| This paper concerns the soft map f from a Riemannian manifold to a probability space that minimizes the Dirichlet energy. First we give the explicit formula from any Riemannian manifold M to Prob(R). Secondly we discuss the map from M to Prob(Rd), prove the classic boundary condition implies classic solution. Then we proceed to the map from M to Prob(N), where N is a Riemannian manifold, and shows that if N is non-positive curvature, simply-connected, f has classic boundary condition, then f is classic solution and a harmonic map. Counter-examples are given when some of the above conditions are not fulfilled. In the last part we restrict the discussion in Gaussian measures. Using the Riemannian structure of the space of Gaussian measures, we prove an old result with a new method. We also show the soft map from M to non-degenerate Gaussian measures on R d is harmonic map, give the explicit formula for the soft map in a special case. |
