New methods for variational problems in Riemannian geometry

黎曼几何中变分问题的新方法

• 批准号: RGPIN-2017-06068
• 负责人: Nabutovsky, Alexander
• 金额: $ 2.19万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635350
• 关键词: New; methods; variational; problems; Riemannian

Riemannian manifolds are multidimensional generalizations of surfaces. A well-known open question of R. Thom "What is the best, or the nicest, or the optimal Riemannian metric on a given smooth manifold?" invites us to look for shapes that are less curved that all nearby shapes. Earlier we discovered that each high dimensional manifold, even a sphere, admits infinitely many such locally optimal shapes (=Riemannian metrics) that are very different from each other and from a ``standard" shape (e.g. from a round sphere). These shapes are geometric manifestations of some poorly understood algebraic phenomena (e.g. the existence of very short but highly non-trivial presentations of the trivial group). We plan to continue investigating these locally optimal shapes (especially, in dimension 4 which is relevant for Quantum Gravity) as a part of our broader study of geometry and combinatorics of spaces of Riemannian structures with various bounds on geometry. In particular, ``the least curved" can be understood in a number of natural but different ways (corresponding to different Riemannian functionals). We know that the locally optimal Riemannian metrics exist for some of these functionals, but would like to prove their existence for some others. We would like to find out if some vestiges of these phenomena exist in dimension 3.

黎曼流形是曲面的多维推广。R. Thom的一个著名的开放问题是“在给定的光滑流形上，什么是最好的，或者最好的，或者最优的黎曼度规？”这让我们去寻找比所有附近的形状更不弯曲的形状。前面我们发现，每个高维流形，甚至一个球体，都允许无限多个这样的局部最优形状（=黎曼度量），它们彼此之间和与“标准”形状（例如，与圆球）非常不同。这些形状是一些鲜为人知的代数现象的几何表现（例如，存在非常短但高度非平凡的平凡群表示）。我们计划继续研究这些局部最优形状（特别是与量子引力相关的4维），作为我们更广泛的几何和黎曼结构空间组合学研究的一部分。特别是，“最不弯曲”可以用许多自然但不同的方式来理解（对应于不同的黎曼泛函）。我们知道局部最优黎曼度量对于某些泛函是存在的，但是我们想要证明它们对于其他泛函的存在性。我们想知道这些现象的一些痕迹是否存在于第三维度。