Stability for nonlocal curvature functionals

非局部曲率泛函的稳定性

• 批准号: EP/W014807/1
• 负责人: Julian Scheuer
• 金额: $ 4.75万
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW014807%2F1
• 关键词: Stability; nonlocal; curvature; functionals

In Geometry and Analysis, the curvature of and its connection to the shape of a surface is one of the most widely researched topics. During the past 20 years, their significance has been highlighted through Fields Medal awards to Perelman (2006) and Figalli (2018). This project is about analytical and geometrical aspects of curvature. Let us first have a look at the geometric aspects of curvature. On the local level, the curvature of a surface at a point can intuitively be visualized by the amount and direction of the bending of a surface near this point. This notion of curvature, in the following called "local curvature", is the classical notion and for more than 200 years it has been a major theme in Analysis, how local curvature determines the shape of the surface. For example, if the local curvature is constant among all of the surface points in a suitable sense, then the surface must be a flat plane or a piece of a round sphere. Relaxing the hypothesis in this statement, it is also true that if the curvature is "almost" constant, then the surface must be "close" to a round sphere, where those terms have to be defined precisely to make a rigorous mathematical statement. Questions of this sort are on the edge of current research.This project is about extending questions of the described flavour to new notions of curvature. For example, another notion of curvature of a closed surface could be, how big a ball touching at a surface point may at most be in order to fit into the region that is enclosed by the surface. Contrary to the local curvature, which only depends on the shape of the surface "nearby" a points, this new notion of curvature depends on the global shape of the surface and can not be measured by small inhabitants of the surface. Hence we call such notions "non-local curvature". Research in this area has just started and most results have been developed during the past 10 years. There are many possible ways to define notions of non-local curvature and for a few particular examples, this project intends to explore their connection to the global shape of the surface. Coming to the analytical aspects of curvature, the local curvature is resembled by the second derivative of a parametrisation of the surface, which should not come as a surprise given that it represents the bending of the surface. Hence this branch of research is closely related to the study of partial differential equations of second order.In Analysis there is a non-local version of derivatives, which are usually call "fractional derivatives". Those are defined using suitable integrals over the whole domain of a function. Hence, as local curvature is defined via classical derivatives of a function, by analogy it is tempting to define non-local curvature by fractional derivatives of a function. This is precisely what we are aiming to explore in this project and we hope it will trigger broad interest in the scientific community working in Geometric Analysis.

在几何与分析中，曲面的曲率及其与曲面形状的关系是最广泛研究的课题之一。在过去的20年里，通过授予佩雷尔曼（2006年）和菲加利（2018年）菲尔兹奖，他们的重要性得到了突出。这个项目是关于曲率的分析和几何方面。让我们首先看一下曲率的几何方面。在局部层面上，曲面在某一点的曲率可以通过曲面在该点附近弯曲的量和方向直观地可视化。这个曲率的概念，在下面被称为“局部曲率”，是一个经典的概念，200多年来，它一直是《分析》的一个主要主题，局部曲率如何决定表面的形状。例如，如果在适当的意义上，所有曲面点之间的局部曲率是恒定的，则该曲面必须是一个平面或圆球的一部分。在这个陈述中放松假设，如果曲率“几乎”是恒定的，那么表面必须“接近”一个圆球，这些项必须被精确定义以做出严格的数学陈述。这类问题处于当前研究的边缘。这个项目是关于将所描述的问题扩展到曲率的新概念。例如，一个封闭表面的曲率的另一个概念可以是，一个球在一个表面点上接触的最大可能是多大，以适应被表面包围的区域。与局部曲率相反，局部曲率只取决于“附近”a点的表面形状，这种新的曲率概念取决于表面的整体形状，并且不能由表面的小居民测量。因此我们称这种概念为“非局部曲率”。这方面的研究才刚刚开始，大部分成果是在过去10年里取得的。有许多可能的方法来定义非局部曲率的概念，对于一些特殊的例子，本项目打算探索它们与表面的整体形状的联系。在曲率的解析方面，局部曲率类似于曲面参数化的二阶导数，这并不奇怪，因为它代表了曲面的弯曲。因此，这一研究分支与二阶偏微分方程的研究密切相关。在分析中有一种非局部版本的导数，通常称为“分数阶导数”。它们是用函数整个定义域上合适的积分来定义的。因此，由于局部曲率是通过函数的经典导数来定义的，通过类比，它很容易通过函数的分数阶导数来定义非局部曲率。这正是我们在这个项目中所要探索的，我们希望它能引发科学界对几何分析工作的广泛兴趣。