| Résume | Let (M, g) be a closed negatively curved Riemannian manifold. For surfaces, we know since Hamilton that the normalized Ricci flow (NRF) defines a flow (gt) of negatively curved metrics of constant area which converges to the unique hyperbolic metric in the conformal class of g. Geometrically, the NRF "simplifies" the curvature of (M, g) when t → +∞ . Dynamically, one can understand this simplification by studying the topological entropy of the geodesic flow. Indeed, Manning showed in 2004 that the topological entropy decreases along the NRF on surfaces. In the same paper, he asked if an analogous results holds in higher dimension, in the neighborhood of a hyperbolic metric. I will explain that the answer is yes. The proof relies on a combination of geometrical ideas and analytical ideas coming from microlocal analysis. This is joint work with Karen Butt and Alena Erchenko. |
