Complexity and variational problems in Rienmannian geometry

我的研究对象是黎曼流形，它可以被看作是三维空间中曲面在更高维度上的推广。我感兴趣的是具有“最优”形状的黎曼流形，也就是说，它比所有邻近的黎曼流形曲线都要小。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。（例如，平面上的测地线是直线，圆球上的测地线是大圆，等等）。我们知道，如果流形是封闭的（如球体或环面），那么每两个点都可以由无限多个不同的测地线连接起来。这些测地线变得越来越长。我相信并将尝试证明，对于每一个n，我们可以用n和流形上点之间的最大距离来估计第n个测地线的长度。但是这里不需要关于流形曲率的信息。最后，我将研究封闭测地线网的形状。当一个人试图弄清楚如何用最少的电缆连接几个电话站（在一个表面的不同点上）时，测地线是一种相当神秘的物体，自然会出现。

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.