Mathematical Sciences: Some Constructions of Canonical Geometric Objects

Kapouleas教授将研究微分几何中的几个主题：常平均曲率曲面、调和映射、真正的爱因斯坦流形和曲面的分类。他将使用一种“修补”的方法来构造具有规定拓扑性质的恒定平均曲率曲面。“修补”过程也将被用来尝试构造真正的爱因斯坦流形的新例子。修补过程的全部用途尚不清楚，研究的目标之一是探索该工具可能有用的其他领域。这项建议所支持的研究应该会产生一种从局部信息中获取全局信息的新方法。到目前为止，获得曲面上偏微分方程整体解的标准方法涉及到对方程可能整体解的抽象空间进行处理。在这项研究中使用的方法包括把解决方案的小部分放在一起，就像一个人把拼图拼在一起一样。该方法已经在一类问题上取得了成功，首席研究员现在将尝试将该方法扩展到其他类别。

Professor Kapouleas will study several topics in differential geometry: constant mean curvature surfaces, harmonic maps, real Einstein manifolds, and the classification of surfaces. He will use a method of "patching" to construct constant mean curvature surfaces with prescribed topological properties. The "patching" process will also be used to try to construct new examples of real Einstein manifolds. The full extent of the usefulness of the patching process is not known and one of the goals of the research is to explore other areas in which this tool might be useful. The research supported by this proposal should result in a new method of obtaining global information from local information. The standard method up to this point for obtaining global solutions to partial differential equations on surfaces involved working with the abstract space of possible global solutions to the equation. The method to be used in this research involves putting small parts of solutions together much like one puts together a jigsaw puzzle. The method has already been successful on one class of problems and the principal investigator will now try to extend the method to other classes.