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

在过去的十年里，随着极小曲面理论的新技术的引入，黎曼几何经历了一场革命。在最近与M.Micallef的联合工作中，首席调查员一直站在这些事态发展的最前列。在这项工作中，他们加强了Klingenberg-Berger球面定理。这涉及到对最小两球面的莫尔斯理论的使用。John Moore建议继续应用这些创新技术来寻找黎曼流形的曲率和拓扑之间的进一步关系。他还将研究欧几里德空间中的低余维子流形。提出的问题既有趣又困难。它们的解决方案不仅意义重大，而且这些新技术的发展也非常重要。

During the last decade Riemannian geometry has undergone a revolution following the introducton of new techniques from the theory of minimal surfaces. In recent joint work with M.Micallef, the principal investigator has been at the forefront of these developments. In this work they strengthened the Klingenberg- Berger sphere theorem. This involved use of Morse theory for minimal two-spheres. John Moore proposes to continue applying these innovative techniques to find further relationships between curvature and topology for Riemannian manifolds. He will also be studying submanifolds of low codimension in Euclidean space. The proposed problems are both interesting and difficult. Not only would their solution be of significance, but the development of these new techniques is of great importance.