论文标题
Alexandrov表面上的指标均匀收敛，并具有有限的积分曲率
Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature
论文作者
论文摘要
我们证明，在A.D. Alexandrov的意义上，在封闭的表面上，指标$ g_k $的均匀收敛性，假设曲率测量$ \ MATHBB {k} _ {k} _ {k_k} _ {g_k} =μ^1_k-k-μ^1_k-μ^2_k $，$ unity $ usely usus ususunty n n n n n ney nne n.对于$μ^1，μ^2 $的测量薄弱，每个点的$μ^1 $小于$2π$（无尖）。这是YU的全球版本。 G. Reshetnyak在$ \ Mathbb {C} $中统一的指标收敛的众所周知的结果，并肯定地回答了封闭表面上指标收敛的开放问题。我们还提供了一个分析证明，证明了一个（单数）$ g = e^{2u} g_0 $，具有有界的积分曲率在封闭的riemannian表面$（σ，g_0）$上，可以通过固定同伴类$ [g_0] $中的平滑度量来近似。在距离功能，曲率测量和共形因子方面的％。在具有不同的保形类别和完整的非相表面上的封闭表面上的结果也得到了。
We prove uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures $\mathbb{K}_{g_k}=μ^1_k-μ^2_k$, where $μ^1_k,μ^2_k$ are nonnegative Radon measures converging weakly to measures $μ^1,μ^2$ respectively, and $μ^1$ is less than $2π$ at each point (no cusps). This is the global version of Yu. G. Reshetnyak's well-known result on uniform convergence of metrics on a domain in $\mathbb{C}$, and answers affirmatively the open question on the metric convergence on a closed surface. We also give an analytic proof of the fact that a (singular) metric $g=e^{2u}g_0$ with bounded integral curvature on a closed Riemannian surface $(Σ,g_0)$ can be approximated by smooth metrics in the fixed conformal class $[g_0]$. % in terms of distance functions, curvature measures and conformal factors. Results on a closed surface with varying conformal classes and on complete noncompact surfaces are obtained as well.
