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。他建议研究不是时间对称的约束方程的解，着眼于研究一般的彭罗斯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是很重要的。舍恩教授还建议研究子流形的构造，这些子流形是由特殊的拉格朗日校准形式校准的。他计划将几何测度论的思想应用到这个问题上。他还计划研究维度大于2的哈密顿定常子流形，特别是哈密顿定常切锥。舍恩教授打算研究在覆盖下保持稳定的稳定极小曲面，希望证明这些曲面在一般情况下是全纯的。最后，他打算研究常Gauss-Kronecker曲率的Minkowski空间中超曲面的测地完备性性质。舍恩教授的项目将使人们更好地理解广义相对论爱因斯坦方程的解。更好的理论理解对于解决方案的精确数值模拟的成功至关重要。数值模拟对于预测动态情况下产生的引力辐射的性质很重要，美国国家科学基金会目前有一个大型项目，LIGO，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。舍恩教授建议的其余工作涉及使用几何方法来理解物理情况下出现的不同尺寸的表面界面的行为，如肥皂膜和肥皂泡。这些自然的几何对象可以用来描述它们所居住的空间的微妙属性。这些空间出现在物理模型中，例如弦理论，在那里它们扮演着基本的角色。

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.