Quantitative Homotopy Theory and Geometric Inequalities

我的大多数结果都是各种几何不等式的证明。虽然这些不等式是在黎曼几何的背景下证明的，但很多时候它们可以推广到更一般的度量空间，即我们可以测量点之间距离的空间。黎曼几何是欧几里德三维空间中曲面几何的推广。这些表面的一些例子是一个圆球，（我们都知道地球的表面有球体的形状），或一个环面，形状看起来像一个甜甜圈。如果我们在放大镜下观察一个表面上一点的邻域，它看起来就像一个平面。因此，我们的很多直觉都来自于欧几里得几何。在曲面的每一点上都有一种方法来测量切向量之间的内积。曲面上内积的概念推广为黎曼度规。它为我们提供了一种方法来测量点之间的距离，来定义流形的直径（即流形上点对之间的最大距离），它的体积，它的曲率等。它也给了我们一种将直线的概念推广到黎曼流形的方法，得到了测地线的概念，即流形上最直的曲线。当直线在两个方向上无限延伸时，测地线可以自我闭合，当它们平滑地闭合时，就会产生周期性的测地线，就像球面上的赤道或测地线环路一样。我以前的许多结果，以及提出的问题，都涉及到测地线段、测地线回路或周期测地线的长度如何与流形的其他参数（如直径、体积，有时是曲率）相关。其他我感兴趣的最小物体是最小表面。最小表面的一个原型是肥皂泡的表面。我对最小曲面面积的估计也很感兴趣。另一个方向是证明各种最小对象的存在性，当由于拓扑原因不能立即遵循时。事实证明，我们用来估计最小对象“大小”的一些方法有时也适用于我们试图在非紧化流形上建立最小对象的存在性，即在某些自然几何约束下伸展到无穷大的流形。例如，我们想要研究的一个自然问题是由V. Bangert提出的：设M是有限体积的完全非紧黎曼流形，M上是否总是有一个周期测地线？

Most of my results are proofs of various geometric inequalities. While these inequalities are proven in the setting of Riemannian Geometry, a lot of times they ***can be generalized to more general metric spaces, i. e. spaces in which we can measure distances between points. Riemannian Geometry is the generalization of the geometry of surfaces that lie in a Euclidean three dimensional space. Some of the examples of such surfaces are a round sphere, (as we all know the surface of the Earth has the shape of the sphere), or a torus, the shape that looks like a donut. If we look at the neighbourhood of a point of a surface under the magnifying glass, it will look like a plane. Therefore, a lot of our intuition comes from the Euclidean geometry. At each point of a surface there is a way to measure an inner product between the tangent vectors. This notion of the inner product on a surface generalizes as the Riemannian metric. It gives us a way to measure the distance between points, to define the diameter of a manifold, (i. e. the maximal distance between pairs of points on a manifold), its volume, its curvature, etc. It also give us a way to generalize the notion of a straight line to Riemannian manifolds, obtaining a notion of geodesics, i.e. the straightest curves on a manifold. While the straight line stretches infinitely in both directions, geodesics can close on themselves, resulting in a periodic geodesic, when they do so smoothly, just like the Equator on a sphere, or in a geodesic loop. Many of my previous results, as well as the proposed problems deal with how the lengths of geodesic segments, geodesic loops, or periodic geodesics relate to the other parameters of the manifold, such as its diameter, volume, or some times its curvature. Other minimal objects that are of interest to me are minimal surfaces. A prototype of a minimal surface is a surface of a soap bubble. I am also interested in estimates of the areas of minimal surfaces. Another direction is to prove the existence of various minimal objects, when it does not immediately follow for the topological reasons. It turns out that some of the methods that we use to estimate the "size" of minimal objects some times also works when we try to establish the existence of minimal objects on noncompact manifolds, i.e. the manifolds that stretch to infinity, under certain natural geometric restraints. For example, one of the natural questions that we want study is the following question attributed to V. Bangert: Let M be a complete noncompact Riemannian manifold of finite volume, is there always a periodic geodesic on M?