Metric geometry and Finsler structures of low regularity

在整个20世纪，芬瑟几何一直是微分几何中的一个重要学科。世纪。最近，随着新方法、新问题和新方向的发展，以及与数学其他部分的有趣联系，数学引起了人们的浓厚兴趣。然而，许多重要的芬斯勒流形实例不满足这些光滑或凸性条件，许多标准方法失效。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利普希茨，而芬斯勒度量是一个可测量的，可能是退化的函数。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/或非强凸芬斯勒流形的几何和解析性质及其在几何中的作用。几何分析与应用数学。这些技术从度量几何中汲取了很多，因此我们的研究计划包含了一些形式上属于度量几何的主题。更具体地说，我们将调查以下主题：发展了弱度量空间中的可整流曲线理论。研究弱度量空间到弱巴拿赫空间的等距嵌入。阐明了Lipschitz结构和弱Finsler结构流形上的弱度量之间的关系。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”。调查各种自然的拉普拉斯人

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