This project is concerned with nonlinear parabolic partial differential equations (PDE) and systems satisfied by geometric objects evolving by geometrically natural quantities such as curvature. These PDE are used in programs to evolve given geometries toward ones which are in suitable senses "optimal" or "canonical," and which are thus amenable to classification. But because these PDE generically develop singularities, a classification of those singular behaviors is necessary for the successful completion of those programs. Analyzing formation of (finite- or infinite-time) singularities is the unifying goal of this project. A key approach is the use of matched asymptotics, a technique that can provide the most precise description of the set of points on which a solution becomes singular, and of the behavior of the solution in a space-time neighborhood of that singularity. Major objectives of this proposal include: (1) removing symmetry hypotheses in asymptotic singularity analysis for mean curvature flow (MCF) and Ricci flow (RF), thereby proving that certain singularity profiles are "universal" in a rigorous sense; (2) constructing and analyzing (non-generic) Type-II RF singularities, which form more slowly than the natural parabolic rate and thus feature faster curvature blow-up; (3) constructing codimension-2 RF singularities and studying their asymptotics, genericity, and stability; (4) constructing and studying new examples of RF local singularity formation for complex surfaces (and complex manifolds of higher dimension) with applications to the classification of singularity models in those dimensions; (5) studying stability (properly understood) of product structures and related curvature conditions preserved by RF in low dimensions; (6) showing that singular profiles of geometric PDE depend continuously on their initial data, with applications to topology; and (7) studying formation and stability of infinite-time RF singularity models under distinct convergence schemes designed to provide asymptotics at temporal infinity, along with other geometric information.
The theory of geometric partial differential equations (PDE) has surprising similarities with nonlinear hyperbolic and dispersive equations. Furthermore, the PDE that arise in curvature flows are remarkably similar to equations that model heat propagation, the movement of oil in shale and thin films, combustion in porous media, and certain effects in plasma physics. In all of these applications, the underlying models are fundamentally nonlinear, a property which causes the associated PDE to develop various critical or singular behaviors. The utility of these models requires a precise mathematical understanding of these behaviors. This project will further develop mathematical techniques, particularly matched asymptotic expansions, that should help the analysis of singularity formation in these varied systems and applications.
