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，但它们*很多时候可以推广到更一般的度量空间，即我们可以测定点之间的距离的空间。黎曼几何是欧氏三维空间中曲面几何的推广。这种表面的一些例子是圆形的球体(我们都知道地球表面的形状是球体)，或者环面，看起来像一个甜甜圈的形状。如果我们在放大镜下看一个曲面的一个点的邻域，它会看起来像一个平面。因此，我们的很多直觉都来自欧几里得几何。在曲面的每个点上，都有一种方法来测量切线向量之间的内积。曲面上的内积的概念概括为黎曼度量。它给了我们一种方法来测量点之间的距离，定义流形的直径(即流形上两点之间的最大距离)，它的体积，它的曲率等。它还给我们一种方法来推广直线的概念到黎曼流形，得到测地线的概念，即流形上最直的曲线。虽然直线在两个方向上无限延伸，但测地线可以自己闭合，当它们平滑地进行时，会产生一条周期性的测地线，就像球面上的赤道一样，或者在测地线回路中。我以前的许多结果以及提出的问题都涉及测地线段、测地线环或周期测地线的长度如何与流形的其他参数相关，如流形的直径、体积或其曲率的某些倍。我感兴趣的其他极小对象是极小曲面。极小曲面的原型是肥皂泡的曲面。我还对极小曲面面积的估计感兴趣。另一个方向是证明各种极小对象的存在，当它由于拓扑原因而不是立即跟随时。事实证明，当我们试图证明非紧致流形(即在某些自然几何约束下延伸到无穷远的流形)上的极小对象的存在时，我们有时用来估计极小对象“大小”的一些方法也是有效的。例如，我们想要研究的自然问题之一是V.邦格特提出的下列问题：设M是有限体积的完备非紧黎曼流形，M上是否总有周期测地线？

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?