最小全曲率的极小曲面与平面多段弧的模空间
Minimal Surfaces of Least Total Curvature and Moduli Spaces of Plane Polygonal Arcs
摘要：
本文证明了亏格g>1的完备极小曲面的存在性，它使亏格的全曲率最小。我们的方法是首先确定这个问题（维尔斯特拉斯高维期间）的问题，找到一个特定类型的多边形弧在复杂的域：弧之间交替水平和垂直段，和两个互补的区域承认一个共形，顶点保持地图。这对互补的域代表了维尔斯特拉斯数据片段的平面结构。然后，我们发现这样的弧内的候选多边形弧探索区域之间的共形几何的差异的模空间。该论点是足够强大的，它推广到证明其他类型的极小曲面的存在。（这些曲面将在即将发表的论文中描述;这里描述的曲面扩展了Chen-Gackstätter的工作，并做了Espírito Santo，我们的论证代表了一个独立于K. Sato.）
英文摘要：
We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular type of polygonal arc in the complex domain: the arc alternates between horizontal and vertical segments, and the two complementary regions admit a conformal, vertex-preserving map. The pair of complementary domains represent flat structures for pieces of the Weierstrass data. We then find such an arc within a moduli space of candidate polygonal arcs by exploring differences in conformal geometry between the regions. The argument is sufficiently robust that it generalizes to prove the existence of other types of minimal surfaces. (Those surfaces will be described in a forthcoming paper; the surfaces described here extend work of Chen-Gackstätter and do Espírito Santo, and our argument represents a proof independent of one given at about the same time by K. Sato.)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
