Mathematical Sciences: Differential Structures

数学科学：微分结构

• 批准号: 8702359
• 负责人: Eugenio Calabi
• 金额: $ 56.44万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1991-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702359&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Differential; Structures

Eugenio Calabi will continue studying obstructions to the existence of a metric with constant curvature on a compact Riemann surface with finitely many prescribed singularities; Bernstein problems associated with convex surfaces having extremal, affinely invariant area; relations between the singularities of minimal subvarieties and their Morse index. Hermann Gluck will continue his work on minimizing cycles in homogeneous spaces; the Blaschke problem and fibrations of spheres by great spheres; and the topology of the space of Morse- Smale vector fields on surfaces. Wolfgang Ziller will carry out research into Einstein metrics, calibrations and the rigidity of symmetric spaces. Calabi's research is in the area of geometric variational problems. The problems he will study concern extremal Kahler metrics, the affine Plateau problem and a variational theory of geodesics and minimal subvarieties. Gluck will be continuing the search for the subvarieties of Lie groups and homogeneous spaces which are volume minimizing in their homology classes. He will also work on the famous Blaschke conjecture that every Blaschke manifold is isometric to a round sphere or projective space. His work on Morse-Smale vector fields is concerned with finding the homotopy type of the components of the set of such fields on the two-sphere. Ziller's work on Einstein metrics is concerned with finding information about manifolds which admit such metrics, that is, having constant Ricci curvature. His work on symmetric spaces involves making local changes on the metric whilst preserving the bounds on the curvature. Such changes result in metrics isometric to the original metric. He will also use the method of calibrations to investigate which subgrassmannians have minimal volume in their homology class.

Eugenio Calabi将继续研究 紧空间上常曲率度量的存在性 具有多个指定奇点的Riemann曲面 与凸曲面相关的伯恩斯坦问题 极值，仿射不变面积; 极小子簇的奇性及其莫尔斯指标 赫尔曼·格鲁克将继续他的工作，最大限度地减少周期， 齐性空间; Blaschke问题和纤维化 球的大领域;和拓扑空间的莫尔斯- 曲面上的Smale向量场。 沃尔夫冈·齐勒将对爱因斯坦进行研究 度量，校准和对称空间的刚性。 卡拉比的研究领域是几何变分 问题他将研究的问题涉及极端卡勒 度量，仿射高原问题和变分理论， 测地线和极小子簇。格鲁克将继续 李群与齐性空间的子簇搜索 其在其同源类中体积最小化。他将 我还研究了著名的布拉施克猜想， 流形与球面或射影空间等距。他 Morse-Smale向量场的工作是寻找 上的此类字段的集合的分量的同伦类型 两个球体齐勒关于爱因斯坦度量的工作涉及到 找到关于允许这种度量的流形的信息， 即具有恒定的Ricci曲率。他在对称性 空间涉及对度量进行局部更改， 保持曲率的边界。这种变化导致 度量与原始度量等距。他还将使用 校准的方法，以调查哪些subgrassmannians有 最小的体积。