Differential geometric study of conformal and CR invariant theory via the ambient metric construction due to Fefferman and Graham

共形几何和CR几何是刚性结构，它们可以通过曲率和积分不变量来区分。不变量理论可以通过抛物Cartan几何/牵引演算来研究。另一种方法是环境度规构造（英语：Ambient metric construction），它基本上等同于庞加莱-爱因斯坦模型。通过这些结构，不变量，如GJMS-算子和Q-曲率可以定义为纯数学研究的目的，在共形和CR几何。此外，“全息关系”的物体上的（共形）边界和内部的庞加莱模型的“体积”可以建立。这在物理学中很有趣，其中全息原理（量子引力）在AdS/CMT对应中找到了具体的表现形式，其目的是将弦理论/超引力与超对称共形场论联系起来。该项目的主要目标是在弯曲的情况下实现费曼-格雷厄姆环境模型和庞加莱-爱因斯坦模型。对于此类模型，应计算共形和CR不变量（另有正式定义）的显式表达式。我们的目标是建立“全息”的边界上的物体之间的关系和庞加莱-爱因斯坦模型的内部。特别是，我们要求庞加莱空间的泰勒展开式和GJMS-算子和Q-曲率的显式公式。另一个问题涉及几何泊松变换的调和解的边界问题的曲线庞加莱模型的积分公式。这也是一项任务，涉及对称性，如解决方案的超定不变微分方程（如扭量形式/旋量）的边界上的几何对象的庞加莱“散装”。

Conformal and CR geometries are rigid structures, which are distinguishable by curvature and integral invariants. The invariant theory can be studied via parabolic Cartan geometry/tractor calculus. An alternative approach developed by Fefferman/Graham is the ambient metric construction, which is basically equivalent to the Poincare-Einstein model. Via these constructions invariants such as the GJMS-operators and Q-curvature can be defined for the purpose of purely mathematical studies in conformal and CR geometry. Moreover, "holographic relations" of objects on the (conformal) boundary and the interior "bulk" of the Poincare model can be established. This is of interest in physics, where the holographic principle (of quantum gravity) finds a concrete manifestation in connection with the AdS/CFT-correspondence, which aims to relate string theory/sup er gravity with sup er symmetric conformal field theories. This project considers as its main goal (geometric) realisations of Fefferman-Graham ambient and Poincare-Einstein models in curved situations. For such models explicit expressions for conformal and CR invariants (which are otherwise formally defined) shall be calculated. We aim to establish "holographic" relations between objects on the boundary and on the interior of the Poincare-Einstein model. In particular, we ask for Taylor expansions of Poincare spaces and explicit formulae for GJMS-operators and Q-curvature. A further question concerns geometric Poisson transformations for harmonic solutions of boundary problems on curved Poincare models in terms of integral formulae. It is also a task to relate symmetries such as solutions of overdetermined invariant differential equations (e.g. twistor forms/spinors) on the boundary to geometric objects on the Poincare "bulk".