Rigidity, stability and deformations in nearly parallel G2 geometry

在我们的项目中，我们研究了一类重要的7维爱因斯坦流形：近平行g2流形的性质。这个几何有两个主要元素，爱因斯坦度规和一类结合子流形。我们计划获得更好的理解，特别是通过研究变形，这些元素和几何本身。我们的项目目标可以分为三个部分。在A部分我们想要证明爱因斯坦度规的稳定性或者找到暗示这种稳定性的一般条件。在B部分，我们计划研究近平行G2结构的变形。有一个重要的例子，其中存在的变形仍然不能排除。我们计划回答这个问题，并在此基础上展示几乎平行的G2结构的一般刚度（即不存在变形）。在C部分，我们计划研究结合子流形。我们的目标是在几种自然几何条件下进行分类。我们还计划寻找新的例子并研究变形。在这里，我们想要展示刚性（例如在曲率条件下），但也要展示几何性质，例如在变形下体积的保存。

In our project we study properties of an important class of 7-dimensional Einstein manifolds: nearly parallel G2-manifolds. This geometry has two main elements, the Einstein metric and the class of associative submanifolds. We plan to obtain a better understanding, in particular by the study of deformations, of these elements and of the geometry itself.The goals of our project can be grouped in three parts.In part A we want to prove the stability of the Einstein metric or to find conditions implying this stability in general.In part B we plan to study deformations of the nearly parallel G2 structure. There is one important example where the existence of deformations still could not be ruled out. We plan to answer this question and based on this to show in general the rigidity (i.e. non existence of deformations) of nearly parallel G2 structures.In part C we plan the study of associative sub manifolds. We aim at a classification under several natural geometric conditions. Also we plan to find new example and to study deformations. Here we want to show rigidity (e.g. under curvature conditions), but also geometric properties, e.g. the preservation of the volume under deformations.