Mean curvature flows in higher codimensions

高维平均曲率流平均曲率流是子流形的体积泛函的梯度流。解析性质是一类非线性偏微分方程的抛物型方程组，其平稳相对应于最小子流形。极小超曲面和超曲面的平均曲率流已经被广泛研究了几十年，但许多在数学物理和拓扑中的应用需要对高维情况的理解。本项目拟对高协维的最小子流形和平均曲率流进行系统的研究。三个直接的目标是理解高余维最小曲面系统、Calabi-Yau流形中的拉格朗日平均曲率流和黎曼流形之间映射的平均曲率流变形。最小表面就像肥皂膜，它们是最小面积的表面。平均曲率流是一种在空间中以其面积减少最快的方式移动灶台表面的演化过程。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，如金属合金界面的运动。理解这种演化的行为对于以最有效的方式简化和平滑复杂的表面非常重要。这一过程是一个几何演化方程，其目的是使一个几何对象变形到其最优形状。这个项目的目标是应用这个过程来研究弦理论空间Calabi-Yau流形中的某些特殊表面。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，称为特殊拉格朗日量。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.