A generalization of the Ahlfors-Schwarz lemma
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- by Scott Wolpert PDF
- Proc. Amer. Math. Soc. 84 (1982), 377-378 Request permission
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
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 377-378
- MSC: Primary 53A30; Secondary 32H15
- DOI: https://doi.org/10.1090/S0002-9939-1982-0640235-5
- MathSciNet review: 640235