Quantitative Homotopy Theory and Geometric Inequalities

定量同伦理论和几何不等式

• 批准号: RGPIN-2018-04523
• 负责人: Rotman, Regina
• 金额: $ 1.46万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713084
• 关键词: Quantitative; Homotopy; Theory; Geometric; Inequalities

Most of my results are proofs of various geometric inequalities. While these inequalities are proven in the setting of Riemannian Geometry, a lot of times they can be generalized to more general metric spaces, i. e. spaces in which we can measure distances between points. Riemannian Geometry is the generalization of the geometry of surfaces that lie in a Euclidean three dimensional space. Some of the examples of such surfaces are a round sphere, (as we all know the surface of the Earth has the shape of the sphere), or a torus, the shape that looks like a donut. If we look at the neighbourhood of a point of a surface under the magnifying glass, it will look like a plane. Therefore, a lot of our intuition comes from the Euclidean geometry. At each point of a surface there is a way to measure an inner product between the tangent vectors. This notion of the inner product on a surface generalizes as the Riemannian metric. It gives us a way to measure the distance between points, to define the diameter of a manifold, (i. e. the maximal distance between pairs of points on a manifold), its volume, its curvature, etc. It also give us a way to generalize the notion of a straight line to Riemannian manifolds, obtaining a notion of geodesics, i.e. the straightest curves on a manifold. While the straight line stretches infinitely in both directions, geodesics can close on themselves, resulting in a periodic geodesic, when they do so smoothly, just like the Equator on a sphere, or in a geodesic loop. Many of my previous results, as well as the proposed problems deal with how the lengths of geodesic segments, geodesic loops, or periodic geodesics relate to the other parameters of the manifold, such as its diameter, volume, or some times its curvature. Other minimal objects that are of interest to me are minimal surfaces. A prototype of a minimal surface is a surface of a soap bubble. I am also interested in estimates of the areas of minimal surfaces. Another direction is to prove the existence of various minimal objects, when it does not immediately follow for the topological reasons. It turns out that some of the methods that we use to estimate the "size" of minimal objects some times also works when we try to establish the existence of minimal objects on noncompact manifolds, i.e. the manifolds that stretch to infinity, under certain natural geometric restraints. For example, one of the natural questions that we want study is the following question attributed to V. Bangert: Let M be a complete noncompact Riemannian manifold of finite volume, is there always a periodic geodesic on M?

我的大部分结果都是各种几何不等式的证明。虽然这些不等式在黎曼几何的背景下得到了证明，但很多时候它们 可以推广到更一般的度量空间，即e.我们可以测量点之间距离的空间。黎曼几何是欧几里得三维空间中表面几何的概括。此类表面的一些示例是圆形球体（众所周知，地球表面具有球体的形状）或环面（看起来像甜甜圈的形状）。如果我们在放大镜下观察曲面上一点的邻域，它看起来就像一个平面。因此，我们的很多直觉都来自于欧几里得几何。 在曲面的每个点处，都有一种方法可以测量切向量之间的内积。曲面上的内积概念可概括为黎曼度量。它为我们提供了一种测量点之间距离、定义流形直径（即流形上点对之间的最大距离）、体积、曲率等的方法。它还为我们提供了一种将直线概念推广到黎曼流形的方法，获得测地线的概念，即流形上最直的曲线。虽然直线在两个方向上无限延伸，但测地线可以自行闭合，当它们平滑地闭合时，就会产生周期性测地线，就像球体上的赤道或测地线环中一样。我之前的许多结果以及提出的问题都涉及测地线段、测地线环或周期性测地线的长度如何与流形的其他参数（例如其直径、体积或有时其曲率）相关。我感兴趣的其他最小物体是最小表面。最小表面的原型是肥皂泡的表面。我也对最小曲面面积的估计感兴趣。另一个方向是证明各种最小物体的存在性，但由于拓扑原因它不能立即遵循。事实证明，当我们试图在非紧流形（即在某些自然几何约束下延伸到无穷大的流形）上建立最小对象的存在性时，我们用来估计最小对象“大小”的一些方法有时也有效。例如，我们要研究的自然问题之一是 V. Bangert 提出的以下问题：设 M 是有限体积的完全非紧黎曼流形，M 上是否总存在周期性测地线？