We present a simple analytic model of the interaction of cold convective downwelling currents with an endothermic phase change. The model describes the ponding and lateral spreading of downflows along the phase transition interface. A simple comparison of the vertical forces on the ponding material provides a necessary condition for a downflow to penetrate the phase boundary. This condition is fundamentally dependent on the geometry of the downflow. For planar downwellings, the model predicts a minimum ponding time before the structure can penetrate the phase boundary. For columnar (axisymmetric) downflows, there is no minimum time of spreading required before penetration can proceed. The model thus provides an explanation for the observation that in numerical models of three-dimensional convection with an endothermic phase change, cylindrical downflows penetrate the phase interface while planar ones do not. Since descending slabs in the Earth's mantle display a wide spectrum of geometries between planar and cylindrical (given various trench curvatures, as well as intersections of two or more subduction zones), this phenomenon may explain, in part, why some slabs appear to extend into the lower mantle while others are deflected at the 660 km discontinuity. ¿ American Geophysical Union 1993