Geometric evolution equations and analysis in calibrated geometry

点击翻译按钮获取中文摘要

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.