Large time behaviour of solutions of the Ricci flow equation on R²

Shu-Yu Hsu

Vol. 197 (2001), No. 1, 25–41

DOI: 10.2140/pjm.2001.197.25

Abstract

We will show that if u₀ ∈ L_loc^p(R²) for some constant p > 1, 0 ≤ u₀ ≤ (2∕β)|x|⁻², and u₀(x) − (2∕β)(|x|² + k′)⁻¹ ∈ L¹(R²) for some constants β > 0, k′ > 0, then the rescaled function w(x,t) = e^2βtu(e^βtx,t) of the solution u of the Ricci flow equation u_t = Δlog u, u > 0, in R² × (0,∞), u(x,0) = u₀(x) in R², will converge to ϕ_β,k0(x) = (2∕β)(|x|² + k₀)⁻¹ in L¹(R²) as t →∞ where k₀ > 0 is a constant chosen such that ∫ _R²(u₀ − ϕ_β,k0)dx = 0. Moreover if u₀ satisfies in addition the condition ϕ_β,k1 ≤ u₀ ≤ ϕ_β,k2 for some constants k₁ > 0, k₂ > 0, then w will converge uniformly to ϕ_β,k0 on every compact subset of R² as t →∞.

Milestones

Received: 26 January 1999

Revised: 17 August 1999

Published: 1 January 2001