Metric geometry and functions of bounded variation

自然界中出现的物体很少有光滑的外表。从分形物体到多孔介质，控制热量和压力等量耗散的方程都有非光滑分量。这种非光滑物体也作为某些光滑物体的极限出现。为了理解和利用这些物体的行为，我们需要从黎曼几何中去除平滑假设。对于这样的非光滑对象，我们需要研究非光滑能量最小化的行为及其与底层空间或对象的几何结构的联系，即结构性质。这个关于度量几何和有界变分函数的项目通过探索具有度量的底层度量空间几何与最小表面积集的性质之间的相互联系，有助于实现这一目标。这里研究的对象是度量空间，配备了加倍的度量（也就是说，一个球的度量与半径加倍的同心球的度量相当），这样，一个球上的利普希茨连续函数的方差可以根据它在该球上的能量的平均值来控制（使用其局部振荡计算）。在此背景下，本项目旨在研究局部最小表面积集的可整流性、孔隙度、形状和自然尺寸等特性。对度量空间的分析有多种来源复解析函数的研究、控制混合材料断裂的微分方程和有限变形的相关映射、工程中的控制理论和相关的carno - caratheodory空间、著名的庞加莱猜想的研究是这一研究领域的一些基础。然而，与欧几里得情形不同的是，最小表面（具有最小表面能的表面）的结构性质被很好地理解，在自然界中出现的非光滑环境以及物理和工程的控制理论问题中，最小表面的结构被理解得很少。该项目旨在探索这种结构，并扩展我们对非光滑环境中最小表面的认识。这项研究的结果将有助于进一步理解由于热和压力等自然效应而改变的物体的行为变化。除了提高我们对非光滑物体的认识外，本项目进行的研究也将有助于度量空间分析理论的持续发展；这一发展理论的一大好处是，由于经典欧几里得分析中可用的许多结构工具在非光滑环境中不可用，因此必须开发新的工具和方法，使理论更容易为更广泛的受众所接受。

Objects occurring in nature rarely are smooth in appearance. From fractal objects to porous media, the equations that govern dissipation of quantities such as heat and pressure have non-smooth components. Such non-smooth objects also occur as limits of certain smooth objects. To understand and exploit the behavior of such objects, we need to remove the smoothness assumptions from Riemannian geometry. For such non-smooth objects we need to study behaviors of non-smooth energy minimizers and their connection to the geometry, that is, the structural properties, of the underlying space or object. This project on metric geometry and functions on bounded variation contributes to this goal by exploring interconnections between the geometry of the underlying metric space equipped with a measure, and the properties of sets of minimal surface areas. The study of objects here are metric spaces equipped with a measure that is doubling (that is, measure of a ball is comparable to the measure of a concentric ball of double the radius) and such that the variance of a Lipschitz continuous function on a ball can be controlled in terms of the average value of its energy (computed using its local oscillation) on that ball. In this setting, this project aims to study properties such as rectifiability, porosity, shape, and natural dimension of sets of locally minimal surface areas. Analysis on metric measure spaces arose from many sources; the study of complex analytic functions, differential equations governing fractures in mixed material and associated mappings of finite distortion, control theory in engineering and associated Carnot-Caratheodory spaces, the study of the famous Poincare conjecture, are some of the roots of this field of study. However, unlike in the Euclidean situation where structural properties of minimal surfaces (surfaces with smallest surface energy) are well understood, in the non-smooth setting that arise in nature and in control theory problems of physics and engineering, the structure of minimal surfaces is poorly understood. This project seeks to explore such structures and expand our knowledge of minimal surfaces in a non-smooth setting. The results of this study will be useful in further understanding the change in the behavior of objects that are transformed due to natural effects such as heat and pressure. In addition to advancing our knowledge of non-smooth objects, the study conducted in this project will also contribute to the ongoing development of the theory of analysis on metric spaces; a great benefit of this developing theory is that since much of the structural tools available in classical Euclidean analysis are not available in the non-smooth setting, new tools and methods have necessarily to be developed, making the theory more accessible to a wider audience.