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— The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller geodesic flow. These results have been obtained jointly with J.-C. Yoccoz [MY] and G. Forni, A. Zorich [FMZ1], [FMZ2]. Résumé. — Le flot géodésique de Teichmüller est le flot obtenu par déformation quasiconforme des structures de surface de Riemann. Le but de cet exposé est montrer la fort connexion entre la géométrie du fibré de Hodge (un fibré vectoriel audessus de l’espace de modules de surfaces de Riemann) et la dynamique du flot géodésique de Teichmüller. En particulier, on fournira des critères géométriques (basé sur les formules variationnelles derivés par G. Forni) pour detecté certaines orbites speciales (“totalement dégénérées”) du flot géodésique de Teichmüller. Ces resultats sont en colaboration avec J.-C. Yoccoz [MY] et G. Forni, A. Zorich [FMZ1], [FMZ2].

The Teichmüller geodesic flow and the geometry of the Hodge bundle