NEW METHODS FOR VARIATIONAL PROBLEMS IN RIEMANNIAN GEOMETRY

黎曼几何变分问题的新方法

• 批准号: 155879-2012
• 负责人: Nabutovsky, Alexander
• 金额: $ 2.55万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569498
• 关键词: NEW; METHODS; VARIATIONAL; PROBLEMS; RIEMANNIAN

Riemannian manifolds are multidimensional generalizations of surfaces in the Euclidean spaces. We would like to study ``optimal" and ``locally optimal" shapes of various manifolds (such as as, for example, the round sphere). We are developing a new approach to their construction based on mathematical logic. We are also interested in extremal objects on general Riemannian manifolds, such as the soap-film like minimal surfaces, or periodic geodesics that resemble the equator of a round sphere. Previously known topological methods used to prove the existence of such objects yield little information about their properties. Therefore we are developing new methods that can be used to obtain new information about complexity of these extremal objects. Our methods involve looking at the complexities of slicings of Riemannian manifolds into closed curves and complexities of contracting closed curves or spheres on a Riemannian manifold to a point. These problems are also of an independent interest to geometers.

黎曼流形是欧氏空间中曲面的多维推广。 我们想研究各种流形的“最优”和"局部最优”形状（如 例如，圆形球体）。我们正在开发一种新的方法， 数学逻辑。我们也对一般黎曼流形上的极值对象感兴趣， 例如肥皂膜状的最小表面，或类似于赤道的周期测地线。 一个圆球。以前已知的拓扑方法用来证明这种对象的存在 关于它们的属性的信息很少。因此，我们正在开发新的方法， 用于获得关于这些极端对象的复杂性的新信息。我们的方法包括 研究黎曼流形切割成封闭曲线的复杂性 将黎曼流形上的闭曲线或球面收缩为一点。这些问题 也是几何学家的独立兴趣。