论文标题
关于双曲线和抗DE保姆空间中完整表面的Weyl问题
On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces
论文作者
论文摘要
经典的Weyl问题（由Lewy，Alexandrov，Pogorelov和其他问题解决）询问是否在$ \ r^3 $的独特凸面的边界上诱导了球体上的任何曲率$ k \ geq 0 $。 Alexandrov在1950年代将答案扩展到双曲线空间中的表面，并用``双重''陈述``dual''陈述，用其边界的第三种基本形式描述了凸体（例如，其二面角，为理想的多面体）。 我们以$ \ hh^3 $的形式描述了Weyl问题的三个猜想概括，并以与当代几何形状相关的方式对无限的凸子集和凸表面进行了双重概括，因为许多最近的结果和众所周知的开放问题可以被视为特殊情况。一个重点是具有``薄''渐近边界的凸形域，例如一个准圆形 - 该问题的这一部分与克莱琳基团的理论密切相关。第二个方向是朝具有``厚''理想边界的凸子集，例如磁盘的不相交联合 - 在这里发现与复杂分析中的问题的连接，例如Koebe Circle域的猜想。第三个方向是在$ \ hh^3 $中朝着无限区域的完整，凸面磁盘，并在双曲线末端的表面 - 与圆形包装上的问题或双曲线磁盘上的握把的连接。在Anti-De保姆几何形状中提出了类似的陈述，这是一种有趣的新现象的双曲线几何表弟，以及Minkowski和Half-Pipe几何形状。我们还主要根据最近的工作收集一些部分新结果。
The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature $K\geq 0$ on the sphere is induced on the boundary of a unique convex body in $\R^3$. The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a ``dual'' statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later proved. We describe three conjectural generalizations of the Weyl problem in $\HH^3$ and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a number of recent results and well-known open problems can be considered as special cases. One focus is on convex domain having a ``thin'' asymptotic boundary, for instance a quasicircle -- this part of the problem is strongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a ``thick'' ideal boundary, for instance a disjoint union of disks -- here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete, convex disks of infinite area in $\HH^3$ and surfaces in hyperbolic ends -- with connections to questions on circle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recent works.
