Isoperimetric filling problems and the large scale geometry of metric spaces

拟议的研究位于度量几何和度量空间分析的交叉点上。该项目的主要目的是通过对高维等周不等式的研究，加深对各类度量空间的大规模几何的理解。该项目的第一部分和主要部分集中在Alexandrov意义下的非正曲率的单连通测地度量空间，称为Hadamard空间。在这种情况下，一个特定的目标是在高维等周函数(和其他填充不变量)的行为与各种秩概念之间建立联系。如果成功，这项研究将回答Gromov关于线性等周不等式的猜想，该猜想断言在适当的余紧Hadamard空间中，在以下维度上的欧几里德行为和在欧几里德等级以上的维度上的线性行为。所提出的方法依赖于几何测度论和度量空间分析的方法，特别是最近由Ambrosio和Kirchheim发展起来的度量流的概念。因此，该项目的第二个重要部分将是进一步发展度量流理论，该理论提供了度量空间中曲面的适当概念，并为研究几何分析中的许多其他问题提供了强大的工具，例如涉及能量最小化的问题。等周不等式在几何群论中也起着重要的作用。对于有限表示的群，一维等周函数衡量了字问题的复杂性，在过去的15年里得到了广泛的研究。另一方面，高维等周函数的研究直到最近才成为一个非常活跃的研究领域。该项目的最后一部分研究了幂零李群背景下的填充问题。等周问题的研究有很长的历史。它不仅启发了许多数学家，而且还导致了许多新的数学理论。在最经典的情况下--古希腊人已经知道了--等周不等式断言，平面上的一条闭合曲线包围的面积不超过相同周长的圆盘的面积。高维类似物关注的是k维闭曲面被(k，1)维曲面填充得多好的问题。填充问题出现在许多不同的数学领域，包括分析、几何、概率论和群论。本课题的目的是研究曲率对等周填充问题的影响。这一研究将特别有助于对非正曲线奇异空间几何的更深层次的理解。例如，在台球轨迹的数学研究中会出现这种情况。该项目建议使用几何测度论领域的方法，处理奇异曲面的研究。这一理论在很大程度上受到普劳图问题的启发，普劳图问题要求存在一个具有规定边界的极小表面(或肥皂膜)。这个问题的高维类比是海流理论发展的起点，如今海流理论的许多应用远远超出了高原问题。

The proposed research is located at the intersection of metric geometry and analysis on metric spaces. The principal aim of the project is to gain a deeper understanding of the large scale geometry of various classes of metric spaces through the study of higher-dimensional isoperimetric inequalities. The project's first and main part focuses on simply connected geodesic metric spaces of non-positive curvature in the sense of Alexandrov, called Hadamard spaces. In this setting, a particular goal is to establish connections between the behavior of higher-dimensional isoperimetric functions (and other filling invariants) and various notions of rank. If successful, the research will result in an answer to Gromov's conjecture on linear isoperimetric inequalities which asserts Euclidean behavior in the dimensions below and linear behavior in the dimensions above the Euclidean rank in a proper cocompact Hadamard space. The proposed approach relies on methods from geometric measure theory and analysis on metric spaces, in particular on the notion of metric currents, the theory of which has recently been developed by Ambrosio and Kirchheim. An important second part of the project will thus be to further develop the theory of metric currents, which provides a suitable notion of surfaces in a metric space and also gives powerful tools in the study of many other problems in geometric analysis, e.g. such involving energy minimization. Isoperimetric inequalities also play a fundamental role in geometric group theory. For finitely presented groups one-dimensional isoperimetric functions measure the complexity of the word problem and have been extensively studied over the past 15 years. The study of higher-dimensional isoperimetric functions, on the other hand, has only recently become a very active field of research. A last part of the project investigates filling problems in the context of nilpotent Lie groups.The study of isoperimetric problems has a long history. Not only has it inspired many mathematicians, but it has also led to many new theories in mathematics. In the most classical setting-known already to the Ancient Greeks-the isoperimetric inequality asserts that a closed curve in the plane encloses an area no larger than that of a disc of the same circumference. Higher-dimensional analogues are concerned with the question of how well a k-dimensional closed surface can be filled with a (k+1)-dimensional surface. Filling problems appear in many different fields of mathematics, including analysis, geometry, probability theory and group theory. The aim of the present project is to study the effects that curvature has on the isoperimetric filling problem. The investigation will in particular lead to a deeper understanding of the geometry of non-positively curved singular spaces. Such arise for example in the mathematical study of billiard trajectories. The project proposes to use methods from the field of geometric measure theory, dealing with the study of singular surfaces. This theory was to a large extent inspired by Plauteau's problem, which asks for the existence of a minimal surface (or a soap film) with prescribed boundary. Higher-dimensional analogues of this question were the starting point in the development of the theory of currents, which nowadays has many applications going far beyond Plateau's problem.