论文标题
非长度表面的多面体近似和均匀化
Polyhedral approximation and uniformization for non-length surfaces
论文作者
论文摘要
我们证明，具有局部有限的hausdorff 2量的任何度量表面（即具有边界的2个manifold）的任何度量表面是具有受控几何形状的多面体表面的Gromov-Hausdorff极限。我们使用此结果以及经典的统一定理，以证明具有有限的Hausdorff的2个指标同构的任何度量表面同构2-估计量承认Riemann Sphere的弱准文字参数化，从而回答了Rajala-Wenger的问题。这些结果先前是由作者建立的，假设度量表面带有长度度量。作为推论，我们获得了Bonk-kleiner的统一定理的新证明，用于准确和Rajala的相互表面。另一个推论是简化了相互表面的定义，它回答了关于最小假设的拉贾拉问题，在该假设下，公制表面与欧几里得域相当。
We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala-Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk-Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.
