Mean curvature flows in higher codimensions

高维中的曲率流平均曲率流是子流形的体积泛函的梯度流。解析性质是一个抛物型的非线性偏微分方程组，其定态相对应于极小子流形。几十年来，极小超曲面和超曲面的平均曲率流得到了广泛的研究，但在数学物理和拓扑学中的许多应用都需要理解高余维情形。本项目旨在系统地研究高次余维中的极小子流形和平均曲率流。三个直接的目标是了解高维极小曲面系统，Calabi-Yau流形上的拉格朗日平均曲率流，以及黎曼流形之间映射的平均曲率流的形变。极小曲面就像肥皂膜，它们是最小面积的曲面。平均曲率流是一个曲面在空间中运动，使其面积减小最快的演化过程。这是一个非常自然但高度非线性的过程，它模拟了物理现象，例如界面的运动，为金属合金提供信息。了解这种演化的行为对于以最有效的方式简化和平滑复杂曲面非常重要。这个过程，作为一个几何演化方程，倾向于使几何对象变形到其最佳形状。这个项目的一个目标是应用这个过程来研究Calabi-Yau流形中的某些特殊曲面，即弦理论空间。人们认为，Calabi-Yau流形的底层结构被编码在这些特殊的表面上，称为特殊拉格朗日。该方案的成功将对图像处理、材料科学、数学物理等非线性问题产生重要影响。

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.