$\u214b^3$ 中无界凸域的 Weyl 问题
The Weyl problem for unbounded convex domains in $\HH^3$
AI总结：
本文将 Alexandrov 关于 $\u214b^3$ 中有界凸域边界度量实现的结果推广到无界凸域，通过引入包含理想边界的完整共形结构与诱导度量的组合数据，证明了在温和曲率条件下广泛数据可实现。
摘要：
设 $K\subset \HH^3$ 是 $\u214b^3$ 中具有光滑、严格凸边界的凸子集。则 $\partial K$ 上的诱导度量具有曲率 $K>-1$。Alexandrov 证明了若 $K$ 有界，则它由边界上的诱导度量唯一确定，并且任何曲率满足 $K>-1$ 的光滑度量都可以实现。我们在此提出将该结果的存在性部分推广到 $\u214b^3$ 中的无界凸域。此时 $\partial K$ 上的诱导度量显然不足以确定 $K$。然而，可以考虑包含 $K$ 的理想边界在内的更丰富的边界数据。具体地，我们考虑由 $K$ 的边界上的完整共形结构（在 $\u214b^3$ 的 Poincaré 模型中）以及 $\partial K$ 上的诱导度量共同组成的数据。我们证明，满足温和曲率条件的广泛一类“合理”的此类数据，都可以在 $\u214b^3$ 中某个凸子集的边界上实现。
英文摘要：
Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the full conformal structure on the boundary of $K$ (in the Poincaré model of $\HH^3$), together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$.
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