Stability for nonlocal curvature functionals

非局部曲率泛函的稳定性

• 批准号: EP/W014807/2
• 负责人: Julian Scheuer
• 金额: $ 1.71万
• 依托单位: Goethe University Frankfurt
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW014807%2F2
• 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.

在几何和分析中，曲面的曲率及其与曲面形状的连接是最广泛研究的主题之一。在过去的20年里，他们的重要性通过向佩雷尔曼（2006年）和菲加利（2018年）颁发菲尔兹奖而得到了强调。这个项目是关于曲率的分析和几何方面。让我们先来看看曲率的几何方面。在局部水平上，曲面在一点处的曲率可以通过曲面在该点附近的弯曲量和方向直观地可视化。这种曲率的概念，在下文中称为“局部曲率”，是经典的概念，200多年来，它一直是分析中的一个主要主题，即局部曲率如何决定曲面的形状。例如，如果在适当的意义上，所有表面点之间的局部曲率是恒定的，则该表面必须是平坦的平面或圆形球体的一部分。放松这个陈述中的假设，如果曲率是“几乎”恒定的，那么曲面必须“接近”一个圆形球体，其中这些术语必须精确定义以做出严格的数学陈述。这类问题是当前研究的边缘问题，这个项目是关于将所描述的味道问题扩展到曲率的新概念。例如，闭合表面的曲率的另一个概念可以是，在表面点处接触的球最多可以有多大，以便适合由表面包围的区域。相反的局部曲率，这只取决于表面的形状“附近”的一个点，这个新的概念的曲率取决于整体形状的表面，不能测量的小居民的表面。因此，我们称这种概念为“非局部曲率”。这方面的研究刚刚开始，大多数成果是在过去10年中取得的。有许多可能的方法来定义非局部曲率的概念，对于一些特定的例子，本项目旨在探索它们与曲面整体形状的联系。在曲率的分析方面，局部曲率类似于曲面参数化的二阶导数，这不应该让人感到惊讶，因为它代表了曲面的弯曲。因此，这个分支的研究是密切相关的偏微分方程的二阶。在分析有一个非本地版本的衍生物，这通常被称为“分数阶导数”。这些定义使用合适的积分在整个域的功能。因此，由于局部曲率是通过函数的经典导数定义的，因此通过类比，可以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，我们希望它能引起从事几何分析的科学界的广泛兴趣。