Uniqueness of the Ricci flow on complete noncompact manifolds

By Binglong Chen, Xiping Zhu. 2005

223 citations recorded by the source corpus.

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/765acb55a207f50912e51940f7361649c857c29c

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Citations in the corpus

Evolution of an extended Ricci flow system (2008, background): “However, the result in [18] strongly suggests that solutions satisfying the above bounds are unique.”
Recent developments on the Hamilton’s Ricci Flow (2007, methodology): “Since the surgery was done on the δ-neck, the solution on the surgery region is close to a standard solution by the uniqueness theorem of Ricci flow [24].”
Ancient solutions to the Ricci flow in dimension $3$ (2018, background): “By the uniqueness result in [12], (M,g(t)) is a steady gradient Ricci soliton for all t.”
Notes on Perelman's papers (2006, background): “The technical issue, which we do not address here, is to show that a solution to the harmonic heat flow will exist for some time interval [0, T ) with uniformly bounded derivatives; see [21].”
Geometrisation of 3-Manifolds (2010, methodology): “By the Chen-Zhu uniqueness theorem [18], we deduce that the limit flow (M∞, {g∞(t)}t∈[b−1/2,0]) is the cylindrical flow up to parabolic scaling.”
Stability of Euclidean space under Ricci flow (2007, background): “Uniqueness of solutions to Ricci flow with bounded curvature on non-compact manifolds is discussed in [2, 8].”
How to produce a Ricci Flow via Cheeger-Gromoll exhaustion (2011, background): “Because of the uniqueness of the Ricci flow (see [14]) it follows that (C, g∞(t), x0) is isometric to (C, λ(2)g0, x0) for all t .”
Uniqueness and pseudolocality theorems of the mean curvature flow (2007, methodology): “We remark that the uniqueness of the Ricci flow has been established by Zhu and the first author in [5].”
Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature (2005, methodology): “Then by the uniqueness theorem in [ 10 ], we have ˜ R(x, t) =”
Pseudolocality for the Ricci Flow and Applications (2007, background): “unded on M× [0,T). If this is true, then the curvature is also uniformly bounded on M× [0,T) by the argument in [34, §7]. From this it is easy to see that the solution g(t) can be extended beyond Tby [31, 5]. This is a contradiction. Since thecurvature |Rm(x,t)|isuniformly boundedin (M\S)×[0,T) and ”
Existence of Ricci Flows of Incomplete Surfaces (2010, background): “The contribution of Chen-Zhu [CZ06] to Theorem 1.”
On quotients of spaces with Ricci curvature bounded below (2017, unknown): “By the uniqueness results of Hamilton [39] and Chen-Zhu [26], it is known that the Ricci flow for metrics of bounded curvature preserves the isometry group of the solution, i.”
Strong Uniqueness of the Ricci Flow (2007, background): “1 may be viewed as a strong generalization of [4], and we call it a strong uniqueness theorem (see the extrinsic version in [3]).”