Variational and Parabolic Phenomena in Differential Geometry

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Differential Geometry studies shapes of spaces through distances and angles. Mathematically the concept of curvature plays the central role. Partial differential equations (PDEs) arise naturally from many fundamental questions in Differential Geometry, notably in understanding a geometric space in terms of its curvature or finding an optimal geometric structure on a given space. The PI will focus on two important examples: minimal submanifolds and geometric flows. Minimal submanifolds are subspaces that locally minimize area (or volume). They have been studied since the work of Euler and Lagrange, yielding many applications to geometry and other related fields such as low-dimensional topology and general relativity. Moreover, they are important models for many interesting non-linear phenomena in nature and a variety of ideas developed in their study have turned out to be key in the calculus of variations, geometric PDEs, and mathematical physics. A geometric flow, on the other hand, is a process of deforming a given geometry through a parabolic system of PDEs coming from curvature. A prominent example is the flow by the Ricci curvature, or Ricci flow, which has had seminal consequences to the geometrization of three-dimensional spaces by the work of Hamilton and Perelman. One of the tenets of this research project is that the existence of minimal submanifolds of certain topological and Morse index type can impose restrictions on the curvature of the ambient manifold, and vice-versa. The PI will investigate several questions relating the Morse index and the topology of minimal submanifolds in presence of positive curvature. Another aspect is the singularity formation phenomena in geometric flows, with a focus on the profiles of singularities and their rigidity and stability properties.