度量曲面的拟共形几乎参数化
Quasiconformal almost parametrizations of metric surfaces
摘要：
我们寻找局部有限Hausdorff $2$ -测度的二维度量曲面$X$上的极小条件，在此条件下，$X$容许具有良好几何和分析性质的（几乎）参数化.仅假设$X$是局部测地的，我们证明了有限边界长度的$X$中的Jordan域允许拟共形几乎参数化.如果$X$满足一些进一步的条件，那么这样的几乎参数化可以升级为几何拟共形同胚或拟对称同胚。特别是，我们恢复Rajala最近的拟共形一致化定理的特殊情况下，$X$是局部测地线以及Bonk-Kleiner的拟对称一致化定理。在此基础上，我们在$X$的近似极小假设下，建立了$X$中生成Jordan曲线的Sobolev圆的存在性，并证明了能量极小元的连续性.
英文摘要：
We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally geodesic, we show that Jordan domains in $X$ of finite boundary length admit a quasiconformal almost parametrization. If $X$ satisfies some further conditions then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala's recent quasiconformal uniformization theorem in the special case that $X$ is locally geodesic as well as Bonk-Kleiner's quasisymmetric uniformization theorem. On the way we establish the existence of Sobolev discs spanning a given Jordan curve in $X$ under nearly minimal assumptions on $X$ and prove the continuity of energy minimizers.
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