Ergodic Theory and Dynamical Systems

The volume entropy of a surface decreases along the Ricci flow

a1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (e-mail: [email protected])

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The volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path $(g_t, t\geq0)$ that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing.

(Received June 5 2003)
(Revised June 23 2003)