Mean curvature flows in higher codimensions

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ABSTRACTMEAN CURVATURE FLOW IN HIGHER CODIMENSIONSThe mean curvature flow is the gradient flow of the volumefunctional of submanifolds. The analytic nature is a parabolic system of nonlinear partial differential equations whose stationary phase corresponds to minimal submanifolds. Minimal hypersurfaces and mean curvature flows ofhypersurfaces have been studied extensively for decades but manyapplications in mathematical physics and topology call forunderstanding of the higher codimensional case. This projectproposes to embark on a systematic investigation of minimal submanifolds and mean curvature flows in higher codimensions. Three immediate goals are to understand the minimal surface system in higher codimension, the Lagrangian mean curvature flow in Calabi-Yau manifolds and the deformation of maps between Riemannian manifolds by the mean curvature flow.Minimal surfaces are like soap films, they are surfaces of leastarea. The mean curvature flow is an evolution process which movesTa surface in space in such a way that its area is decreased mostrapidly. This is a very natural yet highly nonlinear process andit models physics phenomena such as the motion of an interface informing metallic alloys. Understanding the behavior of this evolution is important for simplifying and smoothing complicated surfaces in the most efficient way. This process, as a geometric evolution equation, tends to deform a geometric object to its optimal shape. A goal of this project is to apply this process to study certain special surfaces in Calabi-Yau manifolds, the spaces of string theory. It is believedthat underlying structures of Calabi-Yau manifolds are encoded inthese special surfaces, called special Lagrangians. The success ofthis proposal will have important impact on image processing,material science, mathematics physics and other nonlinear problems.