Mathematical Sciences: Constant Mean Curvature Surfaces / Isoparametric Spectral Geometry

布鲁斯所罗门将在两个公平的研究 独立的项目，虽然所采用的技术将 在这两种情况下是相似的。第一部分是关于常数均值的研究 曲率曲面这里的大部分工作都是在最低限度的 曲面，零曲率的情况。近年来， 温特激发了人们对非零曲率的兴趣。 这源于温特构造了一个反例， 霍普夫的长期猜想，所有的肥皂泡是 球体。现在越来越清楚，有一个丰富的理论， 这些表面类似于但不同于最小的 表面。第二个领域涉及等参数的研究 表面。它们具有这样的性质： 每一点的曲率都是一样的。对这些表面的兴趣 是因为它们的行为方式 齐性空间所罗门会把注意力集中在 球面等参极小曲面的谱理论 第一个主题将涉及调查的模量 完备常平均曲率曲面空间的意图 这里是比较和对比这个理论与最小 表面。等参线谱性质的研究 表面是重要的，因为它们在许多方面是自然的， 对称空间的后继者，它们提供了唯一的 其他设置，其中拉普拉斯算子的谱是 知道的特征值实际上是整数，所以很自然 来研究这种观察的几何后果。 研究这些表面的关键在于， 分解成相互正交的测地线叶理， 环境球体的子球体。

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.