双曲空间和AdS空间中的Weyl问题

经典Weyl问题研究三维欧氏空间中给定边界度量的有限凸域的实现问题。Thurston猜想、Mess猜想和几何Koebe猜想可视为双曲和AdS情形的广义Weyl问题，它们涉及到Teichmuller理论、双曲几何、AdS几何和凸几何等研究方向。这些猜想的存在性部分在多数情形已被证明，唯一性部分仍然是公开问题。 基于这些猜想的发展现状和趋势，我们将研究三维双曲空间和AdS空间中无限凸域的Weyl问题，包括拟Fuchs双曲情形、双曲“万有”情形和拟Fuchs AdS情形。通过“形变”方法和“逼近”思想，我们拟解决对应凸域的存在唯一性问题，将上述猜想发展成更一般的理论，并应用于曲面形变和拟圆几何特征化的研究。

The classical Weyl problem is the realization problem of bounded convex domains with prescribed metric on the boundary in the 3-dimensional Euclidean space. Thurston Conjecture, Mess Conjecture and Geometric Koebe Conjecture can be viewed as generalized Weyl problems in the hyperbolic and AdS cases, which involve various research areas, including Teichmuller theory, hyperbolic geometry, AdS geometry and convex geometry. The existence parts of these conjectures have been proven in most cases, while the uniqueness parts are still open. Based on the development status and trends of these conjectures, we will study the Weyl problems of unbounded convex domains in the 3-dimensional hyperbolic and AdS spaces, including the quasifuchsian hyperbolic case, “universal”hyperbolic case and quasifuchsian AdS case. By using the “deformation”argument and“approaching”ideas, we plan to solve the existence and uniqueness problems for corresponding convex domains, develop a more general theory to understand the above conjectures, and apply it to study surface deformations and the geometric characterization of quasi-circles.