Abstract

We will show that if u₀ in L_loc^p(R²) for some constant p > 1, 0 ≤ u₀ ≤ (2 ∕ β)|x|⁻², and u₀(x) − (2 ∕ β)(|x|² + k′)⁻¹ in 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 →∞.

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