Curvature flows on surfaces

Prompted by recent work of Xiuxiong Chen, a unified approach to the Hamilton-Ricci and Calabi flows on a closed, compact surface is presented, recovering global existence and exponentially fast asymptotic convergence from concentration-compactness results for conformal metrics. Mathematics Subject Classification (2000): 35K22 (primary), 35K55, 58G11 (secondary). 1. – Introduction Let (M, g0) be a compact Riemann surface without boundary. Consider the normalized Hamilton-Ricci flow (1) ∂g ∂t = rg − Ric = (r − R)g, where R is the scalar curvature of g with average r and where Ric = Rg is the Ricci curvature of g. Since Ric is proportional to g, the flow (1) generates a flow of conformal metrics g(t) of fixed volume. Hamilton [14] and Chow [11] established global existence and exponential convergence for this flow. The most difficult case is the case when M is the sphere S2. For this case a simpler proof of the above result was later given by Bartz-Struwe-Ye [4] along the lines of Ye’s [22] proof of the corresponding result for the Yamabe flow in higher dimensions. Also consider the Calabi flow (2) ∂g ∂t = g K · g, where g is the Laplace-Beltrami operator on (M, g) – with the analysts’ sign! – and where K = R/2 is the Gauss curvature. Again, (2) generates a flow of Pervenuto alla Redazione il 30 ottobre 2000 e in forma definitiva il 29 giugno 2001.