorporating the mentorship and research training of graduate students. This project has two components. In one direction, the PI will build on his past collaborative work, using methods from the theory of unique continuation for elliptic and parabolic equations to study the classification problem for noncompact shrinking solitons and further questions pertaining to finite-time singularity formation in dimensions four and higher. In another direction, the PI will seek to localize methods developed in his prior work on problems of uniqueness and unique continuation for geometric flows in order to extend them to Ricci flows with potentially unbounded curvature and other singular features. The main questions concern uniqueness and issues of preservation of structure (symmetry, restricted holonomy, positivity of curvature) for general complete flows, the long-time existence and analyticity of solutions within special subclasses, and the asymptotics of solutions near fixed points of the equation. A theory flexible enough to accommodate such nontraditional solutions could potentially find important geometric applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.