An analytic approach to the normalized Ricci flow-like equation

This paper is concerned with a parabolic equation with a non-local term defined on a compact two-dimensional Riemannian surface ΩΩ. If the total mass of the solution, λλ, is equal to 8π8π and ΩΩ is the standard sphere S2S2, it is a Hamilton’s normalized Ricci flow. We obtain the global in time existence of the solution to this problem for 0<λ≤8π0<λ≤8π. If 0<λ<8π0<λ<8π, the orbit is compact while for λ=8πλ=8π, there is a time sequence along which the solution converges to a stationary solution.