Differential Geometry and Partial Differential Equations

舍恩教授提议在微分几何和广义相对论领域进行研究。他建议研究非时间对称约束方程的解，着眼于研究一般的Penrose不等式以及分析约束流形的稳定性。后一个主题对于理解真空爱因斯坦方程的数值稳定性很重要。Schoen教授还建议研究用特殊的拉格朗日定标形式标定子流形的构造。他计划将几何测量理论的思想应用于这个问题。他还计划研究大于2维的哈密顿平稳子流形；特别地，我们将研究哈密顿静止正切锥。Schoen教授打算研究在覆盖物下保持稳定的稳定最小曲面，希望证明它们在一般情况下是全纯的。最后，他打算研究Minkowski空间中具有常高斯-克罗内克曲率的超曲面的测地线完备性。舍恩教授的项目将有助于更好地理解爱因斯坦广义相对论方程的解。更好的理论理解对于解的精确数值模拟的成功是必不可少的。数值模拟对于预测动态情况下产生的引力辐射的性质非常重要，NSF目前有一个大型项目，LIGO，正试图测量这种辐射。他认为，该项目的理论工作将对数值计算有所帮助。舍恩教授提出的其余工作包括使用几何方法来理解表面界面的行为，例如在物理情况下出现的不同尺寸的肥皂膜和肥皂泡。这些自然的几何物体可以用来描述它们所在空间的微妙属性。这些空间出现在物理模型中，比如弦理论，它们在其中起着基本的作用。

Professor Schoen is proposing research in the areas of Differential Geometry and General Relativity. He is proposing to study solutions of the constraint equations which are not time symmetric with an eye to the study of the general Penrose inequality as well as an analysis of the stability of the constraint manifold. The latter topic is important for understanding numerical stability for the vacuum Einstein equations. Professor Schoen is also proposing to study the construction of submanifolds which are calibrated by the special lagrangian calibrating form. He plans to apply ideas from geometric measure theory to this problem. He also plans to study hamiltonian stationary submanifolds in dimension greater than two; in particular, the hamiltonian stationary tangent cones will be studied. Professor Schoen intends to investigate stable minimal surfaces which remain stable under coverings with the hope of showing that these are holomorphic in general situations. Finally, he intends to study geodesic completeness properties of hypersurfaces in Minkowski space of constant Gauss-Kronecker curvature. Professor Schoen's project will lead to a better understanding of solutions of the Einstein equations of General Relativity. A better theoretical understanding is essential for the success of accurate numerical modeling of solutions. Numerical modeling is important for predicting the nature of the gravitational radiation which arises from dynamic situations, and NSF currently has a large project, LIGO, which is attempting to measure this radiation. He believes that the theoretical work of this project will be helpful for the numerics. The remainder of Professor Schoen's proposed work involves the use of geometric methods to understand the behavior of surface interfaces, such as soap films and soap bubbles, of varying dimension which arise in physical situations. These natural geometric objects can be used to describe subtle properties of the spaces in which they reside. These spaces arise in physical models such as string theory where they play a basic role.