The aim of this project is to study geometric flows and the relationship between curvature and topology. Regarding geometric flows, the principal investigator plans to study higher codimension mean curvature flow, which is the gradient flow for the area functional. More precisely, we plan to study mean curvature flow deformation of Lagrangian submanifolds. In his thesis, the principal investigator showed that finite time singularities are unavoidable, i.e., they occur in many cases where experts were hoping they would not occur, and then he proved the optimal result about the infinitesimal behavior of singularities. We plan to investigate other settings on which we can understand singularities and also to understand the size of the singular set at the time of the first singularity. This last question is very challenging and a satisfactory answer would be a breakthrough in the field. Regarding the relationship between curvature and topology we plane to investigate which constant scalar curvature metrics does a 3-manifold admit. More specifically, we intent to extend the investigator's previous work and hope to unveil a large class of manifolds $L$ for which $M$ and $M#L$ have the same ``type'' of constant positive scalar curvature metrics.
The underlying philosophy of many problems in geometric analysis is to given a geometric object being able to find another geometric object carrying the same type of information but having better properties. For instance, if one is studying the paths that go from A to B, the best possible path would be the one with shortest lenght. Both problems addressed in this research project obey this guiding principle. In the first one we try to use a heat -equation flow method to deform certain kinds of Lagrangian submanifolds into those that are still Lagrangian but have the least area possible. This is expected to have very nice applications in mathematical physics (more precisely in the SYZ conjecture). In the second problem we try to understand which geometric information does a constant scalar curvature metric carry. It is known that for surfaces these metrics determine its topological type. For 3-manifolds it is know that constant Ricci curvature determines the manifold. It is an important open problem to understand, for 3-dimensional manifolds, the information that can be extracted from constant scalar curvature metrics.
