RICHARD BAMLER - RICCI FLOW LECTURE NOTES NOTES BY OTIS CHODOSH AND CHRISTOS MANTOULIDIS Contents 1. Introduction to Ricci flow 2 2. Short time existence 3 3. Distance distorion estimates 5 4. Uhlenbeck's trick 7 5. Evolution of curvatures through Uhlenbeck's trick 10 6. Global curvature maximum principles 12 7. Curvature estimates and long time existence 15 8. Vector bundle maximum principles 16 9. Curvature pinching and Hamilton's theorem 20 10. Hamilton-Ivey pinching 23 11. Ricci Flow in two dimensions 24 12. Radially symmetric flows in three dimensions 24 13. Geometric compactness 25 14. Parabolic, Li-Yau, and Hamilton's Harnack inequalities 29 15. Ricci solitons 32 16. Gradient shrinkers 34 17. F, W functionals 36 18. No local collapsing, I 38 n 19. Log-Sobolev inequality on R 39 20. Local monotonicity and L-geometry 40 21. No local collapsing, II 43 22. κ-solutions 44 22.1. Comparison geometry 45 22.2. Compactness 50 22.3. 2-d κ-solitons 50 22.4. Qualitative description of 3-d κ-solutions 51 References 53 Date: June 8, 2015. 1 2 NOTES BY OTIS CHODOSH AND CHRISTOS MANTOULIDIS We would like to thank Richard Bamler for an excellent class. Please be aware that the notes are a work in progress; it is likely that we have introduced numerous typos in our compilation process, and would appreciate it if these are brought to our attention. Thanks to James Dilts for pointing out an incorrect statement concerning curvature pinching. 1. Introduction to Ricci flow The history of Ricci flow can be divided into the "pre-Perelman" and the "post-Perelman" eras. The pre-Perelman era starts with Hamilton who first wrote down the Ricci flow equation [Ham82] and is characterized by the use of maximum principles, curvature pinching, and Harnack estimates. These tools also led to the proof of the Differentiable Sphere Theorem by Brendle and Schoen [BS09]. The post-Perelman era is characterized by the use of functionals (the W and F functionals), L geometry, blow up analysis, singularity models, and comparison geometry. Combined with Ricci flow with surgery, these tools helped complete the proof of the Poincar´e conjecture and the geometrization conjecture; [Per02], [Per03b], [Per03a]. A Ricci flow is a family pgtqtPI of Riemannian metrics on a smooth manifold, parametrized by a 1 time interval I Ă R and evolving by Btgt “ ´2 Ricgt Remark 1.1. In harmonic local coordinates around a point p, the Ricci tensor at p is 1 Ric ppq “ ´ ∆pg qppq ij 2 ij and so Ricci flow resembles a heat flow evolution. Example 1.2. If pM; gq is Einstein, i.e. Ricg “ λg, then gt fi p1 ´ 2λtqg is a Ricci flow with g0 “ g, because Btgt “ ´2λg “ ´2 Ricg “ ´2 Ricgt 1 Note that in this example Rmgt “ 1´2λt Rmg and Ricgt “ Ricg. When λ ¡ 0 Ricci flow can only 1 be defined up to time Tmax “ 2λ , after which it becomes extinct. 2 Figure 1. Shrinking round sphere, S . If λ “ 0, Ricci flow is static and can be defined for all times. 2 Figure 2. Static flat torus, T . If λ ă 0, Ricci flow is expanding and can be defined for all positive times. 1 2 1 2 Example 1.3. If pgt q, pgt q are Ricci flows on M1, M2 respectively, then gt ` gt is a Ricci flow on M1 ˆ M2. 1 st Our convention here is that Ricij “ g Rmistj , and Rmijji is a sectional curvature. BAMLER - RICCI FLOW - LECTURE NOTES 3 2 2 Figure 3. Expanding hyperbolic surface, T #T . 2 Figure 4. Evolving product manifold S ˆ R, gt “ p1 ´ 2tqgS2 ` gR. ´1 Remark 1.4 (Parabolic rescaling). If pgtq is a Ricci flow, then so are pλ gλtq and pgt`t0 q. Remark 1.5 (Invariance under diffeomorphisms). Note that the Ricci flow equation is invariant under diffeomorphisms, i.e. if Φ : M Ñ M is a diffeomorphism and pgtq is a Ricci flow, then so is ˚ pΦ gtq. That is, ˚ ˚ BtΦ gt “ ´2 RicrΦ gts: The infinitesimal version of this, assuming Φ is generated by a vector field X, is BtLX gt “ ´2pD Ricgt qrLX gts: From this equation we see that the Ricci flow cannot be strongly parabolic. Here is a heuristic reason: Assume that pgtq is a smooth solution to Ricci flow and consider a vector field X which is highly oscillating. Then LX gt is very likely also highly oscillating. But we expect parabolic equations to have a smoothing effect, which is not the case here. There is another heuristic reason to explain that Ricci flow is not strongly parabolic. If g were a Ricci flat metric then there would be no evolution, and hence LX gt would be in the kernel of the linearized Ricci operator so the kernel would be infinite dimensional. If the equation were strongly parabolic, then the right hand side would be elliptic and should have a finite dimensional kernel. In summary the diffeomorphism invariance of Ricci flow breaks strong parabolicity, so to prove short time existence we had better couple our evolution equation with a separate evolving diffeo- morphism. 2. Short time existence If we write out the Ricci flow equation in local coordinates, we get st 2 st 2 st 2 (2.1) Btgij “ 4ggij ` g Bijgst ´ g Bsigtj ´ g Bsjgti ` lower order terms which is not strongly parabolic. This is related to the problem pointed out above: if pgtq is a 1 ˚ 1 Ricci flow and gt “ Φ gt where Φ is rough and close to the identity, then gt will stay rough. ´1˚ The idea is to show existence of a related flow gt “ pΦtq gt and then switch back to gt. The following is known as de Turck's trick, after [DeT83]. Fix an arbitrary background metricg ¯ on M andr consider diffeomorphisms Φt : M Ñ M and a family of Riemannian metrics pgtqtPr0,τq that evolve according to system r BtΦt “ ∆Φ˚g ;g¯Φt (2.2) t t Btgt “ ´2 Ricgt ´L Φ Φ´1 gt r pBt tq˝ t r r r 4 NOTES BY OTIS CHODOSH AND CHRISTOS MANTOULIDIS ˚ Lemma 2.1. If we write gt “ Φt gt, then the system (2.2) is equivalent to BtΦt “ ∆gt;g¯Φt r Btgt “ ´2 Ricgt Proof. We just observe that: ˚ Btgt “BtpΦt gtq ˚ ˚ “ Φ L ´1 gt ` Φ Btgt t BtΦt˝Φt t ˚r “ ´2Φt Ricgt “ ´2 Ricgt : r r r So we have reduced the proof of short time existence for Ricci flow to proving short time existence for another system. We analyze that system and show that the evolution is strongly parabolic. Keep in mind that the setup is: Φt : M ÝÝÑ M with the target manifold endowed with the metricsg ¯, gt, and the domain endowed with the pullback ˚ metrics Φt gt. Let r, r be the Levi Civita connections with respectr to gt,g ¯, and let p P M be some fixed point at which wer seek to compute the Laplacian r ∆ ˚ Φtppq P T Mr Φt gt;g¯ Φtppq Let peiq be an orthonormal frame at ther point Φtppq with respect to gt, with reipΦtppqq “ 0. Then BtΦtppq “ ∆ ˚ Φtppq “ re eipΦtppqq “ re ei ´ re ei pΦtpprqq Φt gt;g¯ i i ir i i ¸ ¸ ´ ¯ The point is that the expressionr above is now tensorial, because r ´ r is a two tensor. We'll use the Koszul formula to rewrite this in terms of the tension field associated with the harmonic heat flow. Suppose that X, Y , Z are arbitrary vector fields such that rX “r rY “ rZ “ 0 at Φtppq. Then: 2gtprX Y ´ rX Y; Zq “ ´2gtprX Y; Zq “ ´XgtpY; Zq ´ Y gtpX; Zq ` ZgtpX; Y q “ ´prX gtqpY; Zq ´ prY gtqpX; Zq ` prZ gtqpX; Y q r r r r r r r Plug in the ei for X, Y , Z (which is compatible with the assumption rX “ rY “ rZ “ 0 due to tensoriality) in the Koszul formula above:r r r BtΦtppq “ ∆ ˚ Φtppq Φt gt;g¯ st 1 “ g r ´pr gqpe q ` prgqpe ; e q t es t 2 s t s;t ¸ ˆ ˙ fi ´Xrg¯pgtq r r So from (2.2) we obtain the Ricci de Turck equation r Btgt “ ´2 Ricgt `LXg¯pgtqgt We can now use that r r r r pLY gtqpA; Bq “ Y gtpA; Bq ´ gtprY; As;Bq ´ gtpA; rY; Bsq “ prY gtqpA; Bq ` gtprAY; Bq ` gtpA; rBY q r r r r so in local coordinates LXg¯pgtqgt is r r r st st 2 gt Bsigtj ` g Bsjgti ´ Bijgij ` lower order terms r r r r r r r BAMLER - RICCI FLOW - LECTURE NOTES 5 By examining (2.1) we conclude that in local coordinates Ricci de Turck is Btgt “ ∆gt gt ` lower order terms and is indeed strongly parabolic. r r r Lemma 2.2 (Short time existence). Assume pM; gq is a closed Riemannian manifold. There is τ ¡ 0 and a Ricci flow pgtqtPr0,τq such that g0 “ g. This flow is unique. Proof. For existence, first solve the Ricci de Turck equation with initial data g0 to obtain gt, then integrate the evolution of Φt in (2.2) with initial data Φ0 ” id to obtain Φt. By the previous lemma, ˚ gt “ Φt gt is a Ricci flow starting at g0. r For uniqueness, given gt solve the harmonic map heat flow r BtΦt “ ∆gt;g¯Φt ´1 ˚ The Φt are diffeomorphisms for a short amount of time. Then gt “ pΦt q gt solves (2.2), whose solution is however unique by the parabolicity of Ricci de Turck.