Propagation and extinction of fronts for semilinear parabolic equations on Riemannian manifolds

We study the Cauchy problem for the semilinear heat equation on Riemannian manifolds. Propagation and extinction of solutions are addressed, supposing that the nonlinear forcing term is either of KPP type or of bistable type. In particular, we highlight the influence both of sectional curvatures and of Ricci curvature on such phenomena.