Geometric evolution equations and analysis in calibrated geometry

给定空间中的一条闭曲线，在所有跨越该曲线的曲面中，有一个曲面的面积最小。这样的曲面称为极小曲面。极小曲面在数学的其他领域也有重要的应用，在物理学中也是如此，例如广义相对论。对于三维空间中的给定表面，可以尝试使其连续变形，以使表面积最快地减小。最有效的方法是每次都在垂直于表面的方向上进行变形。这种随时间变化的演化过程就是所谓的平均曲率流。为了推广这个过程，可以移动更高维的“表面”，这在数学中被称为流形，以类似的方式，即使在弯曲的环境空间。我们想研究这个流和最小曲面，其中有一些来自周围空间的额外的漂亮特征。我们还想研究一类特殊的极小曲面，即所谓的拉格朗日曲面的方程。例如，我们想知道这些曲面一般有多光滑，以及在什么条件下拉格朗日极小曲面必须是平面。我们还感兴趣的一类特殊的映射发送到另一个流形和满足一定的微分方程。这些映射携带几何信息，在比较两个流形时很有用。

Given a closed curve in space, among all surfaces spanning the curve there is one with least area. Such a surface is called a minimal surface. Minimal surfaces have important applications in other areas of mathematics, and in physics as well, for example in general relativity. For a given surface in 3-dimensional space, one can try to deform it continuously in such a way that the surface area decreases most rapidly. The most effective way of doing this is to take deformations in the direction which is perpendicular to the surface each time. This time dependent evolution process is the so-called mean curvature flow. To generalize this procedure, one can move higher dimensional "surfaces", which are called manifolds in mathematics, in a similar fashion, even in a curved ambient space. We want to study this flow and minimal surfaces with some extra nice features coming from the ambient space. We also want to study the equations which govern a special class of minimal surfaces, the so called Lagrangian ones. For instance, we want to know how smooth these surfaces are in general and under what conditions the Lagrangian minimal surfaces must be flat planes. We are also interested in a special class of mappings which send one manifold into another and satisfy certain differential equations. These mappings carry geometric information and are useful in comparing two manifolds.