Mathematical Sciences: Constant Mean Curvature Surfaces / Isoparametric Spectral Geometry

Bruce Solomon将在两个相当独立的项目中进行研究，尽管在这两个项目中使用的技术将是相似的。第一个是常平均曲率曲面的研究。这里的大部分工作都是在最小曲面上完成的，在零曲率的情况下。近年来，温特的工作激发了人们对非零曲率情况的兴趣。这源于温特构建了一个反例，以反驳霍普夫长期存在的猜想，即所有的肥皂泡都是球体。现在越来越清楚的是，有一个丰富的理论，这些表面类似，但不同于最小表面。第二个领域涉及等参曲面的研究。它们的性质是每一点的主曲率是相同的。对这种表面的兴趣源于这样一个事实，即它们的行为方式与同质空间非常相似。所罗门将把注意力集中在球体等参最小曲面的光谱理论上。第一个主题将涉及对完全常平均曲率曲面的模空间的研究。这里的目的是将这个理论与最小曲面的理论进行比较和对比。等参曲面的谱性质的研究是重要的，因为它们在许多方面是对称空间的自然继承者，并且它们提供了已知拉普拉斯算子谱的唯一其他设置。特征值实际上是整数，因此研究这种观察的几何结果是很自然的。研究这些表面的关键是，它们通过环境球的子球分解成相互正交的测地线叶状。

Bruce Solomon will carry out research in two fairly independent projects, although the techniques to be employed will be similar in both cases. The first is the study of constant mean curvature surfaces. Most work here has been done on minimal surfaces, the case of zero curvature. In recent years the work of Wente has stimulated interest in the case of non-zero curvature. This arose from Wente's construction of a counterexample to Hopf's long standing conjecture that all soap bubbles are spheres. It is now becoming clear that there is a rich theory of these surfaces analogous but different from that of minimal surfaces. The second area involves the study of isoparametric surfaces. These have the property that the distinct principal curvatures at each point are the same. Interest in such surfaces stems from the fact that they behave in a fashion very similar to the homogeneous spaces. Solomon will focus his attention on the spectral theory of isoparametric minimal surfaces in spheres. The first topic will involve an investigation of the moduli space of complete constant mean curvature surfaces. The intention here is to compare and contrast this theory with that of minimal surfaces. The study of spectral properties of isoparametric surfaces is important since they are in many ways the natural successors to the symmetric spaces and they provide the only other setting in which the spectrum of the Laplace operator is known. The eigenvalues are in fact integers and so it is natural to investigate the geometric consequences of such an observation. The key to the study of these surfaces is the fact that they decompose into mutually orthogonal geodesic foliations by subspheres of the ambient sphere.