4 papers
Preservation of F-convexity under the heat flow
Kazuhiro Ishige, Troy Petitt, Paolo Salani
We introduce the notion of F-convexity as a general extension of power convexity. We characterize the F-convexities preserved under the heat flow in the n-dimensional Euclidean spa…
Riemannian starshape and capacitary problems
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restr…
Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds
Kazuhiro Ishige, Asuka Takatsu, Haruto Tokunaga
We prove 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 everywher…
Eventual concavity properties of the heat flow
Kazuhiro Ishige
The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of…