Complexity and variational problems in Rienmannian geometry

本文的研究对象是黎曼流形，它可以看作是三维空间中曲面在高维空间中的推广。我感兴趣的黎曼流形，有一个“最佳”的形状，也就是说，是较少弯曲比所有附近的黎曼流形。我的另一个研究对象是黎曼流形上的测地线。测地线是黎曼流形上最直的线。(For例如，平面上的测地线是直线，圆球上的测地线是大圆，等等）。众所周知，如果一个流形是封闭的（如球面或环面），那么每两个点可以由无限多个不同的测地线连接。这些测地线变得越来越长。我相信，并将试图证明，对于每一个n可以估计长度的第n个这些测地线在n和最大距离点之间的流形。但这里不需要有关流形曲率的信息。最后，我将研究封闭测地线网的形状。测地线网是一种非常神秘的物体，当人们试图弄清楚如何用最少的电缆连接几个电话站（在表面的不同点）时，它会自然出现。

The object of my study are Riemannian manifolds, which can be regarded as a generalization of surfaces in three-dimensionalspace for higher dimensions. I am interested in Riemannian manifolds that have an "optimal" shape, that is, are less curved than all nearby Riemannian manifolds. My other object of studyis geodesics on Riemannian manifolds. Geodesics are the straightest lines on Riemannian manifolds. (For example, geodesics on a plane are straight lines, geodesics on theround sphere are big circles, and so on). It is known that if a manifold is closed (like a sphere or a torus), then every two points can be connected by infinitely many distinct geodesics. These geodesics become longer and longer. I beleive and will try to prove that for every n one canestimate the length of the nth of these geodesics in terms of n and the maximal distance between points on the manifold. But no information about curvature of the manifold is necessary here. Finally, I am going to study shapes of closed geodesic nets. Geodesic nets are quite a mysterious class of objects that naturally appears, when one tries to figure out how to connect several phone stations (at different points of a surface) by the minimal amount of  cable.