Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces

Abstract

We prove the sharp local $L^1-L^{\infty }$ smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known $L^p-L^{\infty }$ estimate for $p>1$ . It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.

Keywords: Logarithmic fast diffusion equation; Ricci flow; Smoothing estimate.

Fig. 1

Conformal factors of hyperbolic metrics h on B and $h_{\mathit{\rho }}$ on $B_{\mathit{\rho }}$

Fig. 2

Smoothing function $\mathit{\gamma }$