Problems in Global Riemannian Geometry

小行星9704369 这个项目属于黎曼几何领域。更具体地说，调查是考虑结构的isopectral集的度量，可能存在的度量在欧几里得空间具有相同的散射行为，建设isopectral度量的几何性质不同，在某些方面，以及在何种程度上的频谱的薛定谔运营商决定的潜力。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，以及带边界流形上的Ricci流也将被追求。 黎曼流形是曲面的高维推广。这样的流形在理论物理学中有着众所周知的应用。一个黎曼流形拥有一个距离的概念，或一个度量。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联的基本对象。这个研究项目的大部分内容都是关于“黎曼流形的几何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?"