Hele-Shaw flow on weakly hyperbolic surfaces Haakan HedenmalmAnders Olofsson 35R3535Q3531A0531C1253B2076D27Hele-Shaw flowbiharmonic Green functionmean value property We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains are more or less determined implicitly by a mean value property for harmonic functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and obtain essentially the same conclusions while imposing a weaker curvature condition on the surface. Incidentally, the curvature condition is the same as the one that appears in Hedenmalm and Perdomo's paper \cite{HP}, where the problem of finding smooth area minimizing surfaces for a given curvature form under a natural normalizing condition was considered. Probably there are deep reasons behind this coincidence. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2651 10.1512/iumj.2005.54.2651 en Indiana Univ. Math. J. 54 (2005) 1161 - 1180 state-of-the-art mathematics http://iumj.org/access/