Monotonicity of hyperbolic curvature under univalent mappings

Xiangyang Liu, David Minda
1991 Annales Academiae Scientiarum Fennicae Series A I Mathematica  
We investigate the behavior of the hyperbolic (geodesic) curvature of a path on a Riemann surface when the surface increases. Suppose X is a hyperbolic Riemann surface and åx(o,Z) denotes the hyperbolic curvature at the point a of a smooth path 7. We determine a necessary and sufficient geometric condition for the existence of a finite constant K(X) such that kx(a,t) < åv(f(o),f o7) whenever lc;(a,T) > /((X) and /: X*Y is an (injective) conformal embedding of X into another hyperbolic Riemann
more » ... yperbolic Riemann surface Y. The constant is independent of / and Y. In particular, this monotonicity property of the hyperbolic curvature holds for any simply connected surface X with /((X) = 2;this special case is due to B. Flinn and B. Osgood. They raised the question of considering the problem of monotonicity for hyperbolic curvature for more general surfaces.
doi:10.5186/aasfm.1991.1616 fatcat:qhyvuvs3x5dvfhcxf2ogvm54pq