The modulus of a doubly connected region and negatively curved metrics

Abstract

Ahlfors introduced the geodesic curvature-area method and used it to obtain sharp upper and lower bounds for the conformal radius of a simply connected region. The method has a striking resemblance to extremal length except that it is limited to C ²-metrics. By employing a different technique, Minda extended the upper bound to SK(k) metrics—upper semicontinuous metrics with generalized curvature at most — k≤0. Later, Minda used the geodesic curvature-area method to obtain a sharp upper bound for the modulus of a doubly connected region. Here we extend the bound to the class of SK(λ) metrics by using the length-area method in conjunction with an isoperimetric inequality. There are strong analogies between this result and classical facts concerning quadratic differentials and logarithmic area.

AMS (MOS):

• 30C99

¹The author was supported by the National Science Foundation grant MCS 8201131.

¹The author was supported by the National Science Foundation grant MCS 8201131.

Notes

¹The author was supported by the National Science Foundation grant MCS 8201131.