Yamabe flows and extremal entropy

We introduce curvature-normalized versions of the Yamabe flow on complete manifolds with
negative scalar curvature, for which we show long time existence of solutions and convergence
of these flows to complete Yamabe metrics.

We apply them to study the extrema of the topological entropy in conformal classes and off
er an entropy rigidity theorem for convex-cocompact surfaces: extrema of the entropy are the
metrics whose closed geodesics coincide with those of the unique hyperbolic metric conformally
equivalent to the initial one. On convex-cocompact manifolds of higher dimension we show that l
ocal extrema of the entropy haveconstant scalar curvature on their non-wandering set.

All of this is joint work with Samuel Tapie, University of Nantes.