PDE Problems in Geometry

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The proposed research is concerned with problems in the field of geometric flows, in particular the Ricci flow and the mean curvature flow. Specifically, we plan to study the evolution of symplectomorphisms under the mean curvature flow. In particular, we want to find condition that guarantee that the mean curvature flow evolves a given symplectomorphism to a biholomorphic isometry. The PI also plans to study self-similar solutions to the Ricci flow. The PI has recently obtained a uniqueness result for the Bryant soliton under a noncollapsing assumption. It seems interesting to try to construct self-similar solutions which are collapsed, but not rotationally symmetric. Finally, the PI plans to study the gap between the first and second eigenvalue of a Schroedinger operator, building on recent work of Andrews and Clutterbuck.The main goal of this project is to approach problems in differential geometry using tools from analysis, especially partial differential equations. This approach has been successful in the past, and has led to the solution of several major open in the problems in the field, including Min-Oo's Conjecture, the Compactness Conjecture for the Yamabe problem, and the Differentiable Sphere Theorem. The Lagrangian mean curvature flow offers potential applications to mathematical physics, due to its connection with the Strominger-Yau-Zaslow conjecture.