Mathematical Sciences: Constant Mean Curvature Surfaces / Isoparametric Spectral Geometry

布鲁斯·所罗门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。第一个是对常平均曲率曲面的研究。这里的大部分工作都是在零曲率的极小曲面上完成的。近年来，Wente的工作激发了人们对非零曲率情形的兴趣。这源于温特构建了一个反例来反驳霍普夫长期以来的猜想，即所有的肥皂泡都是球状的。现在越来越清楚的是，这些曲面有丰富的理论，类似于极小曲面，但又不同于极小曲面。第二个领域涉及等参曲面的研究。它们具有这样的性质，即在每个点上不同的主曲率是相同的。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均匀空间非常相似。所罗门将把注意力集中在球面中等参极小曲面的谱理论上。第一个主题将涉及到完备常平均曲率曲面的模空间的研究。这里的目的是将该理论与极小曲面的理论进行比较和对比。等参曲面的光谱性质的研究很重要，因为它们在许多方面是对称空间的自然继承者，它们提供了拉普拉斯算子光谱已知的唯一另一种设置。本征值实际上是整数，因此研究这种观测的几何后果是很自然的。研究这些表面的关键是它们被周围球体的子球体分解成相互垂直的测地线叶理。

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.