Uniqueness of weak solutions to the Ricci flow

Transcription

1 Uniqueness of weak solutions to the Ricci flow Richard H Bamler (based on joint work with Bruce Kleiner, NYU) September 2017 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

2 ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

3 Structure of Lecture Series 1 Introduction Preliminaries on Ricci flows Statement of main results Detailed analysis of Blow-ups Introduction of toy case 2 Stability analysis 3 Comparing singular Ricci flows slides available at Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

4 Preliminaries on Ricci flow Basics Singularity analysis (blow-ups, κ-solutions etc.) Ricci flow with surgery Kleiner and Lott s construction of RF spacetimes Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

5 Ricci flow, basics Ricci flow: (M n, (g t ) t [0,T ) ) t g t = 2 Ric gt, g 0 = g ( ) Hamilton 1982 If M is compact, then ( ) has a unique solution. If T <, then max M Rm g t t T Speak: (g t ) develops a singularity at time T. Goal of this talk Theorem (Ba., Kleiner, 2016) Any (compact) 3-dimensional (M 3, g) can be evolved into a unique (canonical), weak Ricci flow defined for all t 0 that flows through singularities. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

6 Ricci flow, basics Ricci flow: (M n, (g t ) t [0,T ) ) t g t = 2 Ric gt, g 0 = g ( ) Important Examples: If Ric g = λg, then g t = (1 2λt)g round shrinking cylinder T = 1 2λ M = S 2 R g t = 2tg S 2 + g R Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

7 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

8 Singularity analysis, general case General case: Perelman 2002, very imprecise form These are (essentially) all 3d singularities To make this statement more precise, we have to discuss: the No Local Collapsing Theorem geometric convergence of RFs (blow-up analysis) qualitative classification of κ-solutions ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

9 Singularity analysis, general case Consider a general Ricci flow (M n, (g t ) t [0,T ) ) Goal: Rule out S 1 (ε) Σ 2 -singularity models No Local Collapsing Theorem (Perelman 2002) There is a constant κ(n, T, g 0 ) > 0 such that (M n, (g t ) t [0,T ) ) is κ-noncollapsed at scales < 1, i.e. for any (x, t) M [0, T ) and 0 < r < 1: Rm (, t) < r 2 on B(x, t, r) = B(x, t, r) t κr n Corollary Rm (, t) < r 2 on B(x, t, r) = inj(x, t) > c(κ)r n ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

10 Singularity analysis, general case Blowup analysis: Choose (x i, t i ) M [0, T ) s.t. Q i := Rm (x i, t i ) i Rm < CQ i on P i = P(x i, t i, A i Q 1/2 i ) = B(x i, t i, A i Q 1/2 i ) [t i (A i Q 1/2 i ) 2, t i ] for A i parabolic ball parabolic rescaling: g i t := Q i g Q 1 i t+t i ( Rm < C on P(x i, 0, A i ), NLC = inj(x i, 0) > c > 0) After passing to a subsequence: (M, g0, i C x i ) CG (M, g 0, x) i i.e. diffeomorphisms onto their images ψ i : B M (x, 0, A i ) M i, ψ i (x) = x i, A i s.t. ψ i g i 0 C loc g i ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

11 Singularity analysis, general case After passing to a subsequence: (M, g i 0, x i ) C CG (M, g 0, x) i i.e. diffeomorphisms onto their images ψ i : B M (x, 0, A i ) M i, ψ i (x) = x i, A i s.t. ψi g0 i C loc g i Using Rm < C on P(x i, 0, A i ), we can show that m ψ i g i t are locally uniformly bounded for all i. So after passing to a subsequence ψ i g i t C loc i g t, where (g t ) t (,0] is an ancient Ricci flow. Hamilton s convergence of RFs: (M, (g i t ) t [ TQi,0], x i ) C HCG (M, (g t ) t (,0], x) i (M, (g t ) t (,0], x) models the flow near (x i, t i ) for large i ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

12 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

13 Singularity analysis, general case Rotationally symmetric example Extinction shrinking sphere M = S 3, g t = 4tg S 3 Neck pinch shrinking cylinder M = S 2 R, g t = 2tg S 2 + g R Cap Bryant soliton (M, (g t )) = (M Bry, (g Bry,t ) t R ) M Bry = R 3, g Bry,t = dr 2 + f 2 t (r)g S 2 f t (r) r as r steady soliton equation: Ric = 2 f = 1 2 L f g = g Bry,t = Φ t g Bry,0 ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

14 Singularity analysis, general case General case: Perelman 2002, imprecise form All blow-ups (M, (g t ) t (,0] ) are κ-solutions. κ-solution: ancient flow (M, (g t ) t (,0] ) s.t. sec 0, R > 0, Rm < C on M (, 0] κ-noncollapsed at all scales: Rm < r 2 on B(x, t, r) = B(x, t, r) t κr 3 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

15 Singularity analysis, general case Qualitative classification of κ-solutions extinction cap neck M S 3 /Γ isometry types not classified M R 3 M K S 2 (0, ) asymptotic to a warped product asymptotic to round cylinder after rescaling (more later) M S 2 R / Γ g t = 2tg S 2 + g R Brendle 2011 If (M, (g t )) is also a steady soliton, then it is isometric to the Bryant soliton. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

16 Singularity analysis, general case (M, g) Riemannian manifold, x M point curvature scale: ρ(x) = ( 1 3 R(x)) 1/2 (M, g, x) pointed Riem. manifold (model space) (M, g, x) local ε-model at x if there is ψ : B(x, 0, ε 1 ) M such that ψ(x) = x and ρ 2 (x)ψ g g C [ε 1 ] < ε. (M, (g t ) t [0,T ) ) Ricci flow... satisfies ε-canonical neighborhood assumption at scales < r 0 if all (x, t) with ρ(x, t) < r 0 are locally ε-modeled on the final time-slice (M, g 0, x) of a pointed κ-solution. Perelman 2002, precise form (M, (g t ) t [0,T ) ) satisfies the ε-canonical nbhd assumption at scales < r(ε). Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

17 Ricci flow with surgery Ricci flow with surgery: surgery (M 1, g 1 t ), t [0, T 1 ], (M 2, g 2 t ), t [T 1, T 2 ], (M 3, g 3 t ), t [T 1, T 2 ],... U i M i U + i M i+1 ϕ i : (U i, gt i = i ) (U + i, g i+1 T i ) surgery scale δ 1 Theorem (Perelman 2003) This process can be continued indefinitely. No accumulation of T i. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

18 Ricci flow with surgery Note: surgery process is not canonical (depends on surgery parameters) Perelman: It is likely that [... ] one would get a canonically defined Ricci flow through singularities, but at the moment I don t have a proof of that. Our approach [... ] is aimed at eventually constructing a canonical Ricci flow, [... ] - a goal, that has not been achieved yet in the present work. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

19 Space-time picture Space-time 4-manifold: M 4 = ( M 1 [0, T 1 ] ϕ1 M 2 [T 1, T 2 ] ϕ2 M 3 [T 2, T 3 ] ϕ3... ) S S = (M 1 {T 1 } U 1 ) (M 2 {T 1 } U + 1 )... Time function: t : M [0, ). Time-slice: M t = t 1 (t) Time vector field: t on M (with t t = 1). Metric g: on the distribution {dt = 0} T M Ricci flow equation: L t g = 2 Ric g (surgery points) (M, t, t, g is called a Ricci flow spacetime. Note: there are holes at scale δ space-time is δ-complete ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

20 Space-time picture The spacetime (M, t, t, g)... satisfies the ε-canonical neighborhood assumption at scales (Cδ, r ε )... is Cδ-complete i.e. {ρ ρ, t T } M is compact for all ρ > Cδ, T > 0 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

21 Theorem (Kleiner, Lott 2014) Given a compact (M 3, g 0 ), there is a Ricci flow space-time (M, t, t, g) s.t.: initial time-slice: (M 0, g) = (M, g 0 ). (M, t, t, g) is 0-complete (i.e. singularity scale δ = 0 ) M satisfies the ε-canonical nbhd assumption at small scales for all ε > 0. (M, t, t, g) flows through singularities at infinitesimal scale Remarks: g is smooth everywhere and not defined at singularities singular times may accummulate (M, t, t, g) arises as limit for δ i 0. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

22 Statement of main results Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

23 Uniqueness Theorem (Ba., Kleiner, 2017) There is a constant ε can > 0 such that: Every Ricci flow space-time (M, t, t, g) is uniquely determined by its initial time-slice (M 0, g 0 ), provided that it is 0-complete and satisfies the ε can -canonical neighborhood assumption below some positive scale. Corollary For every compact (M 3, g 0 ) there is a unique, canonical singular Ricci flow space-time (M, t, t, g) with M 0 = (M 3, g 0 ). ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

24 Continuity of RF space-times continuous family of metrics (g s ) s [0,1] on M {M s } s [0,1] canonical RF space-times ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

25 Continuity of RF space-times continuous family of metrics (g s ) s [0,1] on M {M s } s [0,1] canonical RF space-times ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

26 Applications Corollary (Ba., Kleiner) Every continuous/smooth family (g s ) s Ω of Riemannian metrics on a compact manifold M 3 gives rise to a continuous/smooth family of Ricci flow space-times. Generalized Smale Conjecture Isom(S 3 /Γ) Diff(S 3 /Γ) is a homotopy equivalence. Hatcher 1983: Case Γ = 1,... Conjecture R + (S 3 ) = {metrics g on S 3 R g > 0} is contractible. Marques 2012: π 0 (R + (S 3 )) = 0. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

27 Uniqueness Theorem (Ba., Kleiner, 2017) There is a constant ε can > 0 such that: Every Ricci flow space-time (M, t, t, g) is uniquely determined by its initial time-slice (M 0, g 0 ), provided that it is 0-complete and satisfies the ε can -canonical neighborhood assumption below some positive scale. Ingredients of proof 1 Blow-up analysis of almost singular part (talks 1+2) 2 Linear stability theory (talk 2) 3 Spatial Extension Principle (talk 3) ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

28 More blow-up analysis & Toy Case Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

29 Recap κ-solutions Recall: κ-solutions model high curvature / low ρ regions κ-solutions R 3 are not completely classified However, any κ-solution that is also a steady soliton is isometric to the Bryant soliton (M Bry, g Bry ) modulo rescaling. Hamilton Brendle 2011 t R 0 on every κ-solution (M, (g t )) If t R(x, t) = 0 for some (x, t) M (, 0], then (M, (g t )) is isometric to the Bryant soliton modulo rescaling. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

30 (Strong) ε-necks Recall: shrinking cylinder: ρ = ( 1 3 R) 1/2 (M, g) Riemannian manifold g S 2 R,t = ( 2 3 2t)g S 2 + g R ρ(, 0) 1, ρ(, 1) 2 ε-neck (at scale r): U M such that there is a diffeomorphism ψ : S 2 ( ε 1, ε 1 ) U with r 2 ψ g g S2 R,0 C < ε [ε 1] (S 2 ( ε 1,ε 1 )) ψ(s 2 {0}) central 2-sphere, consisting of centers of U (M, (g t )) Ricci flow strong ε-neck (at scale r): as before, but r 2 ψ g r 2 t+t 0 g S2 R,t C [ε 1] (S 2 ( ε 1,ε 1 ) [ 1,0]) < ε ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

31 (Strong) ε-necks Example: Bryant soliton (M Bry, g Bry, x Bry ) If d t (y, x Bry ) > C Bry (ε), then (y, t) is a center of a strong ε-neck. In a general RF, if (y, t) is a center of an ε (ε)-neck, then it is also a center of a strong ε-neck. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

32 Toy Case We will only consider Ricci flow spacetimes M with the following properties: 1 M (0,T ) comes from a non-singular, conventional RF (M, (g t ) t (0,T ) ) that is κ-noncollapsed at scales < 1. 2 M 0 can be compactified by adding a single point. M 0 = (M X, g0 := lim t 0 g t ) for some X M. 3 ρ min (t) := min M ρ(, t) = ρ(x t, t) is weakly increasing. 4 Every (x, t) M (0, T ) with ρ(x, t) < 1 is one of the following: a center of a strong ε-neck at scale ρ(x, t) locally ε-modeled on (M Bry, g Bry, y) for some y M Bry with d t (y, x Bry ) < C Bry (ε). (ε will be chosen small in the course of the talks.) ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

33 Strategy Let M, M be as before and suppose M 0 = M 0. If we could show that M t = M t for some t > 0, then M = M. Construct N M (comparison domain) and φ : N M (comparison map) that is (1 + η)-bilipschitz Let η 0 and N M, limit isometry between M, M ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

34 Uniqueness in the non-singular case (M, g(0)) = (M, g (0)) = (M, g(t)) = (M, g (t)) Proof Comparison map: φ : M M such that φ g (0) = g(0) Perturbation: h(t) = φ g (t) g(t), h(0) 0 DeTurck s trick: If φ 1 (t) moves by harmonic map heat flow, then t h = g(t) h + 2 Rm g(t) (h)+ h h + h 2 h (Ricci-DeTurck flow) Standard parabolic theory: h(0) 0 = h(t) 0 q.e.d. Later: If h(t) < η lin 1, then t h g(t) h + 2 Rm g(t) (h) where (Rm(h)) ij = R istj h st (linearized Ricci-DeTurck flow) Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

35 Linearized DeTurck flow (g t ) Ricci flow, (h t ) linearized Ricci DeTurck flow Anderson, Chow (2005) h t R h R 2 R R h R Proof: = t h R = 1 ( ) R h R h R 2 R 2 R 1 ( 2 Rm(h, h) R 2 R h 2 Ric h h ) R 2 R R h R Need Rm(h, h)r h 2 Ric 2 Can be checked using an elementary computation. ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

36 Vanishing Theorem Let (M, (g t ) t (,0] be a κ-solution (e.g. shrinking cyliner, Bryant soliton) and (h t ) t (,0] a linearized RDTF such that Then h 0. h CR 1+γ, γ > 0. Proof: h CR 1+γ CC R. Choose (x i, t i ) M (, 0] s.t. R γ (x i, t i ) = R1+γ (x i, t i ) R(x i, t i ) After passing to a subsequence h CR 1+γ, h R (x h i, t i ) sup i M (,0] R =: C 0. (M, (g t+ti ) t (,0], x i ) C 1 h R (x i, t i ) C 1 C 0 > 0 ( ) i C HCG (M, (gt ) t (,0], x ) i (h t+ti ) t (,0] i (h t ) t (,0] h R C 0 (equality at (x, 0)) ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

37 After passing to a subsequence (M, (g t+ti ) t (,0], x i ) C HCG (M, (gt ) t (,0], x ) i (h t+ti ) t (,0] i (h t ) t (,0] h CR 1+γ, h R C 0 (equality at (x, 0)) strong maximum principle = on M (, 0] h R = C 0 R γ = R1+γ R C 1 h = C 1 C 1 0 > 0 R q.e.d. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

38 Linear Non-Linear (g t ) RF background (h t ) solution to non-linear RDTF Observation: Divide by a > 0 t ( h a ) If a i 0 and h i a i t h = h + 2 Rm(h) + h h + h 2 h ( h ) ( h ) ( h ) ( h ) ( h ) = + 2 Rm + a + a a a a a a h, then (assuming certain derivative bounds) t h = h + 2 Rm(h ) 2( h ) a (linearized RDTF) Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

39 (g t ) t [0,T ) κ-noncollapsed RF, ε-canonical nbhd assumption, (h t ) t [0,T ) (non-linear) RDTF Ht h Q := e R E, E > 1 Given α > 0 there are H, ε, η, L > 0 s.t. if then h η on P = P(x, t, Lρ(x, t)), Q(x, t) α sup Q. P Proof: Assume not for E, α, κ. Choose H i, L i, η i, ε i 0. Counterexamples (M i, (g i t )), (h i ), h i η i 0, r i := ρ(x i, t i ) Q(x i, t i ) α sup P i Q = h i (y, t) α 1 e H i (t i t) RE (y, t) R E (x i, t i ) h i (x i, t i ) on P i (M i, (r 2 C i g r 2 i t+t i ), x i ) HCG (M, (gt ) t (,0], x ) i h i h i (x i, t i ) i (h,t) t (,0] (linearized RDTF) ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

40 Q(x i, t i ) α sup P i Q = h i (y, t) α 1 e H i (t i t) RE (y, t) R E (x i, t i ) h i (x i, t i ) on P i (M i, (r 2 C i g r 2 i t+t i ), x i ) HCG (M, (gt ) t (,0], x ) i h i h i (x i, t i ) i (h,t) t (,0] (linearized RDTF) Then h (x, 0) = 1 h (y, t) lim e H i r 2 i t R E (y, t) i Case liminf i r i > 0: h (, t) 0 for t < 0 = h 0 Case liminf i r i = 0: imit is κ-solution h α 1 R E = h 0 q.e.d. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

41 Recap: Strategy Let M, M be as before and suppose M 0 = M 0. Construct N M (comparison domain) and φ : N M (comparison map) that is (1 + η)-bilipschitz Let η 0 and N M, limit isometry between M, M ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

42 Construction of the comparison domain λ = λ(δ n ) to be determined later Choose r comp 1 (comparison scale) t j := j r 2 comp t J = T choose j 0 minimal s.t. ρ min (t j0 ) λr comp There is a comparison domain such that N = (N 1... N j0 ) (N j N J ) M N j = N j [t j 1, t j ] N j t j is central 2-sphere of strong δ n -neck at scale r comp ρ > 1 2 r comp on N 1... N j0 N j = M [t j 1, t j ] for j j ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

43 N j central 2-spheres of strong δ n -necks need λ λ(δ n ) ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

44 Construction of comparison map for t t j0 Goal: Construct comparison map φ : N 1... N j0 M, evolving by harmonic map heat flow s.t. h = φ g g < η ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

45 Construction of comparison map for t t j0 Strategy: φ N0 given since M 0 = M 0 For each j = 0,..., j 0 1 solve HMHF with initial data φ N j t j φ N j+1 : N j+1 M. This works if δ n < δ n (η) and h < η 0 η near N j+1 t j Choose Q such that Q < Q implies h < η. Ensure Q < Q near N j+1 t j. Interior decay estimate: If d tj ( Nt j+1 j, Nt j j ) > Lr comp, then Q < αq, α 1 = h < η 0 near Nt j+1 j ichard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

46 Construction of comparison map at t = t j0 Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

47 Cap extension Cantilever Paradox Where do you feel safer? long cantilever, good engineering short cantilever, sketchy engineering Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

48 Ht h Assume that Q = e R E < Q on N. Bryant Extension Principle ( Cap Extension ) If E > 100 and D D 0 (η), then there is an extension φ : B N M Bry of φ : N M Bry such that φ g Bry g Bry < η Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

49 Proof, concluded Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

50 Proof, concluded Extend φ N j 0 tj0 onto N j0+1 t j0 via Bryant Extension Principe (at time t j0 ) Extend φ j N 0 +1 onto N j N J by solving HMHF (no boundary!). t j0 Control h using Q = e Ht h R E for E E. (Bryant Extension Principle = Q Q at time t j0.) q.e.d. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50

51 The general case Problems that may arise Caps may not always be modeled on Bryant solitons Solution: Perform cap extension at the right time, when t R 0 on M and M N and φ must be chosen simultaneously There may be more than one cusp Solution: Continue the neck and cap extension process after the first cap extension. Curvature of the cap may increase again after cap extension (and there may be an accumulation of singular times) Solution: Perform a cap removal when the curvature exceeds a certain threshold. Ensure that cap extensions and cap removals are sufficiently separated in space and time such that Q has time to recover after a cap extension. Richard H Bamler (based on joint work with Bruce Kleiner, Uniqueness NYU) (Berkeley) of weak solutions to the Ricci flow September / 50