On equivalence of simple closed curves in flat surfaces |
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Authors: | Zong Liang Sun |
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Affiliation: | 1. Department of Mathematics, Shenzhen University, Shenzhen, 518060, P. R. China
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Abstract: | By explicit constructions, we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g > 1, there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that l X (α) ≠ l X (β), ext X (α) ≠ ext X (β) and l q (α) ≠ l q (β), where l X (·), ext X (·) and l q (·) are the hyperbolic length, the extremal length and the quadratic differential length respectively. These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces. |
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Keywords: | Hyperbolic metric quadratic differential metric simple closed curve |
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