Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.05620 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915411373588480 |
|---|---|
| author | Chen, Jingyi Li, Yuxiang |
| author_facet | Chen, Jingyi Li, Yuxiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_05620 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature Chen, Jingyi Li, Yuxiang Differential Geometry 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. |
| title | Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2208.05620 |