| 冯宇,史毅茜,许斌.共形双曲度量的孤立奇点[J].数学年刊A辑,2019,(1):015~26 |
| 共形双曲度量的孤立奇点 |
| Isolated Singularities of Conformal Hyperbolic Metrics |
| Received:November 18, 2017 Revised:June 06, 2018 |
| DOI：10.16205/j.cnki.cama.2019.0002 |
| 中文关键词: Hyperbolic metric, Conical singularity, Cuspsingularity, Developing map |
| 英文关键词:Hyperbolic metric, Conical singularity, Cuspsingularity, Developing map |
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| 中文摘要: |
| Nitsche证明了共形双曲度量的孤立奇点要么是锥奇点, 要么是尖奇点,
二者必其一(Nitsche J. {\"{U}}ber die isolierten singularit{\"{a}}ten der L{\"{o}}sungen von $\Delta u=\rme^{u}$ [J]. {\it Math Z}, 1957, 68(3):316--324.).
本文利用展开映射证明了在孤立奇点附近存在复坐标 $z$,
使得度量要么为$\displaystyle{\frac{4\alpha^2\vert z
\vert^{2\alpha-2}}{(1-\vert z \vert ^{2\alpha})^2}\vert \mathrm{d} z
\vert^2}$, 其中 $\alpha>0$, 要么为$\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|\rmd z|^{2}}$.\\ |
| 英文摘要: |
| Nitsche proved that an isolated singularity of a conformal
hyperbolic metric is either a conical singularity or a cusp one
(Nitsche, J., {\"{U}}ber die isolierten singularit{\"{a}}ten der
L{\"{o}}sungen von $\Delta u=\rme^{u}$, {\it Math. Z}, 1957, vol.\;68, no.\;3, pp.\;316--324.).
The authors prove that there exists a complex coordinate $z$ centered at the singularity where the metric
has the expression of either $\displaystyle{\frac{4\alpha^2\vert z
\vert^{2\alpha-2}} {(1-\vert z \vert ^{2\alpha})^2}\vert \mathrm{d}
z \vert^2}$ with $\alpha>0$ or $\displaystyle{\vert z \vert
^{-2}\big(\ln|z|\big)^{-2}|\rmd z|^{2}}$ by developing map.
\\ |
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