Metric geometry and Finsler structures of low regularity

公制几何和低正则性芬斯勒结构

• 批准号: 390960259
• 负责人: Professor Dr. Vladimir Matveev
• 金额: --
• 依托单位: Lehrstuhl für Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/390960259?language=en
• 关键词: Metric; geometry; Finsler; structures; low

Finlser geometry has been an important subject in differential geometry throughout the 20th. century. It has recently attracted a strong renewed interest with new methods, problems and directions being developed as well as interesting connections with other parts of mathematics. However a number of important examples of Finsler manifolds do not satisfy these smoothness or convexity conditions and many of the standard methods break down. Our project aims to close this gap. In the most extreme setup we merely assume the manifold to be Lipschitz, and the Finsler metrics to be a measurable, possibly degenerate function. In this research project we principally aim at a deeper understanding of the geometry and analytic properties of non smooth and/or non strongly convex Finsler manifolds and their role in geometry. geometric analysis and applied mathematics. The techniques draw alot from metric geometry and therefore our research program contain some topics which formally belong to metric geometry. More specifically we shall investigate the following topics:1. Develop the theory of rectifiable curves in weak metric spaces.2. Study isometric embedding of weak metric spaces to weak Banach spaces.3. Clarify the relation between weak metrics on a manifold that are Lipschitz and weak Finsler structures.4. Describe weak metric spaces admitting a non trivial dilation.5. Study the geodesics and a class of ``special geodesic'' on weak Finsler spaces.6. Investigate the various natural Laplacians

Finlser几何在整个20世纪一直是微分几何中的一个重要课题。世纪。它最近吸引了一个强大的新的兴趣与新的方法，问题和方向正在开发以及有趣的连接与其他部分的数学。 然而，一些重要的例子芬斯勒流形不满足这些光滑性或凸性条件和许多标准的方法打破。我们的项目旨在缩小这一差距。在最极端的情况下，我们仅仅假设流形是Lipschitz，芬斯勒度量是一个可测的、可能退化的函数。在这个研究项目中，我们的主要目的是更深入地了解非光滑和/或非强凸Finsler流形的几何和解析性质及其在几何中的作用。几何分析和应用数学。 这些技术从度量几何中吸取了很多，因此我们的研究计划包含了一些正式属于度量几何的主题。更具体地说，我们将研究以下主题：1。发展了弱度量空间中的可求长曲线理论.研究弱度量空间到弱Banach空间的等距嵌入.阐明流形上的弱度量Lipschitz与弱Finsler结构之间的关系.描述允许非平凡扩张的弱度量空间.研究弱Finsler空间上的测地线和一类"特殊测地线“.研究各种自然拉普拉斯算子