Complexity and Variational Problems in Differential Geometry

我对几何变分微积分的定量方面很感兴趣。在1951年，j.p Serre证明了封闭黎曼流形上的每两个点都可以由无限组不同的测地线连接起来。我想证明它们中的第k个的长度有一个形式为f(n,k)d的上界，其中n是流形的维数，d是流形的直径。我感兴趣的是最短周期测地线的长度和最小曲面的最小面积的类似的无曲率上界。我也对流形上固定点对和测地线网之间的测地线段分布感兴趣。在另一个方向上，我想把我以前关于黎曼泛函摩尔斯地貌的分形特征的结果扩展到涉及里奇曲率下界的尺度不变黎曼泛函。这将涉及证明一些关于黎曼流形的新结果，黎曼流形的曲率从下有界。封闭黎曼流形的概念是封闭曲面的高维推广，如甜甜圈或球体的表面。我们计划研究封闭黎曼流形上各种极值对象的“尺寸”与流形的“尺寸”之间的联系。极值对象的例子包括测地线段（即两点之间最直的曲线），周期测地线（即流形上平滑地靠近自己的直线曲线），测地线（当试图通过最短树连接三个或更多点时产生的对象）和最小表面（即数学模型，如肥皂泡）。在另一个方向上，我们计划研究高维流形的“最佳”形状。我们解决最后一个问题的方法涉及到来自不同数学领域的思想，包括可计算性理论。

I am interested in quantitative aspects of Geometric Calculusof Variations. In 1951 J.P. Serre proved that every two pointson a closed Riemannian manifold can be connected by an infinite setof distinct geodesics. I would like to prove that the lengths of the firstk of them admit an upper bound of the form f(n,k)d, where n is the dimensionand d is the diameter of the manifold. I am interested in similar curvature-free upper bounds for the length of the shortest periodic geodesicand the smallest area of a minimal surface. I am also interestedin distribution of geodesic segments between a fixed pair of pointsand geodesic nets on a manifold. In another direction I would like to extendmy previous results on fractal features of Morselandscapes of Riemannian functionals to scale-invariant Riemannianfunctionals involving a lower bound for the Ricci curvature. This wouldinvolve proving some new results about Riemannian manifolds with Riccicurvature bounded from below.The notion of a closed Riemannian manifold is a higher dimensionalgeneralization of a closed surface, like the surface of a donut, or a sphere.We plan to study connections between ``sizes" of various extremalobjects on a closed Riemannian manifold and the ``size" of the manifold.Examples of extremal objects include geodesic segments (i.e.straightest curves between two points), periodic geodesics (i.e.straight curves on manifolds that smoothly close on themselves), geodesicnets (objects that arize when one tries to connect three or more pointsby a shortest tree) and minimal surfaces (i.e. mathematical modelsof soap bubbles). In another direction we plan to study"optimal" shapes of higher dimensional manifolds. Our approach to this lastquestion involves ideas coming from different areas of Mathematics,including Computability Theory.