Problems in Global Riemannian Geometry

这个项目属于黎曼几何领域。更具体地说，研究者将考虑等谱度量集的结构，欧几里得空间中具有相同散射行为的度量的可能存在，几何性质在某些方面不同的等谱度量的构造，以及薛定谔算符的谱决定势的程度。本文还研究了轨道的谱几何、极小超曲面及其在三维Ricci曲率几何中的应用以及具有边界的流形上的Ricci流。黎曼流形是曲面的高维推广。这种流形在理论物理中有着众所周知的应用。黎曼流形具有距离或度规的概念。通过对度规求导我们可以测量它的曲率。拉普拉斯算子及其谱是与任何黎曼流形相关的基本对象。这个研究项目的大部分是关于“黎曼流形的几何形状在多大程度上是由它的拉普拉斯算子的谱决定的？”

9704369 Gordon This project lies in the area of Riemannian geometry. More specifically, the investigator is to consider the structure of isopectral sets of metrics, the possible existence of metrics in Euclidean space with the same scattering behavior, constructions of isopectral metrics whose geometric properties differ in certain ways, and the extent to which the spectrum of a Schrodinger operator determines the potential. The spectral geometry of orbifolds, minimal hypersurfaces and their applications to the geometry of Ricci curvature in dimension three, and the Ricci flow on manifolds with boundary are also to be pursued. Riemannian manifolds are higher dimensional generalizations of curved surfaces. Such manifolds have well-known applications in theoretical physics. A Riemannian manifold possesses a notion of distance, or a metric. And by taking the derivative of the metric one can measure its curvature. The Laplace operator and its spectrum are a fundamental object associated to any Riemannian manifold. Much of this research project is concerned with the question "To what extent is the geometry of a Riemannian manifold determined by the spectrum of its Laplace operator?"