Quasisymmetric Maps, Doubling Measures, and Geometry of Banach Spaces

这个项目汇集了几何分析、调和分析和非线性泛函分析的主题。连接这三个领域的概念是增生图的概念，它可以被认为是一个实变量的递增函数的深远推广。该项目要解决的一些问题如下：(1)具有非光滑共形度量的欧氏空间的双Lipschitz等价性，特别是与由Riesz势定义的度量的双Lipschitz等价；(2)作为可能非加倍度量的矢量势的拟对称映射的构造；(3)满足加倍条件的更强的各向同性形式的度量的几何性质；(4)Banach空间中近似对偶映射的一致连续性和Lipschitz连续性；(5)关于增生映射和实值函数的复梯度的Beurling-Ahlfors扩张定理的变体。几何和分析中的一个基本问题是找到给定度量空间(即，具有距离概念的集合)的参数化，从而使空间可视化和理解其结构成为可能。(出现这个问题的一个实际情况是构建哺乳动物大脑皮层的平面地图。)黎曼映射定理提供了一种曲面的参数化，该曲面保留了所有角度，但会显著扭曲点之间的距离。有时，可以通过仔细选择曲面所映射到的平面区域的附加变形来减少距离的扭曲。我们目前对欧几里得空间的这种形变的理解甚至在两个维度上都是远远不完整的。首席研究人员对使用增值地图来阐明这一问题很感兴趣。在增生式地图下，空间可以在多个位置显著拉伸或收缩，但它的旋转永远不会超过预定的量。

This project brings together topics from geometric analysis, harmonic analysis, and nonlinear functional analysis. The concept that connects the three areas is the notion of an accretive map, which can be thought of as a far-reaching generalization of an increasing function of one real variable. Some of the problems to be addressed within the project are the following: (1) bi-Lipschitz equivalence of Euclidean spaces with nonsmooth conformal metrics, in particular with metrics defined by Riesz potentials; (2) construction of quasisymmetric maps as vector-valued potentials of possibly non-doubling measures; (3) geometric properties of measures that satisfy a stronger, isotropic form of the doubling condition; (4) uniform continuity and Lipschitz continuity of approximate duality maps in Banach spaces; (5) variants of the Beurling-Ahlfors extension theorem for accretive maps and for complex gradients of real-valued functions. A fundamental problem in geometry and analysis is to find a parametrization of a given metric space (i.e., a set with a notion of distance) that makes it possible to visualize the space and to understand its structure. (One practical situation in which this issue arises is the construction of flat maps of the mammalian cerebral cortex.) The Riemann mapping theorem provides a parametrization of a surface that preserves all angles but that can significantly distort distances between points. This distortion of distances can sometimes be reduced by a carefully chosen additional deformation of the flat region onto which the surface is mapped. Our current understanding of such deformations of Euclidean spaces is far from complete even in two dimensions. The principal investigator is interested in using accretive maps to shed new light on this subject. Under an accretive map a space can be significantly stretched or shrunk at multiple locations, but it is never rotated by more than a predetermined amount.