Metrics on a Surface with Bounded Total Curvature
Keyword(s):
Abstract Let $g=e^{2u}g_{euc}$ be a conformal metric defined on the unit disk of ${{\mathbb{C}}}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty }(D_{\frac{1}{2}})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{\mu (B_r^g(z),g)}{\pi r^2}<\Lambda $ for any $r$ and $z\in D_{\frac{3}{4}}$. Then we use this estimate to study the Gromov–Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.
2021 ◽
Vol 500
(2)
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pp. 125131
2013 ◽
Vol 45
(2)
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pp. 879-899
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2017 ◽
Vol 358
(3)
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pp. 919-994
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2013 ◽
Vol 15
(04)
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pp. 1250057
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2016 ◽
Vol 27
(3)
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pp. 1855-1873
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