Non-positive curvature and cubical surfaces

这个项目的主要目标是展示和分析嵌入（或浸入）在高维立方流形中的高格曲面。为此，我们建立在诸如组合完整概念和分支覆盖等技术的基础上，这些技术是在JZ项目“组合完整”的第一个资助阶段开发的。此外，我们的方法将以基本的方式使用亚历山德罗夫和格罗莫夫意义上的组合曲率的离散概念（见[19]）。为了解决关于立方体（和其他多面体）表面的已知开放问题，我们想要访问更广泛的有趣的候选曲面（特别是高曲率/高属曲面，以及具有极值f向量的曲面）。为此，我们首先对某些高维立方流形中的强规则组合立方曲面（嵌入或浸入）进行了研究和分类。然后，我们可以应用项目Z（3.7）中开发的技术来确定（要找到的）表面是否也可以嵌入到欧几里德空间中。

The main goal of this project is to exhibit and to analyze high genus surfaces that appear embedded (or immersed) in higher-dimensional cubical manifolds. For this we build on techniques such as combinatorial holonomy concepts and branched coverings that were developed in the first funding period of Project JZ, "Combinatorial Holonomy". Additionally, our methods will use discrete concepts of combinatorial curvature in the sense of Alexandrov and Gromov (see [19]) in an essential way.In order to tackle known open problems about cubical (and other polyhedral) surfaces we want to access a wider class of interesting candidates (in particular, high curvature/high genus surfaces, and surfaces with extremal f-vector). For this, we first study and classify strongly regular combinatorial cubical surfaces (embedded or immersed) in certain higher-dimensional cubical manifolds. Then we can apply techniques developed in Project Z (3.7) to decide if the surfaces (to be) found can also be embedded into Euclidean space.