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

这个建议的主要目的是建立黎曼几何中极小对象的长度/面积/体积的无曲率估计，例如(不是最短的)测地线段、闭测地线、测地线环、极小曲面等。特别是，J.P.Serre证明了对于闭黎曼流形上的任何两个点，存在无穷多个连接它们的测地线段。虽然连接任意两点的最短测地线段的长度至多等于流形的直径这一说法并不重要，但研究者建议研究其他测地线线段的长度。例如，找出是否总是存在长度不超过闭流形直径n倍的不同测地线段将是一件有趣的事情。黎曼几何是曲面几何到更高维的推广。黎曼几何的一些中心研究对象是测地线、测地线网、极小曲面和其他极小对象。测地线将直线的概念推广到黎曼几何。直线的两个基本性质是：(1)它是直线；(2)它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，两点之间的距离通过测地线段来最小化。然而，测地线和线段之间的类比并不完美。例如，在闭黎曼流形的情况下，有无限多个测地线段连接任意两点，正如J.P.Serre所证明的那样。此外，一些测地线封闭了自己，变成了周期性的。研究人员建议研究连接闭黎曼流形上任意两点的不同测地线段的长度与该流形的大小之间的关系，该流形的大小由其体积和/或直径表示，定义为两点之间的最大距离。研究人员还建议研究流形上最短闭测地线的长度与流形的大小之间的关系，以及流形的大小与各种其他极小对象之间的关系。

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.