Dynamics of cylindrical singularities in geometric flows and ancient solutions

About This Grant

The main objective of the project is to study geometric flows, more precisely equations in which one evolves a geometric object (for example a metric or a surface in the Euclidean space) in time, expecting it will improve its properties in time, for example become more symmetric, starting resembling familiar objects, such as spheres or cylinders. These geometric equations usually develop singularities in a finite amount of time, after which one cannot expect to have a nice solution to the considered geometric flow. One would like to understand more closely what happens at those singular times, and what the singularities look like. This should help one find a way to define a solution past the singularities. After repeating this finitely many times, one should get in the end very familiar geometric objects, and this could help, for example, to understand and classify all possible topologies of the initial geometric object. The PI expects that new students and postdocs, besides current ones will be trained, and that they will benefit from the research activity. The PI also plans to co-organize workshops in various topics in Geometric Analysis. This project is to study singularity formation in asymptotic sense, and to classify singularities in nonlinear parabolic equations which come from differential geometry problems, such as the evolution of a hypersurface in the Euclidean space by functions of its principal curvatures, and the Ricci flow. The first part of the project is to understand the formation of cylindrical singularities which are expected to be generis and their stability. This includes understanding the dynamics of nondegenerate and degenerate neckpinch singularities, more precisely the denseness and stability of nondegenerate neckpinch singularities in the Ricci flow and the mean curvature flow. The PI plans to understand the formation of cylindrical singularities in the asymptotic sense in both, the Ricci flow and the mean curvature flow. The second part of the project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow, since ancient solutions appear as singularity models in both flows. The PI will combine the PDE techniques and geometric estimates to study ancient solutions of such flows. The PI plans to classify ancient closed noncollapsed solutions to higher dimensional Ricci flow (cases n= 2,3 have been solved), under suitable conditions. More precisely, the PI would like to classify four dimensional noncollapsed Ricci flow ancient solutions whose asymptotic shrinker is a round cylinder or a bubble sheet. One motivation for this classification comes from showing an analogue of the Mean Convex Neighborhood Theorem for the Ricci flow. This could potentially help one to perform surgery in the Ricci flow in dimension four without assuming global curvature conditions initially. The PI will also try to understand mean curvature flow ancient solutions whose tangent flows at negative infinity are generalized cylinders. 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.