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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=683559
• 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. ******In a different direction we plan to study closed minimal surfaces (``soap bubbles") in Riemannian manifolds, and their one-dimensional analogs: geodesic nets and periodic geodesics. A geodesic is a straightest possible curve on a manifold; also, the shortest way to travel between points is always provided by a geodesic. If a geodesic smoothly closes upon itself, it is called periodic. A closed geodesic net consists of finitely many geodesics meeting at their endpoints and satisfying a natural equilibrium condition at every endpoint. The existence of such minimal objects was proven in many situations. Yet the existence proofs are non-constructive and shed little light on the most natural questions such as ``What is the smallest length of a periodic geodesic? a closed geodesic net? What is the smallest area of a minimal surface?". Continuing a pioneering work of M. Gromov and C. Croke, I and R. Rotman proved many theorems answering these and similar questions in different situations. In some cases our upper bounds unexpectedly involve surprisingly little information about the ambient manifold, e.g. only its volume or diameter. Yet many other questions of such nature remain unsolved. They are closely related to questions about geometry of optimal sweep-outs of Riemannian manifolds by cycles, and geometry of spaces of loops and cycles on Riemannian manifolds, where the flabbiness of these huge infinite-dimensional spaces is somewhat tamed by the rigidity stemming from finite-dimensionality of the underlying manifold.**

黎曼流形是曲面的多维推广。 R.在给定的光滑流形上，什么是最好的，或者最好的，或者最优的黎曼度量？“邀请我们去寻找比附近所有形状都更少弯曲的形状。早些时候我们发现，每个高维流形，甚至一个球，承认无限多个这样的局部最优形状（=黎曼度量）是非常不同的彼此和从一个“标准”的形状（例如，从一个圆球）。这些形状是一些很少被理解的代数现象的几何表现（例如，平凡群的非常短但高度非平凡的表示的存在）。我们计划继续研究这些局部最优形状（特别是在与量子引力相关的4维中），作为我们更广泛研究具有各种几何边界的黎曼结构空间的几何和组合学的一部分。特别是，“最小弯曲”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）。我们知道局部最优的黎曼度量存在于这些泛函中的一些，但想证明它们存在于其他一些。我们想知道这些现象是否存在于第三维度。 ** 在不同的方向，我们计划研究黎曼流形中的闭极小曲面（"肥皂泡”），以及它们的一维类似物：测地线网和周期测地线。测地线是流形上可能的最直的曲线;此外，点之间的最短路径总是由测地线提供。如果一条测地线平滑地闭合于自身，则称之为周期线。一个封闭的测地线网是由许多端点相交的测地线组成的，并且每个端点都满足一个自然平衡条件。在许多情况下证明了这种最小物体的存在。 然而，存在性证明是非建设性的，对最自然的问题，如“什么是周期测地线的最小长度？”封闭测地线网最小曲面的最小面积是多少？".继续M. Gromov和C.克罗克，我和R。罗特曼证明了许多定理回答这些和类似的问题在不同的情况下。在某些情况下，我们的上限出乎意料地涉及关于周围流形的很少信息，例如，只有它的体积或直径。然而，许多其他此类性质的问题仍未解决。它们与黎曼流形的最优圈扫出的几何学问题以及黎曼流形上的圈和圈空间的几何学问题密切相关，在这些几何学问题中，这些巨大的无限维空间的软弱性在某种程度上被来自基础流形的有限维性的刚性所驯服。