Curvature-Free Estimates for Extremal Objects in Riemannian Geometry and Quantitative Topology

本提案的主要目标是建立黎曼几何中最小物体的长度/面积/体积的无曲率估计，例如（不是最短的）测地线段，封闭测地线，测地线环路，最小曲面等。特别地，J. P. Serre证明了对于封闭黎曼流形上的任意两点，存在无穷多个测地线段将它们连接起来。虽然连接任意两点的最短测地线段的长度最多等于流形的直径是一个平凡的陈述，但研究者建议研究其他测地线段的长度。例如，找出是否总是存在n个不同的测地线段，其长度最多为封闭流形直径的n倍，这将是很有趣的。黎曼几何是曲面几何向高维的推广。黎曼几何研究的一些中心对象是测地线、测地线网、最小曲面和其他最小物体。测地线将直线的概念推广到黎曼几何中。直线的两个基本性质是：(1)直线；(2)使两点之间的距离最小。我们将测地线定义为位于给定黎曼流形上的“最直”曲线。结果是两点之间的距离被测地线段最小化了。然而，测地线和线段之间的类比并不完美。例如，在封闭黎曼流形的情况下，有无限多个测地线线段连接任意两点，正如J. P. Serre所证明的那样。另外，一些测地线彼此靠近，变得周期性。研究者建议研究连接封闭黎曼流形上任意两点的各种测地线段的长度与该流形的大小之间的联系，该流形由其体积和/或直径表示，定义为两点之间的最大距离。研究者还建议研究流形上最短封闭测地线长度与流形尺寸之间的关系，以及流形尺寸与其他各种最小对象之间的关系。

The main goal of this proposal is to establish curvature-free estimates for the length/area/volume of minimal objects in Riemannian geometry, such as (not the shortest) geodesic segments, closed geodesics, geodesic loops, minimal surfaces, etc. In particular, J. P. Serre had shown that for any two points on a closed Riemannian manifold there exist infinitely many geodesic segments joining them. While it is a trivial statement that the length of a shortest geodesic segment joining any two points equals to at most the diameter of a manifold, the investigator proposes to study the lengths of the other geodesic segments. For example, it would be interesting to find out whether there always exist n distinct geodesic segments of length at most n times the diameter of a closed manifold.Riemannian geometry is a generalization of geometry of surfaces to higher dimensions. Some of the central objects of study of Riemannian geometry are geodesics, geodesic nets, minimal surfaces and other minimal objects. Geodesics generalize the notion of a straight line to Riemannian geometry. The two essential properties of a line are: (1) that it is straight; (2) that it minimizes distance between two points. We define geodesic as "the straightest" curve that lies on a given Riemannian manifold. It turns out that the distance between two points is minimized by a geodesic segment. However, the analogy between geodesic and line segments is not perfect. For example, in the case of a closed Riemannian manifold there are infinitely many geodesic segments connecting any two points, as it was shown by J. P. Serre. Also, some geodesics close on themselves and become periodic. The investigator proposes to study the connection between the lengths of various geodesic segments connecting any two points on a closed Riemannian manifold and the size of this manifold represented by its volume and/or the diameter, defined as the maximal distance between two points. The investigator also proposes to study the relationship between the length of a shortest closed geodesic on a manifold and the size of a manifold, as well as relations between the size of a manifold and various other minimal objects.