A Harmonic Map Flow Associated with the Standard Solution of Ricci Flow

Let (R, g(t)), 0 ≤ t ≤ T , n ≥ 3, be a standard solution of the Ricci flow with radially symmetric initial data g0. We will extend a recent existence result of P. Lu and G. Tian and prove that for any t0 ∈ [0, T ) there exists a solution of the corresponding harmonic map flow φt : (R, g(t)) → (R, g(t0)) satisfying ∂φt/∂t = ∆g(t),g(t0)φt of the form φt(r, θ) = (ρ(r, t), θ) in polar coordinates in R × (t0, T0), φt0 (r, θ) = (r, θ), where r = r(t) is the radial co-ordinate with respect to g(t) and T0 = sup{t1 ∈ (t0, T ] : ‖e ρ(·, t)‖L∞(R+) + ‖∂e ρ/∂r(·, t)‖L∞(R+) < ∞ ∀t0 < t ≤ t1} with e ρ(r, t) = log(ρ(r, t)/r). We will also prove the uniqueness of solution of the harmonic map flow within the class of functions of the form φt(r, θ) = (ρ(r, t), θ), ρ(r, t) = re ρ(r,t), for some function e ρ(r, t). We will also use the same technique to prove that the solution u of the heat equation in (Ω\{0})× (0, T ) has removable singularities at {0} × (0, T ), Ω ⊂ R, m ≥ 3, if and only if |u(x, t)| = O(|x|2−m) locally uniformly on every compact subset of (0, T ). It is known that Ricci flow is a powerful method in studying the geometry of manifolds. A manifold (M, g(t)), 0 ≤ t ≤ T , with an evolving metric g(t) is said to be a Ricci flow if it satisfies ∂ ∂t gij = −2Rij in M × (0, T ). Short time existence of solution of Ricci flow on compact manifold was proved by R. Hamilton [H1] using the Nash-Moser Theorem. Short time existence of solutions of the Ricci flow on complete non-compact Riemannian manifold with bounded curvature was proved by W.X Shi [S1]. Global existence and uniqueness of solutions of the 1991 Mathematics Subject Classification. Primary 35B60, 35K15 Secondary 58J35, 58C99.