Evolutions Equations in Geometry

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Evolution equations are basic objects in the sciences, describing how natural phenomena change over time. For instance, the modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, involves tracking fronts that move with curvature-dependent speed. When a surface evolves with speed that is proportional to the curvature, this results in one of the classic degenerate nonlinear differential equations called the mean curvature flow. For nonlinear equations like this, it is possible that solutions may develop sharp points and corners, so it is natural to ask "what is the regularity of solutions?'' Optimal regularity for mean curvature flow was recently proven by the PI and Minicozzi. The proof weaves together analysis and geometry. It is expected that many of the new ingredients and techniques should lead to many other results for a wide range of equations that the PI will investigate in this project.This project has two parts. The main part concerns geometric evolution equations, like mean curvature flow (MCF) and Ricci flow. It deals with optimal regularity and applications. The PI has, together with Minicozzi, settled a number of long-standing open problems and conjectures for mean curvature flow and expect that the results and ideas developed will have significant applications also to other flows and plan to pursue them. Rigidity of prevalent singularities and uniqueness of blow-ups has had a wide range of applications for mean curvature flow from optimal regularity of the level set equation to optimal estimates on the singular set of the flow. The PI plans on investigating similar conjectures for the Ricci flow. The second part of the project will deal with other (non-geometric) evolution equations that are motivated by questions in social science and engineering. One particular focus will be on a natural evolution equation that describes how the opinions of a group of people evolve as they are influenced by each other.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.