论文标题
Teichmüller空间上的射线结构
Ray structures on Teichmüller Space
论文作者
论文摘要
尽管在Teichmüller空间中的一对点之间可能有许多瑟斯顿度量测量学，但我们发现，通过对测量学的额外的能量最小化约束，将其视为谐波映射射线的限制，我们选择了一个独特的瑟斯顿大地测量。将目标表面扩展到瑟斯顿边界，对于teichmüller空间中的每个点$ y $，从该点$ y $ y $ y $ y $ y $ y $ y $的每个点$ y $都延伸到teichmüller空间，并带有视觉边界TeichmüllerSpace的瑟斯顿边界。 我们首先将Teichmüller空间上的谐波射线结构描述为Teichmüller射线结构和Thurston Geodesic Ray结构之间的几何跃迁。特别是，通过适当地将双曲线表面之间的谐波图的来源（沿着“谐波映射双射线”）之间的谐波映射源，谐波映射通过目标射线会融合到瑟斯顿大地测量。通过适当地退化谐波图的目标，这些谐波图双光线通过域汇合到TeichmüllerGeodesics。然后，我们将这种过渡扩展到从Teichmüller磁盘通过Hopf差速器磁盘到拉伸磁盘的过渡。这些结果适用于边界的表面，解决了此类表面之间拉伸图的问题。
While there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. Extending the target surface to the Thurston boundary yields, for each point $Y$ in Teichmüller space, an "exponential map" of rays from that point $Y$ onto Teichmüller space with visual boundary the Thurston boundary of Teichmüller space. We first depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces (along "harmonic map dual rays"), the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to Teichmüller geodesics. We then extend this transition to one from Teichmüller disks through Hopf differential disks to stretch-earthquake disks. These results apply to surfaces with boundary, resolving a question on stretch maps between such surfaces.
