Mean curvature flow in higher co-dimensions

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In this research project we intend to investigate analytic properties of the mean curvature flow (MCF) in higher co-dimension of submanifolds of a Riemannian or Kählerian manifold and its implications on the topology and geometry of submanifolds. Specifically, one of our main objectives is to use MCF to understand the topology and geometry of smooth maps between Riemannian or Kählerian manifolds. This plan can be achieved by evolving the corresponding graphical submanifold in the product space by the mean curvature flow. A prime motivation for the study of the graphical mean curvature flow is a question posed by Gromov of how the Jacobians of a map between given Riemannian manifolds affects the homotopy type of the map. A second motivation is a general question of Yau of how to deform a symplectomorphism into a homolorphic map. We would like to mention that these problems have attracted the attention of many mathematicians during the last decade. Another aim is to understand the singularities of the MCF. In particular, we are interested to study in detail translating solitons of the mean curvature flow. One of the major problems of singular analysis of MCF is the classification of such objects. Our approach to attack this problem is to combine the analysis of elliptic systems of PDEs with techniques arising from minimal submanifold theory.