UNIQUENESS AND STABILITY OF RICCI FLOW THROUGH SINGULARITIES RICHARD BAMLER AND BRUCE KLEINER Abstract. We show that three dimensional \Ricci flows through singularities" are uniquely determined by their initial data. To- gether with the corresponding existence result of the second au- thor and Lott this verifies a conjecture of Perelman, stating that every compact Riemannian manifold can be evolved to a unique and canonical singular Ricci flow. We furthermore show that such flows depend continuously on their initial data and we establish a stability result that also holds for Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds | in particular the Generalized Smale Conjecture, | which will be described in a subsequent paper. Contents 1. Introduction 2 2. Overview of the proof 13 3. Preliminaries I 26 4. Preliminaries II 31 5. A priori assumptions 37 6. Preparatory results 48 7. Semilocal maximum principle 78 8. Extending maps between Bryant solitons 88 9. Inductive step: extension of the comparison domain. 100 10. Inductive step: extension of the comparison map 122 Date: August 15, 2017. The first author was supported by a Sloan Research Fellowship and NSF grant DMS-1611906. The research was in part conducted while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester. The second author was supported by NSF grants DMS-1405899 and DMS- 1406394, and a Simons Collaboration grant. 1 2 RICHARD BAMLER AND BRUCE KLEINER 11. Proofs of the main results 166 Appendix A. Ricci-DeTurck flow and harmonic map heat flow 178 Appendix B. Properties of Bryant solitons 194 Appendix C. Properties of κ-solutions 195 Appendix D. Smoothing maps 199 References 200 1. Introduction 1.1. Background and overview. The understanding of many as- pects of Ricci flow has advanced dramatically in the last 15 years. This has led to numerous applications, the most notable being Perel- man's landmark proof of the Geometrization and Poincar´eConjectures. Nonetheless, from an analytical viewpoint, a number of fundamental questions remain, even for 3-dimensional Ricci flow. One of these con- cerns the nature of Ricci flow with surgery, a modification of Ricci flow that was central to Perelman's proof. Surgery, an idea initially developed by Hamilton, removes singularities as they form, allowing one to continue the flow. While Perelman's construction of Ricci flow with surgery was spectacularly successful, it is not entirely satisfying due to its ad hoc character and the fact that it depends on a number of non-canonical choices. Furthermore, from the PDE point of view, Ricci flow with surgery does not provide a theory of solutions to the Ricci flow PDE, since surgery violates the equation. In fact, Perelman himself was aware of these drawbacks and drew attention to them in both of his Ricci flow preprints: \It is likely that by passing to the limit in this construc- tion [of Ricci flow with surgery] one would get a canon- ically defined Ricci flow through singularities, but at the moment I don't have a proof of that" | [Per02, p.37]. \Our approach. is aimed at eventually constructing a canonical Ricci flow . a goal, that has not been achieved yet in the present work" | [Per03, p.1]. Motivated by the above, the paper [KL14] introduced a new notion of weak solutions to Ricci flow in dimension 3, and proved the existence of solutions for arbitrary initial data as well as a number of results about their geometric and analytical properties. UNIQUENESS AND STABILITY OF RICCI FLOW 3 The main theorem in this paper implies that the weak solutions of [KL14] are uniquely determined by their initial data. In combination with [KL14], this gives a proof of Perelman's conjecture, and also shows that the initial value problem has a unique weak solution that depends continuously on its initial data, thereby solving the long-standing prob- lem of finding a satisfactory theory of weak solutions to the Ricci flow equation, in the 3-dimensional case. From a broader perspective, it is interesting to compare the results here with work on weak solutions to some other geometric PDEs. Regarding existence and partial regularity there is a vast literature. As with PDEs in general, proving existence of solutions requires a choice of objects and a topology that is strong enough to respect the equation, but weak enough to have some compactness properties; es- tablishing finer structure of solutions (e.g. partial regularity) requires, generally speaking, a mechanism for restricting blow-ups. For mini- mal surfaces, harmonic maps, and harmonic map heat flow, good no- tions of weak solutions with accompanying existence and partial reg- ularity theorems were developed long ago. By contrast, the theory of weak solutions to mean curvature flow, the Einstein equation and Ricci flow, are at earlier stages of development. For mean curvature flow, for instance, there are different approaches to weak solutions | (en- hanced) Brakke flows, level set flow | introduced over the last 40 years [Bra78, ES91, CGG91, Ilm94]. Yet in spite of deep results for mean convex or generic initial conditions [Whi00, Whi03, Whi05, CM16], to our knowledge the best results known for flows starting from a general compact smooth surface in R3 are essentially those of [Bra78], which are presumably very far from optimal. For the (Riemannian) Einstein equation there are many results in the Kahler case and on limits of smooth Einstein manifolds, but otherwise progress toward even a vi- able definition of weak solutions has been rather limited. Regarding uniqueness, our focus in this paper, much less is known. The paper [Ilm95] describes a mechanism for non-uniqueness stemming from the dynamical instability of cones, which is applicable to a number of geometric flows. This provides examples of non-uniqueness for mean curvature flow in high dimensions, while [Whi02] has examples of non- uniqueness for smooth surfaces in R3. Examples for harmonic map heat flow are constructed in [GR11, HHS14], and for Ricci flow in higher di- mensions, there are examples in [FIK03] which suggest non-uniqueness. Since any discussion of uniqueness must refer to a particular class of admissible solutions, the interpretation of some of the above examples is not entirely clear, especially in the case of higher dimensional Ricci 4 RICHARD BAMLER AND BRUCE KLEINER flow where a definition of weak solutions is lacking. In the other direc- tion, uniqueness has been proven to hold in only a few cases: harmonic map heat flow with 2-dimensional domain [Str85], mean convex mean curvature flow [Whi03], and K¨ahler-Ricci flow [ST09, EGZ16]. These theorems rely on special features, like the fact that singularities oc- cur at a discrete set of times, and have a very restricted structure, or comparison techniques specific to scalar equations. The method of proving uniqueness used here is quite different in spirit from earlier work. Uniqueness is deduced by comparing two flows with nearby initial condition, and estimating the rate at which they diverge from one another. Due to the nature of the singularities, which might in principle occur at a Cantor set of times, the flows can only be compared after the removal their almost singular regions. Since one knows nothing about the correlation between the almost singular parts, the crux of the proof is to control the influence of effects emanating from the boundary of the truncated flows. In essence, uniqueness is deduced from a new strong stability property, which says that, roughly speaking, if two flows are initially close, away from their almost singular parts, then they remain so. Surprisingly, a consequence is that the strong stability result applies not just to Ricci flows with surgery and the weak solutions of [KL14], but to flows whose almost singular parts are allowed to evolve in an arbitrary fashion, possibly violating the Ricci flow equation. Together with the examples of non-uniqueness alluded to above, the uniqueness argument here suggests the following tentative criterion for when one might expect, and possibly prove, uniqueness for weak solu- tions to a given geometric flow: (1) There should be a classification of all blow-ups | not just shrinking solitons. (2) The linearized equation should exhibit uniform strict stability on all blow-ups. (3) The blow-ups must have an additional quantitative rigidity prop- erty that allows one to “fill in" solutions to the perturbation equation over recently resolved singularities. From a philosophical point of view (1) and (2) are completely natu- ral, if not obvious. However, implementation of even (1) can be quite difficult; indeed, to date there are few situations where such a classifi- cation is known. For more discussion of (3), we refer the reader to the overview of the proof in Section 2. In the case of 3-dimensional Ricci flow considered here, (3) turns out to be by far the most delicate point, UNIQUENESS AND STABILITY OF RICCI FLOW 5 and is responsible for much of the complexity in the proof. One may hope that the approach used here may be adapted to other situations to prove the stability and uniqueness of weak solutions, for instance mean curvature flow of 2-spheres in R3. We mention that our main result implies that weak solutions to Ricci flow behave well even when one considers continuous families of initial conditions. This leads to new results for diffeomorphism groups of 3-manifolds, which will be discussed elsewhere [BK]. 1.2. History and status quo. We will now give a more detailed overview of Perelman's \Ricci flow with surgery" and the concept of \weak Ricci flows", as introduced by the second author and Lott.