Preservation of concavity properties by the Dirichlet heat flow

We characterize concavity properties preserved by the Dirichlet heat flow in convex domains of the Euclidean space. (This is a joint work with Paolo Salani and Asuka Takatsu.)
Next, we show that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. (This is a joint work with Asuka Takatsu and Haruto Tokunaga.)