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Proceedings of the American Mathematical Society

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A generalization of the Ahlfors-Schwarz lemma

Author: Scott Wolpert
Journal: Proc. Amer. Math. Soc. 84 (1982), 377-378
MSC: Primary 53A30; Secondary 32H15
MathSciNet review: 640235
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Abstract: Consider, for a compact surface, two conformal metrics $ d{s^2}$ and $ d{\sigma ^2}$ of negative Gauss curvature. Assume the curvatures $ K(d{s^2})$ and $ K(d{\sigma ^2})$ satisfy $ K(d{s^2})$. It is concluded that $ d{s^2} \leqslant d{\sigma ^2}$. In particular if the curvature is pinched, $ - {c_1} \leqslant K(d{s^2}) \leqslant - {c_2} < 0$, then the inequality $ {c_1}^{ - 1/2}d{\gamma ^2} \leqslant d{s^2} \leqslant {c_2}^{ - 1/2}d{\gamma ^2}$ follows for $ d{\gamma ^2}$ the constant curvature $ -1$ metric.

References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743

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Article copyright: © Copyright 1982 American Mathematical Society

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