A Flow Tangent to the Ricci Flow via Heat Kernels and Mass Transport
  We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below. 
Nicola Gigli and Carlo Mantegazza
math.FA math.DG