Mathematical Sciences: Applications of Minimal Surfaces to Curvature and Topology of Riemannian Manifolds and Submanifolds

首席研究人员将发展一个类似于闭合测地线的极小两球面的莫尔斯理论。该项目的一个目标是利用研究人员在低能量下关于Morse理论的结果来构造大量最小的低面积两球。研究人员还将利用黎曼流形中的极小两球与黎曼球面上的全纯向量丛之间的关系。并将Hilbert定理推广到奇维欧氏空间中具有常负曲率的n维子流形上。本项目的目标是了解具有低面积的最小曲面。肥皂泡是最小化能量的最小表面的例子。首席研究员将使用莫尔斯理论来研究这些表面及其在更高维度上的推广。这一理论是我们的海拔概念的延伸，盆地和山口经历了不同的数学解释。

The principal investigator will develop a Morse theory of minimal two-spheres analogous to that of closed geodesics. A goal of the project is to construct large numbers of minimal two- spheres of low area using a result of the investigator concerning Morse theory at low energies. The investigator will also exploit the relationship between minimal two-spheres in Riemannian manifolds and holomorphic vector bundles over the Riemann sphere. And an extension of Hilbert's theorem to n-dimensional submanifolds of constant negative curvature in odd-dimensional Euclidean space will also be investigated. This goal of this project is to understand minimal surfaces with low area. Soap bubbles are examples of minimal surfaces which minimize energy. The principal investigator will use Morse theory to study these surfaces and their generalizations to higher dimensions. This theory is an extension or our concept of elevation, where basins and mountain passes undergo separate mathematical interpretations.