Stability of Euclidean space under Ricci flow

Abstract

Abstract. We study the Ricci flow for initial metrics which are C 0 small perturbations of the Euclidean metric on R n . In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.

Stability of hyperbolic space under Ricci flow

“…Similar results and methods to those found in this paper may be found in the authors' paper [8] addressing the stability of Euclidean space under Ricci flow. For further references, we refer to the introduction therein.…”

“…The two flows are related by time dependent diffeomorphisms ϕ t : H n → H n :g(t) := ϕ * t g(t). As in the paper [8], we show that the estimates we obtained for g(t) imply thatg(t) → ψ * h as t → ∞ in the C k -norms. Here ψ is a diffeomorphism, and this diffeomorphism is the C k -limit of the time-dependent diffeomorphisms ϕ t which relate the two flows.…”

“…Here ψ is a diffeomorphism, and this diffeomorphism is the C k -limit of the time-dependent diffeomorphisms ϕ t which relate the two flows. We also show (as in [8]) that ψ → id as |x| → ∞, if the initial metric g 0 satisfies g 0 − h → 0 as x → ∞ (see Theorem 4.2). The proofs of this section are the same (up to some minor modifications) as those of the paper [8].…”

“…The calculations to prove this depend on an eigenvalue estimate for the Laplacian on hyperbolic domains due to McKean [7] and the closeness of the evolving metric to that of hyperbolic space. In contrast to the corresponding Euclidean result [8], we need strict monotonicity of our integral quantity to establish long-time existence. The decay of the L 2 -norm implies that the C 0 -norm of g(t) − h is exponentially decaying in time (Theorem 3.2).…”

“…There we prove short time existence using the same techniques as those presented in [8][9][10], see Theorem 2.1.…”