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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 Riemanndoi:10.5186/aasfm.1991.1616 fatcat:qhyvuvs3x5dvfhcxf2ogvm54pq