Width and finite extinction time of Ricci flow
aa r X i v : . [ m a t h . DG ] J u l WIDTH AND FINITE EXTINCTION TIME OF RICCI FLOW
TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II Introduction
This is an expository article with complete proofs intended for a general non-specialistaudience. The results are two-fold. First, we discuss a geometric invariant, that we callthe width, of a manifold and show how it can be realized as the sum of areas of minimal2-spheres. For instance, when M is a homotopy 3-sphere, the width is loosely speaking thearea of the smallest 2-sphere needed to “pull over” M . Second, we use this to conclude thatHamilton’s Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We havechosen to write this since the results and ideas given here are quite useful and seem to be ofinterest to a wide audience.Given a Riemannian metric on a closed manifold M , sweep M out by a continuous one-parameter family of maps from S to M starting and ending at point maps. Pull thesweepout tight by, in a continuous way, pulling each map as tight as possible yet preservingthe sweepout. We show the following useful property (see Theorem 1.14 below); cf. 12.5 of[Al], proposition 3.1 of [Pi], proposition 3.1 of [CD], [CM3], and [CM1]:Each map in the tightened sweepout whose area is close to the width (i.e., the maximalenergy of the maps in the sweepout) must itself be close to a collection of harmonicmaps. In particular, there are maps in the sweepout that are close to a collection ofimmersed minimal 2-spheres.This useful property that all almost maximal slices are close to critical points is virtuallyalways implicit in any sweepout construction of critical points for variational problems yet itis not always recorded since most authors are only interested in existence of a critical point.Similar results hold for sweepouts by curves instead of 2-spheres; cf. [CM3] where sweep-outs by curves are used to estimate the rate of change of a 1-dimensional width for convexhypersurfaces in Euclidean space flowing by positive powers of their mean curvatures. Theideas are essentially the same whether one sweeps out by curves or 2-spheres, though thetechniques in the curve case are purely ad hoc whereas for sweepouts by 2-spheres additionaltechniques, developed in the 1980s, have to be used to deal with energy concentration (i.e.,“bubbling”); cf. [SaU] and [Jo]. The basic idea in each of the two cases is a local replacementprocess that can be thought of as a discrete gradient flow. For curves, this is now known asBirkhoff’s curve shortening process; see [B1], [B2]. The authors were partially supported by NSF Grants DMS 0606629 and DMS 0405695. Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917; see [B1],[B2], section 2 in [Cr], and [CM3]. In the 1980s Sacks-Uhlenbeck, [SaU], found minimal 2-spheres on generalmanifolds using Morse theoretic arguments that are essentially equivalent to sweepouts; a few years later,Jost explicitly used sweepouts to obtain minimal 2-spheres in [Jo]. The argument given here works equallywell on any closed manifold, but only produces non-trivial minimal objects when the width is positive.
Local replacement had already been used by H.A. Schwarz in 1870 to solve the Dirichletproblem in general domains, writing the domain as a union of overlapping balls, and usingthat a solution can be found explicitly on balls by, e.g., the Poisson formula; see [Sc1]and [Sc2]. His method, which is now known as Schwarz’s alternating method, continues toplay an important role in applied mathematics, in part because the replacements convergerapidly to the solution. The underlying reason why both Birkhoff’s method of finding closedgeodesics and Schwarz’s method of solving the Dirichlet problem converge is convexity. Wewill deviate slightly from the usual local replacement argument and prove a new convexityresult for harmonic maps. This allows us to make replacements on balls with small energy,as opposed to balls with small C oscillation. It is, in our view, much more natural to makethe replacement based on energy and gives, as a bi-product, a new uniqueness theorem forharmonic maps since already in dimension two the Sobolev embedding fails to control the C norm in terms of the energy; see Figure 1.The second thing we do is explain how to use this property of the width to show that on ahomotopy 3-sphere, or more generally closed 3-manifolds without aspherical summands, theRicci flow becomes extinct in finite time. This was shown by Perelman in [Pe] and by Colding-Minicozzi in [CM1]; see also [Pe] for applications to the elliptic part of geometrization. Figure 1.
A conformal map to a longthin surface with small area has little en-ergy. In fact, for a conformal map, thepart of the map that goes to small areatentacles contributes little energy and willbe truncated by harmonic replacement. W The min-max surface.
Figure 2.
The sweepout, the min–maxsurface, and the width W.We would like to thank Fr´ed´eric H´elein, Bruce Kleiner, and John Lott for their comments.1.
Width and finite extinction
On a homotopy 3-sphere there is a natural way of constructing minimal surfaces and thatcomes from the min-max argument where the minimal of all maximal slices of sweepouts is aminimal surface. In [CM1] we looked at how the area of this min-max surface changes underthe flow. Geometrically the area measures a kind of width of the 3-manifold (see Figure2) and for 3-manifolds without aspherical summands (like a homotopy 3-sphere) when the
IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 3 metric evolve by the Ricci flow, the area becomes zero in finite time corresponding to thatthe solution becomes extinct in finite time. Width.
Let Ω be the set of continuous maps σ : S × [0 , → M so that for each t ∈ [0 ,
1] the map σ ( · , t ) is in C ∩ W , , the map t → σ ( · , t ) is continuous from [0 ,
1] to C ∩ W , , and finally σ maps S × { } and S × { } to points. Given a map β ∈ Ω,the homotopy class Ω β is defined to be the set of maps σ ∈ Ω that are homotopic to β through maps in Ω. We will call any such β a sweepout ; some authors use a more restrictivenotion where β must also induce a degree one map from S to M . We will, in fact, be mostinterested in the case where β induces a map from S to M in a non-trivial class in π ( M ).The reason for this is that the width is positive in this case and, as we will see, equal to thearea of a non-empty collection of minimal 2-spheres.The (energy) width W E = W E ( β, M ) associated to the homotopy class Ω β is defined bytaking the infimum of the maximum of the energy of each slice. That is, set(1.1) W E = inf σ ∈ Ω β max t ∈ [0 , E ( σ ( · , t )) , where the energy is given by(1.2) E ( σ ( · , t )) = 12 Z S |∇ x σ ( x, t ) | dx . Even though this type of construction is always called min-max, it is really inf-max. Thatis, for each (smooth) sweepout one looks at the maximal energy of the slices and then takesthe infimum over all sweepouts in a given homotopy class. The width is always non-negativeby definition, and positive when the homotopy class of β is non-trivial. Positivity can, forinstance, be seen directly using [Jo]. Namely, page 125 in [Jo] shows that if max t E( σ ( · , t ))is sufficiently small (depending on M ), then σ is homotopically trivial. One could alternatively define the width using area rather than energy by setting(1.3) W A = inf σ ∈ Ω β max t ∈ [0 , Area ( σ ( · , t )) . The area of a W , map u : S → R N is by definition the integral of the Jacobian J u = p det ( du T du ), where du is the differential of u and du T is its transpose. That is, if e , e isan orthonormal frame on D ⊂ S , then J u = ( | u e | | u e | − h u e , u e i ) ≤ | du | and(1.4) Area( u (cid:12)(cid:12) D ) = Z D J u ≤ E( u (cid:12)(cid:12) D ) . Consequently, area is less than or equal to energy with equality if and only if h u e , u e i and | u e | − | u e | are zero (as L functions). In the case of equality, we say that u is almostconformal . As in the classical Plateau problem (cf. Section 4 of [CM2]), energy is somewhat It may be of interest to compare our notion of width, and the use of it, to a well-known approach to thePoincar´e conjecture. This approach asks to show that for any metric on a homotopy 3-sphere a min-maxtype argument produces an embedded minimal 2-sphere. Note that in the definition of the width it play norole whether the minimal 2-sphere is embedded or just immersed, and thus, the analysis involved in this wassettled a long time ago. This well-known approach has been considered by many people, including Freedman,Meeks, Pitts, Rubinstein, Schoen, Simon, Smith, and Yau; see [CD]. For example, when M is a homotopy 3-sphere and the induced map has degree one. See the remarks after Corollary 3.4 for a different proof.
TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II easier to work with in proving the existence of minimal surfaces. The next proposition,proven in Appendix D, shows that W E = W A as for the Plateau problem (clearly, W A ≤ W E by the discussion above). Therefore, we will drop the subscript and just write W . Proposition 1.5. W E = W A .1.2. Finite extinction.
Let M be a smooth closed orientable 3-manifold and g ( t ) a one-parameter family of metrics on M evolving by Hamilton’s Ricci flow, [Ha1], so(1.6) ∂ t g = − M t . When M is prime and non-aspherical, then it follows by standard topology that π ( M ) isnon-trivial (see, e.g., [CM1]). For such an M , fix a non-trivial homotopy class β ∈ Ω. Itfollows that the width W ( g ( t )) = W ( β, g ( t )) is positive for each metric g ( t ). This positivityis the only place where the assumption on the topology of M is used in the theorem belowgiving an upper bound for the derivative of the width under the Ricci flow. As a consequence,we get that the solution of the flow becomes extinct in finite time (see paragraph 4.4 of [Pe]for the precise definition of extinction time when surgery occurs). Theorem 1.7. [CM1]. Let M be a closed orientable prime non-aspherical 3-manifoldequipped with a metric g = g (0). Under the Ricci flow, the width W ( g ( t )) satisfies(1.8) ddt W ( g ( t )) ≤ − π + 34( t + C ) W ( g ( t )) , in the sense of the limsup of forward difference quotients. Hence, g ( t ) becomes extinct infinite time.The 4 π in (1.8) comes from the Gauss-Bonnet theorem and the 3 / C > ddt (cid:0) W ( g ( t )) ( t + C ) − / (cid:1) ≤ − π ( t + C ) − / and integrate to get(1.10) ( T + C ) − / W ( g ( T )) ≤ C − / W ( g (0)) − π (cid:2) ( T + C ) / − C / (cid:3) . Since W ≥ T sufficiently large, we get the claim.Theorem 1.7 shows, in particular, that the Ricci flow becomes extinct for any homotopy3-sphere. In fact, we get as a corollary finite extinction time for the Ricci flow on all 3-manifolds without aspherical summands (see 1 . Corollary 1.11. ([CM1], [Pe]). Let M be a closed orientable 3-manifold whose primedecomposition has only non-aspherical factors and is equipped with a metric g = g (0).Under the Ricci flow with surgery, g ( t ) becomes extinct in finite time. IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 5
Part of Perelman’s interest in the question about finite time extinction comes from thefollowing: If one is interested in geometrization of a homotopy 3-sphere (or, more generally,a 3-manifold without aspherical summands) and knew that the Ricci flow became extinct infinite time, then one would not need to analyze what happens to the flow as time goes toinfinity. Thus, in particular, one would not need collapsing arguments.One of the key ingredients in the proof of Theorem 1.7 is the existence of a sequence ofgood sweepouts of M , where each map in the sweepout whose area is close to the width(i.e., the maximal energy of any map in the sweepout) must itself be close to a collection ofharmonic maps. This will be given by Theorem 1.14 below, but we will first need a notionof closeness and a notion of convergence of maps from S into a manifold.1.3. Varifold convergence.
Fix a closed manifold M and let Π : G k M → M be theGrassmanian bundle of (un-oriented) k -planes, that is, each fiber Π − ( p ) is the set of all k -dimensional linear subspaces of the tangent space of M at p . Since G k M is compact,we can choose a countable dense subset { h n } of all continuous functions on G k M withsupremum norm at most one (dense with respect to the supremum norm). If ( X , F ) and( X , F ) are two compact (not necessarily connected) surfaces X , X with measurable maps F i : X i → G k M so that each f i = Π ◦ F i is in W , ( X i , M ) and J f i is the Jacobian of f i , thenthe varifold distance between them is by definition(1.12) d V ( F , F ) = X n − n (cid:12)(cid:12)(cid:12)(cid:12)Z X h n ◦ F J f − Z X h n ◦ F J f (cid:12)(cid:12)(cid:12)(cid:12) . It follows easily that a sequence X i = ( X i , F i ) with uniformly bounded areas converges to( X, F ), iff it converges weakly, that is, if for all h ∈ C ( G M ) we have R X i h ◦ F i J f i → R X h ◦ F J f . For instance, when M is a 3-manifold, then G M , G M , and T M/ {± v } areisomorphic. (Here T M is the unit tangent bundle.) If Σ i is a sequence of closed immersedsurfaces in M converging to a closed surface Σ in the usual C k topology, then we can thinkof each surface as being embedded in T M/ {± v } ≡ G M by mapping each point to plus-minus the unit normal vector, ± n , to the surface. It follows easily that the surfaces withthese inclusion maps converges in the varifold distance. More generally, if X is a compactsurface and f : X → M is a W , map, where M is no longer assumed to be 3-dimensional,then we let F : X → G M be given by that F ( x ) is the linear subspace df ( T x X ). (When M is 3-dimensional, then we may think of the image of this map as lying in T M/ {± v } .)Strictly speaking, this is only defined on the measurable space, where J f is non-zero; weextend it arbitrarily to all of X since the corresponding Radon measure on G M given by h → R X h ◦ F J f is independent of the extension.1.4. Existence of good sweepouts. A W , map u on a smooth compact surface D withboundary ∂D is energy minimizing to M ⊂ R N if u ( x ) is in M for almost every x and(1.13) E( u ) = inf { E( w ) | w ∈ W , ( D, M ) and ( w − u ) ∈ W , ( D ) } . The map u is said to be weakly harmonic if u is a W , weak solution of the harmonic mapequation ∆ u ⊥ T M ; see, e.g., lemma 1 . .
10 in [He1].The next result gives the existence of a sequence of good sweepouts. This is a corollary of the Stone-Weierstrass theorem; see corollary 35 on page 213 of [R].
TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
Theorem 1.14.
Given a metric g on M and a map β ∈ Ω representing a non-trivial classin π ( M ), there exists a sequence of sweepouts γ j ∈ Ω β with max s ∈ [0 , E( γ js ) → W ( g ), andso that given ǫ >
0, there exist ¯ j and δ > j > ¯ j and(1.15) Area( γ j ( · , s )) > W ( g ) − δ , then there are finitely many harmonic maps u i : S → M with(1.16) d V ( γ j ( · , s ) , ∪ i { u i } ) < ǫ . One immediate consequence of Theorem 1.14 is that if s j is any sequence with Area( γ j ( · , s j ))converging to the width W ( g ) as j → ∞ , then a subsequence of γ j ( · , s j ) converges to a col-lection of harmonic maps from S to M . In particular, the sum of the areas of these mapsis exactly W ( g ) and, since the maps are automatically conformal, the sum of the energies isalso W ( g ). The existence of at least one non-trivial harmonic map from S to M was firstproven in [SaU], but they allowed for loss of energy in the limit; cf. also [St]. This energyloss was ruled out by Siu and Yau, using also arguments of Meeks and Yau (see ChapterVIII in [SY]). This was also proven later by Jost in theorem 4 . . Upper bounds for the rate of change of width.
Throughout this subsection, let M be a smooth closed prime and non-aspherical orientable 3-manifold and let g ( t ) be aone-parameter family of metrics on M evolving by the Ricci flow. We will prove Theorem1.7 giving the upper bound for the derivative of the width W ( g ( t )) under the Ricci flow. Todo this, we need three things.One is that the evolution equation for the scalar curvature R = R ( t ), see page 16 of [Ha2],(1.17) ∂ t R = ∆ R + 2 | Ric | ≥ ∆ R + 23 R , implies by a straightforward maximum principle argument that at time t > R ( t ) ≥ / [min R (0)] − t/ − t + C ) . The curvature is normalized so that on the unit S the Ricci curvature is 2 and the scalarcurvature is 6. In the derivation of (1.18) we implicitly assumed that min R (0) <
0. If thiswas not the case, then (1.18) trivially holds for any
C >
0, since, by (1.17), min R ( t ) isalways non-decreasing. This last remark is also used when surgery occurs. This is becauseby construction any surgery region has large (positive) scalar curvature.The second thing that we need in the proof is the observation that if { Σ i } is a collectionof branched minimal 2-spheres and f ∈ W , ( S , M ) with d V ( f, ∪ i Σ i ) < ǫ , then for anysmooth quadratic form Q on M we have (the unit normal n f is defined where J f = 0)(1.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f [Tr( Q ) − Q ( n f , n f )] − X i Z Σ i [Tr( Q ) − Q ( n Σ i , n Σ i )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ǫ k Q k C Area( f ) . The last thing is an upper bound for the rate of change of area of minimal 2-spheres.Suppose that X is a closed surface and f : X → M is a W , map, then using (1.6) an easy IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 7 calculation gives (cf. pages 38–41 of [Ha2])(1.20) ddt t =0 Area g ( t ) ( f ) = − Z f [ R − Ric M ( n f , n f )] . If Σ ⊂ M is a closed immersed minimal surface, then(1.21) ddt t =0 Area g ( t ) (Σ) = − Z Σ K Σ − Z Σ [ | A | + R ] . Here K Σ is the (intrinsic) curvature of Σ, A is the second fundamental form of Σ, and | A | isthe sum of the squares of the principal curvatures. To get (1.21) from (1.20), we used that ifK M is the sectional curvature of M on the two-plane tangent to Σ, then the Gauss equationsand minimality of Σ give K Σ = K M − | A | . The next lemma gives the upper bound. Lemma 1.22.
If Σ ⊂ M is a branched minimal immersion of the 2-sphere, then(1.23) ddt t =0 Area g ( t ) (Σ) ≤ − π − Area g (0) (Σ)2 min M R (0) . Proof.
Let { p i } be the set of branch points of Σ and b i > ddt t =0 Area g ( t ) (Σ) ≤ − Z Σ K Σ − Z Σ R = − π − π X b i − Z Σ R , where the equality used the Gauss-Bonnet theorem with branch points (this equality alsofollows from the Bochner type formula for harmonic maps between surfaces given on page 10of [SY] and the second displayed equation on page 12 of [SY] that accounts for the branchpoints). Note that branch points only help in the inequality (1.23). (cid:3)
Using these three things, we can show the upper bound for the rate of change of the width.
Proof. (of Theorem 1.7) Fix a time τ . Below ˜ C denotes a constant depending only on τ butwill be allowed to change from line to line. Let γ j ( τ ) be the sequence of sweepouts for themetric g ( τ ) given by Theorem 1.14. We will use the sweepout at time τ as a comparison toget an upper bound for the width at times t > τ . The key for this is the following claim:Given ǫ >
0, there exist ¯ j and ¯ h > j > ¯ j and 0 < h < ¯ h , thenArea g ( τ + h ) ( γ js ( τ )) − max s Area g ( τ ) ( γ js ( τ )) ≤ [ − π + ˜ C ǫ + 34( τ + C ) max s Area g ( τ ) ( γ js ( τ ))] h + ˜ C h . (1.25)To see why (1.25) implies (1.8), use the equivalence of the two definitions of widths to get(1.26) W ( g ( τ + h )) ≤ max s ∈ [0 , Area g ( τ + h ) ( γ js ( τ )) , and take the limit as j → ∞ (so that max s Area g ( τ ) ( γ js ( τ )) → W ( g ( τ ))) in (1.25) to get(1.27) W ( g ( τ + h )) − W ( g ( τ )) h ≤ − π + ˜ C ǫ + 34( τ + C ) W ( g ( τ )) + ˜ C h .
Taking ǫ → This follows by combining that Area g ( τ ) ( γ js ( τ )) ≤ E g ( τ ) ( γ js ( τ )) by (1.4), max s E g ( τ ) ( γ js ( τ )) → W ( g ( τ )), and W ( g ( τ )) ≤ max s Area g ( τ ) ( γ js ( τ )) by the equivalence of the two definitions of width. TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
It remains to prove (1.25). First, let δ > j , depending on ǫ (and on τ ), be given byTheorem 1.14. If j > ¯ j and Area g ( τ ) ( γ js ( τ )) > W ( g ) − δ , then let ∪ i Σ js,i ( τ ) be the collectionof minimal spheres given by Theorem 1.14. Combining (1.20), (1.19) with Q = Ric M , andLemma 1.22 gives ddt t = τ Area g ( t ) ( γ js ( τ )) ≤ ddt t = τ Area g ( t ) ( ∪ i Σ js,i ( τ )) + ˜ C ǫ k Ric M k C Area g ( τ ) ( γ js ( τ )) ≤ − π − Area g ( τ ) ( γ js ( τ ))2 min M R ( τ ) + ˜ C ǫ (1.28) ≤ − π + 34( τ + C ) max s Area g ( τ ) ( γ js ( τ )) + ˜ C ǫ , where the last inequality used the lower bound (1.18) for R ( τ ). Since the metrics g ( t )vary smoothly and every sweepout γ j has uniformly bounded energy, it is easy to see thatArea g ( τ + h ) ( γ js ( τ )) is a smooth function of h with a uniform C bound independent of both j and s near h = 0 (cf. (1.20)). In particular, (1.28) and Taylor expansion give ¯ h > j ) so that (1.25) holds for s with Area g ( τ ) ( γ js ( τ )) > W ( g ) − δ . In theremaining case, we have Area( γ js ( τ )) ≤ W ( g ) − δ so the continuity of g ( t ) implies that (1.25)automatically holds after possibly shrinking ¯ h > (cid:3) Parameter spaces.
Instead of using the unit interval, [0 , P and required that themaps are constant on ∂ P (or that ∂ P = ∅ ). In this case, let Ω P be the set of continuousmaps σ : S × P → M so that for each t ∈ P the map σ ( · , t ) is in C ∩ W , ( S , M ), themap t → σ ( · , t ) is continuous from P to C ∩ W , ( S , M ), and finally σ maps ∂ P to pointmaps. Given a map ˆ σ ∈ Ω P , the homotopy class Ω P ˆ σ ⊂ Ω P is defined to be the set of maps σ ∈ Ω P that are homotopic to ˆ σ through maps in Ω P . Finally, the width W = W (ˆ σ ) isinf σ ∈ Ω P ˆ σ max t ∈P E ( σ ( · , t )). With only trivial changes, the same proof yields Theorem 1.14for these general parameter spaces. The energy decreasing map and its consequences
To prove Theorem 1.14, we will first define an energy decreasing map from Ω to itself thatpreserves the homotopy class (i.e., maps each Ω β to itself) and record its key properties.This should be thought of as a generalization of Birkhoff’s curve shortening process thatplays a similar role when tightening a sweepout by curves; see [B1], [B2], [Cr], and [CM3].Throughout this paper, by a ball B ⊂ S , we will mean a subset of S and a stereographicprojection Π B so that Π B ( B ) ⊂ R is a ball. Given ρ >
0, we will let ρ B ⊂ S denote Π − B of the ball with the same center as Π B ( B ) and radius ρ times that of Π B ( B ). Theorem 2.1.
There is a constant ǫ > , ∞ ) → [0 , ∞ )with Ψ(0) = 0, both depending on M , so that given any ˜ γ ∈ Ω without non-constantharmonic slices and
W >
0, there exists γ ∈ Ω ˜ γ so that E( γ ( · , t )) ≤ E(˜ γ ( · , t )) for each t andso for each t with E(˜ γ ( · , t )) ≥ W/ The main change is in Lemma 3.39 below where the bound 2 for the multiplicity in (1) becomes dim( P )+1.This follows from the definition of (covering) dimension; see pages 302 and 303 in [Mu]. IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 9 ( B Ψ ) If B is any finite collection of disjoint closed balls in S with R ∪ B B |∇ γ ( · , t ) | < ǫ and v : ∪ B B → M is an energy minimizing map equal to γ ( · , t ) on ∪ B ∂ B , then Z ∪ B B |∇ γ ( · , t ) − ∇ v | ≤ Ψ [E(˜ γ ( · , t )) − E( γ ( · , t ))] . The proof of Theorem 2.1 is given in Section 3. The second ingredient that we will need toprove Theorem 1.14 is a compactness result that generalizes compactness of harmonic mapsto maps that are closer and closer to being harmonic (this is Proposition 2.2 below and willbe proven in Appendix B).2.1.
Compactness of almost harmonic maps.
Our notion of almost harmonic relies ontwo important properties of harmonic maps from S to M . The first is that harmonic mapsfrom S are conformal and, thus, energy and area are equal; see (A) below. The second isthat any harmonic map from a surface is energy minimizing when restricted to balls wherethe energy is sufficiently small; see (B) below.In the proposition, ǫ SU > M ) is the small energy constant from lemma3 . ǫ SU .In particular, any non-constant harmonic map from S to M has energy greater than ǫ SU . Proposition 2.2.
Suppose that ǫ , E > ǫ SU > ǫ and u j : S → M isa sequence of C ∩ W , maps with E ≥ E( u j ) satisfying:(A) Area( u j ) > E( u j ) − /j .(B) For any finite collection B of disjoint closed balls in S with R ∪ B B |∇ u j | < ǫ there isan energy minimizing map v : ∪ B B → M that equals u j on ∪ B ∂B with Z ∪ B B (cid:12)(cid:12) ∇ u j − ∇ v (cid:12)(cid:12) ≤ /j . If (A) and (B) are satisfied, then a subsequence of the u j ’s varifold converges to a collectionof harmonic maps v , . . . , v m : S → M .One immediate consequence of Proposition 2.2 is a compactness theorem for sequences ofharmonic maps with bounded energy. This was proven by Jost in lemma 4 . . C convergence in addition to W , convergence; see theorem 2 . L andyet there is no convergent subsequence (see proposition 4 . Constructing good sweepouts from the energy decreasing map on Ω . GivenTheorem 2.1 and Proposition 2.2, we will prove Theorem 1.14. Let G W +1 be the set ofcollections of harmonic maps from S to M so that the sum of the energies is at most W + 1. Proof. (of Theorem 1.14.) Choose a sequence of maps ˜ γ j ∈ Ω β with(2.3) max t ∈ [0 , E (˜ γ j ( · , t )) < W + 1 j , and so that ˜ γ j ( · , t ) is not harmonic unless it is a constant map. We can assume that
W > γ j ( · , t )) ≤ E(˜ γ j ( · , t )) → γ j ’s gives a sequence γ j ∈ Ω β where each γ j ( · , t ) has en-ergy at most that of ˜ γ j ( · , t ). We will argue by contradiction to show that the γ j ’s havethe desired property. Suppose, therefore, that there exist j k → ∞ and s k ∈ [0 ,
1] withd V ( γ j k ( · , s k ) , G W +1 ) ≥ ǫ > γ j k ( · , s k )) > W − /k . Thus, by (2.3) and the factthat E( · ) ≥ Area( · ), we get(2.4) E(˜ γ j k ( · , s k )) − E( γ j k ( · , s k )) ≤ E(˜ γ j k ( · , s k )) − Area( γ j k ( · , s k )) ≤ /k + 1 /j k → , and, similarly, E ( γ j k ( · , s k )) − Area ( γ j k ( · , s k )) →
0. Using (2.4) in Theorem 2.1 gives(B) If B is any collection of disjoint closed balls in S with R ∪ B B |∇ γ j k ( · , s k ) | < ǫ and v : ∪ B B → M is an energy minimizing map that equals γ j k ( · , s k ) on ∪ B ∂B , then(2.5) Z ∪ B B (cid:12)(cid:12) ∇ γ j k ( · , s k ) − ∇ v (cid:12)(cid:12) ≤ Ψ(1 /k + 1 /j k ) → . Therefore, we can apply Proposition 2.2 to get that a subsequence of the γ j k ( · , s k )’s varifoldconverges to a collection of harmonic maps. However, this contradicts the lower bound forthe varifold distance to G W +1 , thus completing the proof. (cid:3) Constructing the energy decreasing map
Harmonic replacement.
The energy decreasing map from Ω to itself will be given bya repeated replacement procedure. At each step, we replace a map u by a map H ( u ) thatcoincides with u outside a ball and inside the ball is equal to an energy-minimizing map withthe same boundary values as u . This is often referred to as harmonic replacement .One of the key properties that makes harmonic replacement useful is that the energyfunctional is strictly convex on small energy maps. Namely, Theorem 3.1 below gives auniform lower bound for the gap in energy between a harmonic map and a W , map withthe same boundary values; see Appendix C for the proof. Theorem 3.1.
There exists a constant ǫ > M ) so that if u and v are W , maps from B ⊂ R to M , u and v agree on ∂B , and v is weakly harmonic with energy at To do this, first use Lemma D.1 (density of C -sweepouts) to choose ˜ γ j ∈ Ω β so t → ˜ γ j ( · , t ) is continuousfrom [0 ,
1] to C and max t ∈ [0 , E (˜ γ j ( · , t )) < W + j . Using stereographic projection, we can view ˜ γ j ( · , t ) asa map from R . Now fix a j . The continuity in C gives a uniform bound sup t ∈ [0 , sup B |∇ ˜ γ j ( · , t ) | ≤ C for some C . Choose R > πC R ≤ / (2 j ). Define a map Φ : R → R in polar coordinates by:Φ( r, θ ) = (2 r, θ ) for r < R/
2, Φ( r, θ ) = (
R, θ ) for R/ ≤ r ≤ R , and Φ( r, θ ) = ( r, θ ) for R < r . Note that Φis homotopic to the identity, is conformal away from the annulus B R \ B R/ , and on B R \ B R/ has | ∂ r Φ | = 0and | d Φ | ≤
2. It follows that ˜ γ j ( · , t ) = ˜ γ j ( · , t ) ◦ Φ is in Ω β , satisfies (2.3), and has ∂ r ˜ γ j ( · , t ) = 0 on B R \ B R/ .Since harmonic maps from S are conformal (corollary 1 . γ j ( · , t ) is constant on B R \ B R/ and, thus, constant on S by unique continuation (theorem 1 . IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 11 most ǫ , then(3.2) Z B |∇ u | − Z B |∇ v | ≥ Z B |∇ v − ∇ u | . An immediate corollary of Theorem 3.1 is uniqueness of solutions to the Dirichlet problemfor small energy maps (and also that any such harmonic map minimizes energy).
Corollary 3.3.
Let ǫ > u and u are W , weakly harmonicmaps from B ⊂ R to M , both with energy at most ǫ , and they agree on ∂B , then u = u .3.2. Continuity of harmonic replacement on C ( B ) ∩ W , ( B ) . The second conse-quence of Theorem 3.1 is that harmonic replacement is continuous as a map from C ( B ) ∩ W , ( B ) to itself if we restrict to small energy maps. (The norm on C ( B ) ∩ W , ( B ) isthe sum of the sup norm and the W , norm.) Corollary 3.4.
Let ǫ > M = { u ∈ C ( B , M ) ∩ W , ( B , M ) | E( u ) ≤ ǫ } . Given u ∈ M , there is a unique energy minimizing map w equal to u on ∂B and w is in M . Furthermore, there exists C depending on M so that if u , u ∈ M with correspondingenergy minimizing maps w , w , and we set E = E( u ) + E( u ), then(3.6) | E( w ) − E( w ) | ≤ C || u − u || C ( B ) E + C ||∇ u − ∇ u || L ( B ) E / . Finally, the map from u to w is continuous as a map from C ( B ) ∩ W , ( B ) to itself.In the proof, we will use that since M is smooth, compact and embedded, there exists a δ > x in the δ -tubular neighborhood M δ of M in R N , there is a unique closestpoint Π( x ) ∈ M and so the map x → Π( x ) is smooth. Π is called nearest point projection .Furthermore, for any x ∈ M , we have | d Π x ( V ) | ≤ | V | . Therefore, there is a constant C Π depending on M so that for any x ∈ M δ , we have | d Π x ( V ) | ≤ (1 + C Π | x − Π( x ) | ) | V | . Inparticular, we can choose ˆ δ ∈ (0 , δ ) so that | d Π x ( V ) | ≤ | V | for any x ∈ M ˆ δ and V ∈ R N . Proof. (of Corollary 3.4.) The existence of an energy minimizing map w ∈ W , ( B ) wasproven by Morrey in [Mo1]; by Corollary 3.3, w is unique. The continuity of w on B is themain theorem of [Q]. It follows that w ∈ M . Step 1: E ( w ) is uniformly continuous . We can assume that || u − u || C ( B ) ≤ ˆ δ , since(3.6) holds with C = 1 / ˆ δ if || u − u || C ( B ) ≥ ˆ δ . Define a map v by(3.7) v = Π ◦ ( w + ( u − u )) , so that v maps to M and agrees with u on ∂B . Using that | d Π x ( V ) | ≤ | V | for x ∈ M and w maps to M , we can estimate the energy of v by(3.8) E( v ) ≤ (1 + C Π || u − u || C ( B ) ) (cid:2) E( w ) + 2(E( w ) E( u − u )) / + E( u − u ) (cid:3) , where C Π is the Lipschitz norm of d Π in M ˆ δ . Since v and w agree on ∂B , Corollary3.3 yields E( w ) ≤ E( v ). By symmetry, we can assume that E( w ) ≤ E( w ) so that (3.8)implies (3.6). Continuity also essentially follows from the boundary regularity of Schoen and Uhlenbeck, [SU2], exceptthat [SU2] assumes C ,α regularity of the boundary data. Step 2: The continuity of u → w . Suppose that u, u j are in M with u j → u in C ( B ) ∩ W , ( B ) and w and w j are the corresponding energy minimizing maps.We will first show that w j → w in W , ( B ). To do this, set(3.9) v j = Π ◦ ( w + ( u j − u )) , so that v j maps to M and agrees with u j on ∂B . Arguing as in (3.8) and using thatE( w j ) → E( w ) by Step 1, we get that [E( v j ) − E( w j )] →
0. Therefore, applying Theorem 3.1to w j , v j gives that || w j − v j || W , ( B ) →
0. Since || u j − u || C ( B ) ∩ W , ( B ) → ◦ w = w ,it follows that || w − v j || W , ( B ) →
0. The triangle inequality gives || w − w j || W , ( B ) → w j → w in C ( B ). Suppose insteadthat there is a subsequence (still denoted w j ) with(3.10) || w j − w || C ( B ) ≥ ǫ > . Using the uniform energy bound for the w j ’s together with interior estimates for energyminimizing maps of [SU1] (and the Arzela-Ascoli theorem), we can pass to a further subse-quence so that the w j ’s converge uniformly in C on any compact subset K ⊂ B . Finally,as remarked in the proof of the main theorem in [Q], proposition 1 and remark 1 of [Q] implythat the w j ’s are also equicontinuous near ∂B , so Arzela-Ascoli gives a further subsequencethat converges uniformly on B to a harmonic map w ∞ that agrees with w on the boundary.However, (3.10) implies that || w − w ∞ || C ( B ) ≥ ǫ > (cid:3) Corollary 3.4 gives another proof that the width is positive when the homotopy class isnon-trivial or, equivalently, that if max t E( σ ( · , t )) is sufficiently small (depending on M ),then σ is homotopically trivial. Namely, since t → σ ( · , t ) is continuous from [0 ,
1] to C , wecan choose r > σ ( · , t ) maps the ball B r ( p ) ⊂ S into a convex geodesic ball B t in M for every t . If each σ ( · , t ) has energy less than ǫ > σ ( · , t ) outside B r ( p ) by the energy minimizing map with the same boundary values gives ahomotopic sweepout ˜ σ . Moreover, the entire image of ˜ σ ( · , t ) is contained in the convex ball B t by the maximum principle. It follows that ˜ σ is homotopically trivial by contractingeach ˜ σ ( · , t ) to the point σ ( p, t ) via a geodesic homotopy.3.3. Uniform continuity of energy improvement on W , . It will be convenient tointroduce some notation for the next lemma. Namely, given a C ∩ W , map u from S to M and a finite collection B of disjoint closed balls in S so the energy of u on ∪ B B is at most ǫ /
3, let H ( u, B ) : S → M denote the map that coincides with u on S \ ∪ B B and on ∪ B B isequal to the energy minimizing map from ∪ B B to M that agrees with u on ∪ B ∂B . To keepthe notation simple, we will set H ( u, B , B ) = H ( H ( u, B ) , B ). Finally, if α ∈ (0 , α B will denote the collection of concentric balls but whose radii are shrunk by the factor α .In general, H ( u, B , B ) is not the same as H ( u, B , B ). This matters in the proof ofTheorem 2.1, where harmonic replacement on either B or B decreases the energy of u bya definite amount. The next lemma (see (3.12)) shows that the energy goes down a definiteamount regardless of the order that we do the replacements. The second inequality bounds This follows from lemma 4 . . σ ( · , t ) is homotopic to a map in B t and thisfollows from the small energy bound and the uniform lower bound for the energy of any homotopicallynon-trivial map from S given, e.g., in the first line of the proof of proposition 2 on page 143 of [SY]. IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 13 the possible decrease in energy from applying harmonic replacement on H ( u, B ) in termsof the possible decrease from harmonic replacement on u . Lemma 3.11.
There is a constant κ > M ) so that if u : S → M is in C ∩ W , and B , B are each finite collections of disjoint closed balls in S so that theenergy of u on each ∪ B i B is at most ǫ /
3, then(3.12) E( u ) − E [ H ( u, B , B )] ≥ κ (cid:18) E( u ) − E (cid:20) H ( u, B ) (cid:21)(cid:19) . Furthermore, for any µ ∈ [1 / , / u ) − E [ H ( u, B )]) / κ +E( u ) − E [ H ( u, µ B )] ≥ E [ H ( u, B )] − E [ H ( u, B , µ B )] . We will prove Lemma 3.11 by constructing comparison maps with the same boundaryvalues and using the minimizing property of small energy harmonic maps to get upperbounds for the energy. The following lemma will be used to construct the comparison maps.
Lemma 3.14.
There exists τ > M ) so that if f, g : ∂B R → M are C ∩ W , maps that agree at one point and satisfy(3.15) R Z ∂B R | f ′ − g ′ | ≤ τ , then there exists some ρ ∈ (0 , R/
2] and a C ∩ W , map w : B R \ B R − ρ → M so that(3.16) w ( R − ρ, θ ) = f ( R, θ ) and w ( R, θ ) = g ( R, θ ) , and R B R \ B R − ρ |∇ w | ≤ √ (cid:16) R R ∂B R ( | f ′ | + | g ′ | ) (cid:17) / (cid:16) R R ∂B R | f ′ − g ′ | (cid:17) / . Proof.
Let Π and δ > ˆ δ > M ) be as in the proof of Corollary 3.4 and set τ = ˆ δ/ √ π . Since f − g vanishes somewhere on ∂B R , integrating (3.15) gives max | f − g | ≤ ˆ δ .Since the statement is scale-invariant, it suffices to prove the case R = 1. Set ρ = R S | f ′ − g ′ | / [8 R S ( | f ′ | + | g ′ | )] ≤ / w : B \ B − ρ → R N by(3.17) ˆ w ( r, θ ) = f ( θ ) + (cid:18) r + ρ − ρ (cid:19) ( g ( θ ) − f ( θ )) . Observe that ˆ w satisfies (3.16). Furthermore, since f − g vanishes somewhere on S , we canuse Wirtinger’s inequality R S | f − g | ≤ R S | ( f − g ) ′ | to bound R B \ B − ρ |∇ ˆ w | by Z B \ B − ρ |∇ ˆ w | ≤ Z − ρ (cid:20) ρ Z π | f − g | ( θ ) dθ + 1 r Z π ( | f ′ | + | g ′ | )( θ ) dθ (cid:21) r dr ≤ ρ Z π | f ′ − g ′ | ( θ ) dθ + 2 ρ Z π ( | f ′ | + | g ′ | )( θ ) dθ (3.18) = 17 / √ (cid:18)Z S | f ′ − g ′ | Z S ( | f ′ | + | g ′ | ) (cid:19) / . Since | f − g | ≤ ˆ δ , the image of ˆ w is contained in M ˆ δ where we have | d Π | ≤
2. Therefore, ifwe set w = Π ◦ ˆ w , then the energy of w is at most twice the energy of ˆ w . (cid:3) Proof. (of Lemma 3.11.) We will index the balls in B by α and use j for the balls in B ;i.e., let B = { B α } and B = { B j } . The key point is that, by Corollary 3.4, small energyharmonic maps minimize energy. Using this, we get upper bounds for the energy of theharmonic replacement by cutting and pasting to construct comparison functions with thesame boundary values.Observe that the total energy of u on the union of the balls in B ∪ B is at most 2 ǫ / B does not change the map outside these balls and is energynon-increasing, it follows that the total energy of H ( u, B ) on B is at most 2 ǫ / The proof of (3.12) . We will divide B into two disjoint subsets, B , + and B , − , and argueseparately, depending on which of these accounts for more of the decrease in energy afterharmonic replacement. Namely, set(3.19) B , + = { B j ∈ B | B j ⊂ B α for some B α ∈ B } and B , − = B \ B , + . Since the balls in B are disjoint, it follows that(3.20) E( u ) − E( H ( u, B )) = (cid:18) E( u ) − E( H ( u, B , − )) (cid:19) + (cid:18) E( u ) − E( H ( u, B , + )) (cid:19) . Case 1 . Suppose that E( u ) − E (cid:2) H ( u, B , + ) (cid:3) ≥ (cid:0) E( u ) − E (cid:2) H ( u, B ) (cid:3)(cid:1) /
2. Since the ballsin B , + are contained in balls in B and harmonic replacements minimize energy, we get(3.21) E( H ( u, B , B )) ≤ E( H ( u, B )) ≤ E( H ( u, B , + )) , so that (cid:0) E( u ) − E (cid:2) H ( u, B ) (cid:3)(cid:1) / ≤ E( u ) − E( H ( u, B , + )) ≤ E( u ) − E( H ( u, B , B )). Case 2 . Suppose now that(3.22) E( u ) − E( H ( u, B , − )) ≥ (cid:18) E( u ) − E( H ( u, B )) (cid:19) . Let τ > Z S |∇ H ( u, B ) − ∇ u | ≤ τ , since otherwise Theorem 3.1 gives (3.12) with κ = τ /ǫ . The key is to show for B j ∈ B , − that Z B j |∇ H ( u, B ) | − Z B j (cid:12)(cid:12) ∇ H ( u, B , B j ) (cid:12)(cid:12) ≥ Z B j |∇ u | − Z B j (cid:12)(cid:12)(cid:12)(cid:12) ∇ H ( u, B j ) (cid:12)(cid:12)(cid:12)(cid:12) (3.24) − C Z B j |∇ u | + |∇ H ( u, B ) | ! / Z B j |∇ ( u − H ( u, B )) | ! / , where C is a universal constant. Namely, summing (3.24) over B , − and using the inequality | P a j b j | ≤ (cid:0)P a j (cid:1) / (cid:0)P b j (cid:1) / , the bound for the energy of u in B ∪ B , and Theorem 3.1to relate the energy of u − H ( u, B ) to E( u ) − E( H ( u, B )) givesE( u ) − E( H ( u, B , − )) ≤ E( H ( u, B )) − E( H ( u, B , B , − )) + C ǫ / (E( u ) − E[ H ( u, B )]) / ≤ δ E + C ǫ / δ / ≤ ( C + 1) ǫ / δ / , (3.25) IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 15 where we have set δ E = E( u ) − E( H ( u, B , B )) in the last line and the last inequality usedthat δ E ≤ ǫ / < ǫ . Combining (3.22) with (3.25) gives (3.12).To complete Case 2, we must prove (3.24). After translation, we can assume that B j isthe ball B R of radius R about 0 in R . Set u = H ( u, B ) and apply the co-area formula toget r ∈ [3 R/ , R ] (in fact, a set of r ’s of measure at least R/
36) with Z ∂B r |∇ u − ∇ u | ≤ R Z R R/ (cid:18)Z ∂B s |∇ u − ∇ u | (cid:19) ds ≤ r Z B R |∇ u − ∇ u | , (3.26) Z ∂B r ( |∇ u | + |∇ u | ) ≤ R Z R R/ (cid:18)Z ∂B s |∇ u | + |∇ u | (cid:19) ds ≤ r Z B R ( |∇ u | + |∇ u | ) . (3.27)Since B j ∈ B , − and r > R/
2, the circle ∂B r is not contained in any of the balls in B . Itfollows that ∂B r contains at least one point outside ∪ B B and, thus, there is a point in ∂B r where u = u . This and (3.23) allow us to apply Lemma 3.14 to get ρ ∈ (0 , r/
2] and a map w : B r \ B r − ρ → M with w ( r, θ ) = u ( r, θ ), w ( r − ρ, θ ) = u ( r, θ ), and(3.28) Z B r \ B r − ρ |∇ w | ≤ C Z B j |∇ u | + |∇ H ( u, B ) | ! / Z B j |∇ ( u − H ( u, B )) | ! / . Observe that the map x → H ( u, B r )( r x/ ( r − ρ )) maps B r − ρ to M and agrees with w on ∂B r − ρ . Therefore, the map from B R to M which is equal to u on B R \ B r , is equal to w on B r \ B r − ρ , and is equal to H ( u, B r )( r · / ( r − ρ )) on B r − ρ gives an upper bound for theenergy of H ( u , B R )(3.29) Z B R |∇ H ( u , B R ) | ≤ Z B R \ B r |∇ u | + Z B r \ B r − ρ |∇ w | + Z B r |∇ H ( u, B r ) | . Using (3.28) and that ||∇ u | − |∇ u | | ≤ ( |∇ u | + |∇ u | ) |∇ ( u − u ) | , we get Z B R |∇ u | − Z B R |∇ H ( u , B R ) | ≥ Z B r |∇ u | − Z B r |∇ H ( u, B r ) | − Z B r \ B r − ρ |∇ w | ≥ Z B r |∇ u | − Z B r |∇ H ( u, B r ) | − C (cid:18)Z B r |∇ u | + |∇ u | (cid:19) / (cid:18)Z B r |∇ ( u − u ) | (cid:19) / . Since R B R/ |∇ H ( u, B R/ ) | ≤ R B R/ \ B r |∇ u | + R B r |∇ H ( u, B r ) | , we get (3.24). The proof of (3.13) . We will argue similarly with a few small modifications that we willdescribe. This time, let B , + ⊂ B be the balls B j with µB j contained in some B α ∈ B . Itfollows that harmonic replacement on µ B , + does not change H ( u, B ) and, thus,(3.30) E [ H ( u, B )] = E [ H ( u, B , µ B , + )] . Again, we can assume that (3.23) holds. Suppose now that B j ∈ B , − . Arguing as in theproof of (3.24) (switching the roles of u and H ( u, B )), we get Z B j |∇ u | − Z B j (cid:12)(cid:12) ∇ H ( u, µB j ) (cid:12)(cid:12) ≥ Z µB j |∇ H ( u, B ) | − Z µB j (cid:12)(cid:12) ∇ H ( u, B , µ B j ) (cid:12)(cid:12) (3.31) − C Z B j |∇ u | + |∇ H ( u, B ) | ! / Z B j |∇ ( u − H ( u, B )) | ! / . Summing this over B , − and arguing as for (3.25) gives Z |∇ u | − Z |∇ H ( u, µ B ) | ≥ Z |∇ H ( u, B ) | − Z |∇ H ( u, B , µ B , − ) | (3.32) − C ǫ / (E( u ) − E[ H ( u, B )]) / . Combining (3.30) and (3.32) completes the proof. (cid:3)
Constructing the map from ˜ γ to γ . We will construct γ ( · , t ) from ˜ γ ( · , t ) by harmonicreplacement on a family of balls in S varying continuously in t . The balls will be chosenin Lemma 3.39 below. Throughout this subsection, ǫ > M ) given by Theorem 3.1.Given σ ∈ Ω and ǫ ∈ (0 , ǫ ], define the maximal improvement from harmonic replacementon families of balls with energy at most ǫ by(3.33) e σ,ǫ ( t ) = sup B { E( σ ( · , t )) − E( H ( σ ( · , t ) , B )) } , where the supremum is over all finite collections B of disjoint closed balls where the totalenergy of σ ( · , t ) on B is at most ǫ . Observe that e σ,ǫ ( t ) is nonnegative, monotone non-decreasing in ǫ , and is positive if σ ( · , t ) is not harmonic. Lemma 3.34. If σ ( · , t ) is not harmonic and ǫ ∈ (0 , ǫ ], then there is an open interval I t containing t so that e σ,ǫ/ ( s ) ≤ e σ,ǫ ( t ) for all s in the double interval 2 I t . Proof.
By (3.6) in Corollary 3.4, there exists δ > t ) so that if(3.35) || σ ( · , t ) − σ ( · , s ) || C ∩ W , < δ and B is a finite collection of disjoint closed balls where both σ ( · , t ) and σ ( · , s ) have energyat most ǫ , then(3.36) (cid:12)(cid:12)(cid:12)(cid:12) E( H ( σ ( · , s ) , B )) − E( H ( σ ( · , t ) , B )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e σ,ǫ ( t ) / . Here we have used that e σ,ǫ ( t ) > σ ( · , t ) is not harmonic. Since t → σ ( · , t ) is continuousas a map to C ∩ W , , we can choose I t so that for all s ∈ I t (3.35) holds and(3.37) 12 Z S (cid:12)(cid:12) |∇ σ ( · , t ) | − |∇ σ ( · , s ) | (cid:12)(cid:12) ≤ min { ǫ , e σ,ǫ ( t )2 } . Suppose now that s ∈ I t and the energy of σ ( · , s ) is at most ǫ/ B . It followsfrom (3.37) that the energy of σ ( · , t ) is at most ǫ on B . Combining (3.36) and (3.37) gives(3.38) (cid:12)(cid:12)(cid:12)(cid:12) E( σ ( · , s )) − E( H ( σ ( · , s ) , B )) − E( σ ( · , t )) + E( H ( σ ( · , t ) , B )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e σ,ǫ ( t ) . IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 17
Since this applies to any such B , we get that e σ,ǫ/ ( s ) ≤ e σ,ǫ ( t ). (cid:3) Given a sweepout with no harmonic slices, the next lemma constructs finitely many col-lections of balls so that harmonic replacement on at least one of these collections strictlydecreases the energy. In addition, each collection consists of finitely many pairwise disjointclosed balls.
Lemma 3.39. If W > γ ∈ Ω has no non-constant harmonic slices, then we get aninteger m (depending on ˜ γ ), m collections of balls B , . . . , B m in S , and continuous functions r , . . . , r m : [0 , → [0 ,
1] so that for each t :(1) At most two r j ( t )’s are positive and P B ∈B j R r j ( t ) B |∇ ˜ γ ( · , t ) | < ǫ / j .(2) If E(˜ γ ( · , t )) ≥ W/
2, then there exists j ( t ) so that harmonic replacement on r j ( t ) B j ( t ) decreases energy by at least e ˜ γ,ǫ / ( t ) / Proof.
Since the energy of the slices is continuous in t , the set I = { t | E(˜ γ ( · , t )) ≥ W/ } is compact. For each t ∈ I , choose a finite collection B t of disjoint closed balls in S with R ∪ B t |∇ ˜ γ ( · , t ) | ≤ ǫ / γ ( · , t )) − E( H ( γ ( · , t ) , B t )) ≥ e ˜ γ,ǫ / ( t )2 > . Lemma 3.34 gives an open interval I t containing t so that for all s ∈ I t (3.41) e ˜ γ,ǫ / ( s ) ≤ e ˜ γ,ǫ / ( t ) . Using the continuity of ˜ γ ( · , s ) in C ∩ W , and Corollary 3.4, we can shrink I t so that ˜ γ ( · , s )has energy at most ǫ / B t for s ∈ I t and, in addition,(3.42) (cid:12)(cid:12)(cid:12)(cid:12) E( γ ( · , s )) − E( H ( γ ( · , s ) , B t )) − E( γ ( · , t )) + E( H ( γ ( · , t ) , B t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e ˜ γ,ǫ / ( t )4 . Since I is compact, we can cover I by finitely many I t ’s, say I t , . . . , I t m . Moreover, afterdiscarding some of the intervals, we can arrange that each t is in at least one closed interval I t j , each I t j intersects at most two other I t k ’s, and the I t k ’s intersecting I t j do not intersecteach other. For each j = 1 , . . . m , choose a continuous function r j : [0 , → [0 ,
1] so that • r j ( t ) = 1 on I t j and r j ( t ) is zero for t / ∈ I t j . • r j ( t ) is zero on the intervals that do not intersect I t j .Property (1) follows directly and (2) follows from (3.40), (3.41), and (3.42). (cid:3) Proof. (of Theorem 2.1). Let B , . . . , B m and r , . . . , r m : [0 , → [0 , π ) be given by Lemma3.39. We will use an m step replacement process to define γ . Namely, first set γ = ˜ γ andthen, for each k = 1 , . . . , m , define γ k by applying harmonic replacement to γ k − ( · , t ) on the k -th family of balls r k ( t ) B k ; i.e, set γ k ( · , t ) = H ( γ k − ( · , t ) , r k ( t ) B k ). Finally, we set γ = γ m .A key point in the construction is that property (1) of the family of balls gives that onlytwo r k ( t )’s are positive for each t . Therefore, the energy bound on the balls given by property We will give a recipe for doing this. First, if I t is contained in the union of two other intervals, thenthrow it out. Otherwise, consider the intervals whose left endpoint is in I t , find one whose right endpoint islargest and discard the others (which are anyway contained in these). Similarly, consider the intervals whoseright endpoint is in I t and throw out all but one whose left endpoint is smallest. Next, repeat this processon I t (unless it has already been discarded), etc. After at most m steps, we get the desired cover. (1) implies that each energy minimizing map replaces a map with energy at most 2 ǫ / < ǫ .Hence, Corollary 3.4 implies that these depend continuously on the boundary values, whichare themselves continuous in t , so that the resulting map ˜ γ is also continuous in t . Finally,it is clear that ˜ γ is homotopic to γ since continuously shrinking the disjoint closed balls onwhich we make harmonic replacement gives an explicit homotopy. Thus, γ ∈ Ω ˜ γ as claimed.For each t with E(˜ γ ( · , t )) ≥ W/
2, property (2) of the family of balls gives some j ( t ) so thatharmonic replacement for ˜ γ ( · , t ) on r j ( t )2 B j ( t ) decreases the energy by at least e ˜ γ,ǫ / ( t )8 . Thus,even in the worst case where r j ( t ) B j ( t ) is the second family of balls that we do replacementon at t , (3.12) in Lemma 3.11 gives(3.43) E(˜ γ ( · , t )) − E( γ ( · , t )) ≥ κ (cid:18) e ˜ γ,ǫ / ( t )8 (cid:19) . To establish ( B Ψ ), suppose that B is a finite collection of disjoint closed balls in S so thatthe energy of γ ( · , t ) on B is at most ǫ /
12. We can assume that γ k ( · , t ) has energy atmost ǫ / B for every k since otherwise Theorem 3.1 implies a positive lower bound forE(˜ γ ( · , t )) − E( γ ( · , t )). Consequently, we can apply (3.13) in Lemma 3.11 twice (first with µ = 1 / µ = 1 /
4) to getE( γ ( · , t )) − E (cid:20) H ( γ ( · , t ) , B ) (cid:21) ≤ E(˜ γ ( · , t )) − E (cid:20) H (˜ γ ( · , t ) , B ) (cid:21) + 2 κ (E(˜ γ ( · , t )) − E( γ ( · , t ))) / ≤ e ˜ γ,ǫ / ( t ) + 2 κ (E(˜ γ ( · , t )) − E( γ ( · , t ))) / . (3.44)Combining (3.43) and (3.44) with Theorem 3.1 gives ( B Ψ ) and, thus, completes the proof. (cid:3) Appendix A. Bubble convergence implies varifold convergence
A.1.
Bubble convergence and the topology on Ω . We will need a notion of convergencefor a sequence v j of W , maps to a collection { u , . . . , u m } of W , maps which is similar inspirit to the convergence in Gromov’s compactness theorem for pseudo holomorphic curves,[G]. The notion that we will use is a slight weakening of the bubble tree convergencedeveloped by Parker and Wolfson for J -holomorphic curves in [PaW] and used by Parker forharmonic maps in [Pa]. In our applications, the v j ’s will be approximately harmonic whilethe limit maps u i will be harmonic. We will need the next definition to make this precise. S + and S − will denote the northern and southern hemispheres in S and p + = (0 , , p − = (0 , , −
1) the north and south poles.
Definition A.1.
Given a ball B r ( x ) ⊂ S , the conformal dilation taking B r ( x ) to S − is thecomposition of translation x → p − followed by dilation of S about p − taking B r ( p − ) to S − .The standard example of a conformal dilation comes from applying stereographic projec-tion Π : S \ { (0 , , } → R , then dilating R by a positive λ = 1, and applying Π − .In the definition below of convergence, the map u will be the standard W , -weak limitof the v j ’s (see (B1)), while the other u i ’s will arise as weak limits of the composition ofthe v j ’s with a divergent sequence of conformal dilations of S (see (B2)). The condition(B3) guarantees that these limits all arise in genuinely distinct ways, and the condition (B4)means that together the u i ’s account for all of the energy. IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 19
Definition A.2. Bubble convergence . We will say that a sequence v j : S → M of W , maps converges to a collection of W , maps u , . . . , u m : S → M if the following hold:(B1) The v j ’s converge weakly to u in W , and there is a finite set S = { x , . . . , x k } ⊂ S so that the v j ’s converge strongly to u in W , ( K ) for any compact K ⊂ S \ S .(B2) For each i >
0, we get a point x ℓ i ∈ S and a sequence of balls B r i,j ( y i,j ) with y i,j → x ℓ i and r i,j →
0. Furthermore, if D i,j : S → S is the conformal dilationtaking the southern hemisphere to B r i,j ( y i,j ), then the maps v j ◦ D i,j converge to u i as in (B1). Namely, v j ◦ D i,j → u i weakly in W , ( S ) and there is a finite set S i sothat the v j ◦ D i,j ’s converge strongly in W , ( K ) for any compact K ⊂ S \ S i .(B3) If i = i , then r i ,j r i ,j + r i ,j r i ,j + | y i ,j − y i ,j | r i ,j r i ,j → ∞ .(B4) We get the energy equality P mi =0 E( u i ) = lim j →∞ E( v j ) .A.2. Two simple examples of bubble convergence.
The simplest non-trivial exampleof bubble convergence is when each map v j = u ◦ Ψ j is the composition of a fixed harmonicmap u : S → M with a divergent sequence of dilations Ψ j : S → S . In this case, the v j ’s converge to the constant map u = u ( p + ) on each compact set of S \ { p − } and all ofthe energy concentrates at the single point p − = S . Composing the v j ’s with the divergentsequence Ψ − j of conformal dilations gives the limit u = u .For the second example, let Π : S \ { (0 , , } → R be stereographic projection andlet z = x + iy be complex coordinates on R = C . If we set f j ( z ) = 1 / ( jz ) + z = z +1 /jz ,then the maps v j = Π − ◦ f j ◦ Π : S → S are conformal and, therefore, also harmonic.Since each v j is a rational map of degree two, we have E( v j ) = Area( v j ) = 8 π . Moreover,the v j ’s converge away from 0 to the identity map which has energy 4 π . The other 4 π ofenergy disappears at 0 but can be accounted for by a map u by composing with a divergentsequence of conformal dilations; u must also have degree one. In this case, the conformaldilations take f j to ˜ f j ( z ) = f j ( z/j ) = 1 /z + z/j which converges to the conformal inversionabout the circle of radius one.A.3. Bubble convergence implies varifold convergence.Proposition A.3.
If a sequence v j of W , ( S , M ) maps bubble converges to a finite col-lection of smooth maps u , . . . , u m : S → M , then it also varifold converges.Before getting to the proof, recall that a sequence of functions f j is said to convergein measure to a function f if for all δ > { x | | f j − f | ( x ) > δ } goes tozero as j → ∞ ; see [R], page 95. Clearly, L convergence implies convergence in measure.Furthermore, if f j → f in measure and h is uniformly continuous, then h ◦ f j → h ◦ f inmeasure. Finally, we will use the following general version of the dominated convergencetheorem which combines theorem 17 on page 92 of [R] and proposition 20 on page 96 of [R]:(DCT) If f j → f in measure, g j → g in L , and | f j | ≤ g j , then R f j → R f .We will also use that the map ∇ u → J u is continuous as a map from L to L and, thus,Area( u ) is continuous with respect to E( u ). To be precise, if u, v ∈ W , ( S , M ), then(A.4) | J u − J v | ≤ √ |∇ u − ∇ v | / max {|∇ u | / , |∇ v | / } . This follows from the linear algebra fact that if S and T are N × (cid:12)(cid:12) det (cid:0) S T S (cid:1) − det (cid:0) T T T (cid:1)(cid:12)(cid:12) ≤ | T − S | max {| S | , | T | } , where | S | is the sum of the squares of the entries of S and S T is the transpose. Proof. (of Proposition A.3.) For each v j , we will let V j denote the corresponding map to G M . Similarly, for each u i , let U i denote the corresponding map to G M .It follows from (B1)-(B4) that we can choose m + 1 sequences of domains Ω j , . . . , Ω jm ⊂ S that are pairwise disjoint for each j and so that for each i = 0 , . . . , m applying D − i,j to Ω ji gives a sequence of domains converging to S \ S i and accounts for all the energy, that is,(A.6) lim j →∞ Z S \ ( ∪ i Ω ji ) |∇ v j | = 0 . By (A.6), the proposition follows from showing for each i and any h in C ( G M ) that(A.7) Z S h ◦ U i J u i = lim j →∞ Z Ω ji h ◦ V j J v j = lim j →∞ Z D − i,j ( Ω ji ) h ◦ V j ◦ D i,j J ( v j ◦ D i,j ) , where the last equality is simply the change of variables formula for integration.To simplify notation in the proof of (A.7), for each i and j , let v ji denote the restrictionof v j ◦ D i,j to D − i,j (cid:0) Ω ji (cid:1) and let V ji denote the corresponding map to G M .Observe first that J v ji → J u i in L ( S ) by (A.4). Given ǫ > i , let Ω iǫ be the set where J u i ≥ ǫ . Since h is bounded and J v ji → J u i in L ( S ), (A.7) would follow from(A.8) lim j →∞ Z Ω iǫ h ◦ V ji J v ji = Z Ω iǫ h ◦ U i J u i . However, given any δ > W , convergence implies that the measure of(A.9) { x ∈ Ω iǫ | J v ji ≥ ǫ | V ji − U i | ≥ δ } goes to zero as j → ∞ . Since L convergence of Jacobians implies that the measure of { x ∈ Ω iǫ | J v ji < ǫ } goes to zero, it follows that the maps V ji converge in measure to U i onΩ iǫ . Therefore, the h ◦ V ji ’s converge in measure to h ◦ U i on Ω iǫ . Consequently, the generalversion of the dominated convergence theorem (DCT) gives (A.8) and, thus, also (A.7). (cid:3) Appendix B. The proof of Proposition 2.2
The proof of Proposition 2.2 will follow the general structure developed by Parker andWolfson in [PaW] and used by Parker in [Pa] to prove compactness of harmonic maps withbounded energy. The main difficulty is to rule out loss of energy in the limit (see (B4) inthe definition of bubble convergence). The rough idea to deal with this is that energy lossonly occurs when there are very small annuli where the maps are “almost” harmonic and theratio between the inner and outer radii of the annulus is enormous. We will use Proposition Note that | S T T | ≤ | S | | T | , (cid:12)(cid:12) Tr ( S T T ) (cid:12)(cid:12) ≤ | S | | T | , and if X t is a path of 2 × ∂ t det X t = Tr ( X ct ∂ t X t ) where X ct is the cofactor matrix given by swapping diagonal entries and mul-tiplying off-diagonals by −
1. Applying this to X t = ( S + t ( T − S )) T ( S + t ( T − S )) and using the meanvalue theorem gives (A.5). IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 21
B.29 to show that the map must be “far” from being conformal on such an annulus and,thus, condition (A) allows us to rule out energy loss. Here “far” from conformal will meanthat the θ -energy of the map is much less than the radial energy. To make this precise, itis convenient to replace an annulus B e r \ B e r in R by the conformally equivalent cylinder[ r , r ] × S . The (non-compact) cylinder R × S with the flat product metric and coordinates t and θ will be denoted by C . For r < r , let C r ,r ⊂ C be the product [ r , r ] × S .B.1. Harmonic maps on cylinders.
The main result of this subsection is that harmonicmaps with small energy on long cylinders are almost radial. This implies that a sequenceof such maps with energy bounded away from zero is uniformly far from being conformaland, thus, cannot satisfy (A) in Proposition 2.2. It will be used to prove a similar resultfor “almost harmonic” maps in Proposition B.29 and eventually be used when we show thatenergy will not be lost.
Proposition B.1.
Given δ >
0, there exist ǫ > ℓ ≥ δ (and M ) sothat if u is a (non-constant) C harmonic map from the flat cylinder C − ℓ, ℓ = [ − ℓ, ℓ ] × S to M with E( u ) ≤ ǫ , then(B.2) Z C − ℓ,ℓ | u θ | < δ Z C − ℓ, ℓ |∇ u | . To show this proposition, we show a differential inequality which leads to exponentialgrowth for the θ -energy of the harmonic map on the level sets of the cylinder. Once we havethat, the proposition follows. Namely, if the θ -energy in the “middle” of the cylinder was adefinite fraction of the total energy over the double cylinder, then the exponential growthwould force the θ -energy of near the boundary of the cylinder to be too large.The following standard lemma is the differential inequality for the θ -energy that leads toexponential growth through Lemma B.8 below. Lemma B.3.
For a C harmonic map u from C r ,r ⊂ C to M ⊂ R N (B.4) ∂ t Z t | u θ | ≥ Z t | u θ | − M | A | Z t |∇ u | . Proof.
Differentiating R t | u θ | and integrating by parts in θ gives12 ∂ t Z t | u θ | = Z t | u tθ | + Z t h u θ , u ttθ i = Z t | u tθ | − Z t h u θθ , u tt i = Z t | u tθ | − Z t h u θθ , (∆ u − u θθ ) i≥ Z t | u tθ | + Z t | u θθ | − sup M | A | Z t | u θθ | |∇ u | , (B.5)where the last inequality used that | ∆ u | ≤ |∇ u | sup M | A | by the harmonic map equation. The lemma follows from applying the absorbing inequality 2 ab ≤ a / b and noting that R t u θ = 0 so that Wirtinger’s inequality gives R t | u θ | ≤ R t | u θθ | . (cid:3) If u i are the components of the harmonic map u , g jk is the metric on B , and A iu ( x ) is the i -th componentof the second fundamental form of M at the point u ( x ), then page 157 of [SY] gives(B.6) ∆ M u i = g jk A iu ( x ) ( ∂ j u, ∂ k u ) . Remark B.7.
The differential inequality in Lemma B.3 immediately implies that Propo-sition B.1 holds for harmonic functions, i.e., when | A | ≡
0, even without the small energyassumption. The general case will follow by using the small energy assumption to show thatthe perturbation terms are negligible.We will need a simple ODE comparison lemma:
Lemma B.8.
Suppose that f is a non-negative C function on [ − ℓ, ℓ ] ⊂ R satisfying(B.9) f ′′ ≥ f − a , for some constant a >
0. If max [ − ℓ,ℓ ] f ≥ a , then(B.10) Z ℓ − ℓ f ≥ √ a sinh( ℓ/ √ . Proof.
Fix some x ∈ [ − ℓ, ℓ ] where f achieves its maximum on [ − ℓ, ℓ ]. Since the lemma isinvariant under reflection x → − x , we can assume that x ≥
0. If x is an interior point,then f ′ ( x ) = 0; otherwise, if x = ℓ , then f ′ ( x ) ≥
0. In either case, we get f ′ ( x ) ≥ f ( x ) ≥ a , (B.9) gives f ′′ ( x ) ≥ a > f ′ is strictly increasing at x .We claim that f ′ ( x ) > x in ( x , ℓ ]. If not, then there would be a first point y > x with f ′ ( y ) = 0. It follows that f ′ ≥ x , y ] so that f ≥ f ( x ) ≥ a on [ x , y ]and, thus, that f ′′ ≥ a > x , y ], contradicting that f ′ ( y ) ≤ f ′ ( x ).By the claim, f is monotone increasing on [ x , ℓ ] so that (B.9) gives(B.11) f ′′ ≥ f on [ x , ℓ ] . By a standard Riccati comparison argument using f ′ ( x ) ≥ A. t ∈ [0 , ℓ − x ](B.12) f ( x + t ) ≥ f ( x ) cosh( t/ √ ≥ a cosh( t/ √ . Finally, integrating (B.12) on [0 , ℓ ] gives (B.10). (cid:3)
Proof. (of Proposition B.1.) Since we will choose ℓ ≥ ǫ < ǫ SU , the small-energyinterior estimates for harmonic maps (see lemma 3 . C − ℓ, ℓ |∇ u | ≤ C SU Z C − ℓ, ℓ |∇ u | ≤ C SU ǫ . Set f ( t ) = R t | u θ | . It follows from Lemma B.3 that(B.14) f ′′ ( t ) ≥ f ( t ) − M | A | C SU ǫ Z t ( | u θ | + | u t | ) ≥ f ( t ) − C ǫ Z t ( | u t | − | u θ | ) , where C = 2 C SU sup M | A | and we have assumed that C ǫ ≤ / R t ( | u t | − | u θ | ) is constant in t . To see this, differentiate to get(B.15) 12 ∂ t Z t ( | u t | − | u θ | ) = Z t ( h u t , u tt i − h u θ , u tθ i ) = Z t h u t , ( u tt + u θθ ) i = 0 , IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 23 where the second equality used integration by parts in θ and the last equality used that u tt + u θθ = ∆ u is normal to M while u t is tangent. Bound this constant by(B.16) Z t ( | u t | − | u θ | ) = 14 ℓ Z C − ℓ, ℓ ( | u t | − | u θ | ) ≤ ℓ Z C − ℓ, ℓ |∇ u | . By (B.14) and (B.16), Lemma B.8 with a = C ǫ ℓ R C − ℓ, ℓ |∇ u | implies that either(B.17) max [ − ℓ,ℓ ] f < C ǫ ℓ Z C − ℓ, ℓ |∇ u | , or(B.18) Z C − ℓ, ℓ | u θ | = Z ℓ − ℓ f ( t ) dt ≥ √ C ǫ sinh( ℓ/ √ ℓ Z C − ℓ, ℓ |∇ u | . The second possibility cannot occur as long as ℓ is sufficiently large so that we have(B.19) 2 √ C ǫ sinh( ℓ/ √ ℓ > . Using the upper bound (B.17) for f on [ − ℓ, ℓ ] to bound the integral of f gives(B.20) Z C − ℓ,ℓ | u θ | ≤ ℓ max [ − ℓ,ℓ ] f < C ǫ Z C − ℓ, ℓ |∇ u | . The proposition follows by choosing ǫ > C ǫ < min { / , δ } and then choosing ℓ so that (B.19) holds. (cid:3) B.2.
Weak compactness of almost harmonic maps.
We will need a compactness the-orem for a sequence of maps u j in W , ( S , M ) which have uniformly bounded energy andare locally well-approximated by harmonic maps. Before stating this precisely, it is useful torecall the situation for harmonic maps. Suppose therefore that u j : S → M is a sequenceof harmonic maps with E( u j ) ≤ E for some fixed E . After passing to a subsequence, wecan assume that the measures |∇ u j | dx converge and there is a finite set S of points wherethe energy concentrates so that:If x ∈ S , then inf r> (cid:20) lim j →∞ Z B r ( x ) |∇ u j | (cid:21) ≥ ǫ SU . (B.21) If x / ∈ S , then inf r> (cid:20) lim j →∞ Z B r ( x ) |∇ u j | (cid:21) < ǫ SU . (B.22)The constant ǫ SU > C ,α estimates onthe u j ’s in some neighborhood of x . Hence, Arzela-Ascoli and a diagonal argument give afurther subsequence of the u j ’s C -converging to a harmonic map on every compact subsetof S \ S . We will need a more general version of this, where u j : S → M is a sequence of W , maps with E( u j ) ≤ E that are ǫ -almost harmonic in the following sense: In fact, something much stronger is true: The complex-valued function φ ( t, θ ) = ( | u t | − | u θ | ) − i h u t , u θ i is holomorphic on the cylinder (see page 6 of [SY]). This is usually called the Hopf differential. ( B ) If B ⊂ S is any ball with R B |∇ u j | < ǫ , then there is an energy minimizing map v : B → M with the same boundary values as u j on ∂ B with Z B (cid:12)(cid:12) ∇ u j − ∇ v (cid:12)(cid:12) ≤ /j . Lemma B.23.
Let ǫ > ǫ SU . If u j : S → M is a sequence of W , maps sat-isfying ( B ) and with E( u j ) ≤ E , then there exists a finite collection of points { x , . . . , x k } ,a subsequence still denoted by u j , and a harmonic map u : S → M so that u j → u weaklyin W , and if K ⊂ S \ { x , . . . , x k } is compact, then u j → u in W , ( K ). Furthermore,the measures |∇ u j | dx converge to a measure ν with ǫ ≤ ν ( x i ) and ν ( S ) ≤ E . Proof.
After passing to a subsequence, we can assume that: • The u j ’s converge weakly in W , to a W , map u : S → M . • The measures |∇ u j | dx converge to a limiting measure ν with ν ( S ) ≤ E .It follows that there are at most E /ǫ points x , . . . , x k with lim r → ν ( B r ( x j )) ≥ ǫ . We will show next that away from the x i ’s the convergence is strong in W , and u isharmonic. To see this, consider a point x / ∈ { x , . . . , x k } . By definition, there exist r x > J x so that R B rx ( x ) |∇ u j | < ǫ for j ≥ J x . In particular, ( B ) applies so we get energyminimizing maps v jx : B r x ( x ) → M that agree with u j on ∂ B r x ( x ) and satisfy(B.24) Z B rx ( x ) (cid:12)(cid:12) ∇ v jx − ∇ u j (cid:12)(cid:12) ≤ /j . (Here B r x ( x ) is the ball in S centered at x so that the stereographic projection Π x whichtakes x to 0 ∈ R takes B r x ( x ) and B r x ( x ) to balls centered at 0 whose radii differ bya factor of 8.) Since E( v jx ) ≤ ǫ ≤ ǫ SU , it follows from lemma 3 . v jx ’s converges strongly in W , ( B r x ( x )) to a harmonic map v x : B r x ( x ) → M . Combining with the triangle inequality and (B.24), we get(B.25) Z B rx ( x ) (cid:12)(cid:12) ∇ u j − ∇ v x (cid:12)(cid:12) ≤ Z B rx ( x ) (cid:12)(cid:12) ∇ u j − ∇ v jx (cid:12)(cid:12) + 2 Z B rx ( x ) (cid:12)(cid:12) ∇ v jx − ∇ v x (cid:12)(cid:12) → . Similarly, this convergence, the triangle inequality, (B.24), and the Dirichlet Poincar´e in-equality (theorem 3 on page 265 of [E]; this applies since v jx equals u j on ∂ B r x ( x )) give(B.26) Z B rx ( x ) (cid:12)(cid:12) u j − v x (cid:12)(cid:12) ≤ Z B rx ( x ) (cid:12)(cid:12) u j − v jx (cid:12)(cid:12) + 2 Z B rx ( x ) (cid:12)(cid:12) v jx − v x (cid:12)(cid:12) → . Combining (B.25) and (B.26), we see that the u j ’s converge to v x strongly in W , ( B r x ( x )).In particular, u (cid:12)(cid:12) B rx ( x ) = v x . We conclude that u is harmonic on S \ { x , . . . , x k } . Further-more, since any compact K ⊂ S \ { x , . . . , x k } can be covered by a finite number of suchninth-balls, we get that u j → u strongly in W , ( K ).Finally, since u has finite energy, it must have removable singularities at each of the x i ’sand, hence, u extends to a harmonic map on all of S (see theorem 3 . (cid:3) IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 25
B.3.
Almost harmonic maps on cylinders.
The main result of this subsection, Proposi-tion B.29 below, extends Proposition B.1 from harmonic maps to “almost harmonic” maps.Here “almost harmonic” is made precise in Definition B.27 below and roughly means thatharmonic replacement on certain balls does not reduce the energy by much.
Definition B.27.
Given ν > C r ,r , we will say that a W , ( C r ,r , M ) map u is ν -almost harmonic if for any finite collection of disjoint closed balls B in the conformallyequivalent annulus B e r \ B e r ⊂ R there is an energy minimizing map v : ∪ B B → M thatequals u on ∪ B ∂B and satisfies(B.28) Z ∪ B B |∇ u − ∇ v | ≤ ν Z C r ,r |∇ u | . We have used a slight abuse of notation, since our sets will always be thought of as beingsubsets of the cylinder; i.e., we identify Euclidean balls in the annulus with their image underthe conformal map to the cylinder.In this subsection and the two that follow it, given δ >
0, the constants ℓ ≥ ǫ > M and δ . Proposition B.29.
Given δ >
0, there exists ν > δ and M ) so that if m is a positive integer and u is ν -almost harmonic from C − ( m +3) ℓ, ℓ to M with E( u ) ≤ ǫ , then(B.30) Z C − mℓ, | u θ | ≤ δ Z C − ( m +3) ℓ, ℓ |∇ u | . We will prove Proposition B.29 by using a compactness argument to reduce it to the caseof harmonic maps and then appeal to Proposition B.1. A key difficulty is that there is noupper bound on the length of the cylinder in Proposition B.29 (i.e., no upper bound on m ), so we cannot directly apply the compactness argument. This will be taken care of bydividing the cylinder into subcylinders of a fixed size and then using a covering argument.B.4. The compactness argument.
The next lemma extends Proposition B.1 from har-monic maps on C − ℓ, ℓ to almost harmonic maps. The main difference from Proposition B.29is that the cylinder is of a fixed size in Lemma B.31. Lemma B.31.
Given δ >
0, there exists µ > δ and M ) so that if u is a µ -almost harmonic map from C − ℓ, ℓ to M with E( u ) ≤ ǫ , then(B.32) Z C − ℓ,ℓ | u θ | ≤ δ Z C − ℓ, ℓ |∇ u | . Proof.
We will argue by contradiction, so suppose that there exists a sequence u j of 1 /j -almost harmonic maps from C − ℓ, ℓ to M with E( u j ) ≤ ǫ and(B.33) Z C − ℓ,ℓ | u jθ | > δ Z C − ℓ, ℓ |∇ u j | . We will show that a subsequence of the u j ’s converges to a non-constant harmonic map thatcontradicts Proposition B.1. We will consider two separate cases, depending on whether ornot E( u j ) goes to 0. Suppose first that lim sup j →∞ E( u j ) >
0. The upper bound on the energy combined withbeing 1 /j -almost harmonic (and the compactness of M ) allows us to argue as in Lemma B.23to get a subsequence that converges in W , on compact subsets of C − ℓ, ℓ to a non-constantharmonic map ˜ u : C − ℓ, ℓ → M . Furthermore, using the W , convergence on C − ℓ,ℓ togetherwith the lower semi-continuity of energy, (B.33) implies that R C − ℓ,ℓ | ˜ u θ | ≥ δ R C − ℓ, ℓ |∇ ˜ u | .This contradicts Proposition B.1.Suppose now that E( u j ) →
0. Replacing u j by v j = ( u j − u j (0)) / (E( u j )) / gives a sequenceof maps to M j = ( M − u j (0)) / E( u j )) / with E( v j ) = 1 and, by (B.33), R C − ℓ,ℓ | v jθ | > δ >
0. Furthermore, the v j ’s are also 1 /j -almost harmonic (this property is invariant underdilation), so we can still argue as in Lemma B.23 to get a subsequence that converges in W , on compact subsets of C − ℓ, ℓ to a harmonic map v : S → R N (we are using herethat a subsequence of the M j ’s converges to an affine space). As before, (B.33) implies that R C − ℓ,ℓ | v θ | ≥ δ R C − ℓ, ℓ |∇ v | . This time our normalization gives R C − ℓ,ℓ | v θ | ≥ δ so that v contradicts Proposition B.1 (see Remark B.7), completing the proof. (cid:3) B.5.
The proof of Proposition B.29.
Proof. (of Proposition B.29). For each integer j = 0 , . . . , m , let C ( j ) = C − ( j +3) ℓ, (3 − j ) ℓ andlet µ > j -th cylinder C ( j ) is good if therestriction of u to C ( j ) is µ -almost harmonic; otherwise, we will say that C ( j ) is bad .On each good C ( j ), we apply Lemma B.31 to get(B.34) Z C − ( j +1) ℓ, (1 − j ) ℓ | u θ | ≤ δ Z C ( j ) |∇ u | , so that summing this over the good j ’s gives(B.35) X j good Z C − ( j +1) ℓ, (1 − j ) ℓ | u θ | ≤ δ X j good Z C ( j ) |∇ u | ≤ δ Z C − ( m +3) ℓ, ℓ |∇ u | , where the last inequality used that each C i,i +1 is contained in at most 6 of the C ( j )’s.We will complete the proof by showing that the total energy (not just the θ -energy) onthe bad C ( j )’s is small. By definition, for each bad C ( j ), we can choose a finite collection ofdisjoint closed balls B j in C ( j ) so that if v : B j → M is any energy-minimizing map thatequals u on ∂ B j , then(B.36) Z B j |∇ u − ∇ v | ≥ a j > µ Z C ( j ) |∇ u | . Since the interior of each C ( j ) intersects only the C ( k )’s with 0 < | j − k | ≤
5, we can dividethe bad C ( j )’s into ten subcollections so that the interiors of the C ( j )’s in each subcollectionare pair-wise disjoint. In particular, one of these disjoint subcollections, call it Γ, satisfies(B.37) X j ∈ Γ a j ≥ X j bad a j ≥ X j bad µ Z C ( j ) |∇ u | , where the last inequality used (B.36). IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 27
However, since ∪ j ∈ Γ B j is itself a finite collection of disjoint closed balls in the entirecylinder C − ( m +3) ℓ, ℓ and u is ν -almost harmonic on C − ( m +3) ℓ, ℓ , we get that(B.38) µ X j bad Z C ( j ) |∇ u | ≤ ν Z C − ( m +3) ℓ, ℓ |∇ u | . To get the proposition, combine (B.35) with (B.38) to get(B.39) Z C − mℓ, | u θ | ≤ (cid:18) δ + 10 νµ (cid:19) Z C − ( m +3) ℓ, ℓ |∇ u | . Finally, choosing ν sufficiently small completes the proof. (cid:3) B.6.
Bubble compactness.
We will now prove Proposition 2.2 using a variation of therenormalization procedure developed in [PaW] for pseudo-holomorphic curves and later usedin [Pa] for harmonic maps. A key point in the proof will be that the uniform energy bound,(A), and (B) are all dilation invariant, so they apply also to the compositions of the u j ’swith any sequence of conformal dilations of S . Proof. (of Proposition 2.2). We will use the energy bound and (B) to show that a subsequenceof the u j ’s converges in the sense of (B1), (B2), and (B3) of Definition A.2 to a collectionof harmonic maps. We will then come back and use (A) and (B) to show that the energyequality (B4) also holds. Hence, the subsequence bubble converges and, thus by PropositionA.3, also varifold converges.Set δ = 1 /
21 and let ℓ ≥ ǫ > ǫ = min { ǫ / , ǫ } .Step 1: Initial compactness. Lemma B.23 gives a finite collection of singular points S ⊂ S , a harmonic map v : S → M , and a subsequence (still denoted u j ) that convergesto v weakly in W , ( S ) and strongly in W , ( K ) for any compact subset K ⊂ S \ S .Furthermore, the measures |∇ u j | dx converge to a measure ν with ν ( S ) ≤ E and eachsingular point in x ∈ S has ν ( x ) ≥ ǫ .Step 2: Renormalizing at a singular point. Suppose that x ∈ S is a singular point fromthe first step. Fix a radius ρ > x is the only singular point in B ρ ( x ) and R B ρ ( x ) |∇ v | < ǫ /
3. For each j , let r j > y ∈ B ρ − rj ( x ) Z B ρ ( x ) \ B rj ( y ) |∇ u j | = ǫ , and choose a ball B r j ( y j ) ⊂ B ρ ( x ) with R B ρ ( x ) \ B rj ( y j ) |∇ u j | = ǫ . Since the u j ’s convergeto v on compact subsets of B ρ ( x ) \ { x } , we get that y j → x and r j →
0. For each j , letΨ j : R → R be the “dilation” that takes B r j ( y j ) to the unit ball B (0) ⊂ R . By dilationinvariance, the dilated maps ˜ u j = u j ◦ Ψ − j still satisfy (B) and have the same energy. Hence,Lemma B.23 gives a subsequence (still denoted by ˜ u j ), a finite singular set S , and a harmonicmap v so that the ˜ u j ◦ Π’s converge to v weakly in W , ( S ) and strongly in W , ( K ) forany compact subset K ⊂ S \ S . Moreover, the measures |∇ ˜ u j ◦ Π | dx ’s converge to ameasure ν .The choice of the balls B r j ( y j ) guarantees that ν ( S \ { p + } ) ≤ ν ( x ) and ν ( S − ) ≤ ν ( x ) − ǫ . (Recall that stereographic projection Π takes the open southern hemisphere S − to the open unit ball in R .) The key point for iterating this is the following claim: ( ⋆ ) The maximal energy concentration at any y ∈ S \ { p + } is at most ν ( x ) − ǫ / ǫ > ǫ , the only one way that ( ⋆ ) could possibly fail is if v is constant, S is exactly twopoints p + and y , and at most ǫ / ν ( x ) escapes at p + . However, this would imply thatall but at most 2 ǫ / R B ρ ( x ) |∇ u j | is in B t j ( y j ) with t j r j → r j .Step 3: Repeating this. We repeat this blowing up construction at the remaining singularpoints in S , as well as each of the singular points S in the southern hemisphere, etc., toget new limiting harmonic maps and new singular points to blow up at. It follows from ( ⋆ )that this must terminate after at most 3 E /ǫ steps.Step 4: The necks. We have shown that the u j ’s converge to a collection of harmonic mapsin the sense of (B1), (B2), and (B3). It remains to show (B4), i.e., that the v k ’s accountedfor all of the energy in the sequence u j and no energy was lost in the limit.To understand how energy could be lost, it is useful to re-examine what happens to theenergy during the blow up process. At each stage in the blow up process, energy is “takenfrom” a singular point x and then goes to one of two places: • It can show up in the new limiting harmonic map of to a singular point in S \ { p + } . • It can disappear at the north pole p + (i.e., ν ( S \ { p + } ) < ν ( x )).In the first case, the energy is accounted for in the limit or survives to a later stage. However,in the second case, the energy is lost for good, so this is what we must rule out.We will argue by contradiction, so suppose that ν ( S \ { p + } ) < ν ( x ) − ˆ δ for some ˆ δ > δ ≤ ǫ .) Using the notation in Step 1, suppose therefore that A j = B s j ( y j ) \ B t j ( y j ) are annuli with:(B.41) s j → , t j r j → ∞ , and Z A j |∇ u j | ≥ ˆ δ > . There is obviously quite a bit of freedom in choosing s j and t j . In particular, we can choosea sequence λ j → ∞ so that the annuli ˜ A j = B ρ/ ( y j ) \ B t j /λ j ( y j ) also satisfies this, i.e., λ j s j → t j / ( λ j r j ) → ∞ . It follows from (B.41) and the definition of the r j ’s that R ˜ A j |∇ u j | ≤ ǫ ≤ ǫ . However, combining this with Proposition B.29 (with δ = 1 /
21) showsthat the area must be strictly less than the energy for j large, contradicting (A), and thuscompleting the proof. (cid:3) Appendix C. The proof of Theorem 3.1
C.1.
An application of the Wente lemma.
The proof of Theorem 3.1 will use the fol-lowing L estimate for h ζ where ζ is a L ( B ) holomorphic function and h is a W , functionvanishing on ∂B . Proposition C.1. If ζ is a holomorphic function on B ⊂ R and h ∈ W , ( B ), then(C.2) Z B h | ζ | ≤ (cid:18)Z B |∇ h | (cid:19) (cid:18)Z B | ζ | (cid:19) . The estimate (C.2) does not follow from the Sobolev embedding theorem as the productof functions in L and W , is in L p for p <
2, but not necessarily for p = 2. To get aroundthis, we will use the following lemma of H. Wente (see [W]; cf. theorem 3 . . IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 29
Lemma C.3. If B ⊂ R and u, v ∈ W , ( B ), then there exists φ ∈ C ∩ W , ( B ) with∆ φ = h ( ∂ x u, ∂ x u ) , ( − ∂ x v, ∂ x v ) i so that(C.4) || φ || C + ||∇ φ || L ≤ ||∇ u || L ||∇ v || L . Proof. (of Proposition C.1.) Let f and g be the real and imaginary parts, respectively, ofthe holomorphic function ζ , so that the Cauchy-Riemann equations give(C.5) ∂ x f = ∂ x g and ∂ x f = − ∂ x g . Since B is simply connected, (C.5) gives functions u and v on B with ∇ u = ( g, f ) and ∇ v = ( f, − g ). We have(C.6) |∇ u | = |∇ v | = h ( ∂ x u, ∂ x u ) , ( − ∂ x v, ∂ x v ) i = | ζ | . Therefore, Lemma C.3 gives φ with ∆ φ = | ζ | , φ | ∂B = 0, and(C.7) || φ || C + ||∇ φ || L ≤ Z | ζ | . Applying Stokes’ theorem to div( h ∇ φ ) and using Cauchy-Schwarz gives(C.8) Z h | ζ | = Z h ∆ φ ≤ Z |∇ h | |∇ φ | ≤ ||∇ h || L (cid:18)Z h |∇ φ | (cid:19) / . Applying Stokes’ theorem to div( h φ ∇ φ ), noting that ∆ φ ≥
0, and using (C.8) gives(C.9) Z h |∇ φ | ≤ Z | φ | (cid:0) h ∆ φ + |∇ h | |∇ φ | (cid:1) ≤ || φ || C ||∇ h || L (cid:18)Z h |∇ φ | (cid:19) / , so that (cid:0)R h |∇ φ | (cid:1) / ≤ ||∇ h || L || φ || C . Finally, substituting this bound back into (C.8)and using (C.7) to bound || φ || C gives the proposition. (cid:3) C.2.
An application to harmonic maps.Proposition C.10.
Suppose that M ⊂ R N is a smooth closed isometrically embeddedmanifold. There exists a constant ǫ > M ) so that if v : B → M is a W , weakly harmonic map with energy at most ǫ , then v is a smooth harmonic map. Inaddition, for any h ∈ W , ( B ), we have(C.11) Z B | h | |∇ v | ≤ C (cid:18)Z B |∇ h | (cid:19) (cid:18)Z B |∇ v | (cid:19) . Proof.
The first claim follows immediately from F. H´elein’s 1991 regularity theorem forweakly harmonic maps from surfaces; see [He2] or theorem 4 . . . . with Proposition C.1. Following [He1], we can assume that the pull-back v ∗ ( T M )of the tangent bundle of M has orthonormal frames on B and, moreover, that there is afinite energy harmonic section e , . . . , e n of the bundle of orthonormal frames for v ∗ ( T M ) Alternatively, one could use the recent results of T. Rivi`ere, [Ri]. (the frame e , . . . , e n is usually called a Coulomb gauge ). Set α j = h ∂ x v, e j i − i h ∂ x v, e j i for j = 1 , . . . , n . Since e , . . . , e n is an orthonormal frame for v ∗ ( T M ), we have(C.12) |∇ v | = n X j =1 | α j | . On pages 181 and 182 of [He1], H´elein uses that the frame e , . . . , e n is harmonic to constructan n × n matrix-valued function β (i.e., a map β : B → GL ( n, C )) with | β | ≤ C , | β − | ≤ C ,and with ∂ ¯ z ( β − α ) = 0 (where the constant C depends only on M and the bound for theenergy of v ; see also lemma 3 on page 461 in [Q] where this is also stated). In particular, weget an n -tuple of holomorphic functions ( ζ , . . . , ζ n ) = ζ = β − α , so that(C.13) C − | ζ | ≤ | α | = | β ζ | ≤ C | ζ | . The claim (C.11) now follows from Proposition C.1. Namely, using (C.12), the secondinequality in (C.13), and then applying Proposition C.1 to the n holomorphic functions ζ , . . . , ζ n gives(C.14) Z | h | |∇ v | ≤ C Z | h | | ζ | ≤ C Z |∇ h | Z | ζ | ≤ C Z |∇ h | Z |∇ v | , where the last inequality used the first inequality in (C.13) and (C.12). (cid:3) C.3.
The proof of Theorem 3.1.
Proof. (of Theorem 3.1.) Use Stokes’ theorem and that u and v are equal on ∂B to get(C.15) Z |∇ u | − Z |∇ v | − Z |∇ ( u − v ) | = − Z h ( u − v ) , ∆ v i ≡ Ψ . To show (3.2), it suffices to bound | Ψ | by R |∇ v − ∇ u | .The harmonic map equation (B.6) implies that ∆ v is perpendicular to M and(C.16) | ∆ v | ≤ |∇ v | sup M | A | . We will need the elementary geometric fact that there exists a constant C depending on M so that whenever x, y ∈ M , then(C.17) (cid:12)(cid:12) ( x − y ) N (cid:12)(cid:12) ≤ C | x − y | , where ( x − y ) N denotes the normal part of the vector ( x − y ) at the point x ∈ M (the samebound holds at y by symmetry). The point is that either | x − y | ≥ /C so (C.17) holdstrivially or the vector ( x − y ) is “almost tangent” to M .Using that u and v both map to M , we can apply (C.17) to get (cid:12)(cid:12) ( u − v ) N (cid:12)(cid:12) ≤ C | u − v | ,where the normal projection is at the point v ( x ) ∈ M . Putting all of this together gives(C.18) | Ψ | ≤ C Z | v − u | |∇ v | , where C depends on sup M | A | . As long as ǫ is less than ǫ , we can apply Proposition C.10with h = | u − v | to get(C.19) Z | v − u | |∇ v | ≤ C ′ (cid:18)Z |∇| u − v || (cid:19) (cid:18)Z |∇ v | (cid:19) ≤ C ′ ǫ Z |∇ u − ∇ v | . IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 31
The lemma follows by combining (C.18) and (C.19) and then taking ǫ sufficiently small. (cid:3) Combining Corollary 3.3 and the regularity theory of [Mo1], or [SU1], for energy minimiz-ing maps recovers H´elein’s theorem that weakly harmonic maps from surfaces are smooth.Note, however, that we used estimates from [He1] in the proof of Theorem 3.1.
Appendix D. The equivalence of energy and area
By (1.4), Proposition 1.5 follows once we show that W E ≤ W A . The corresponding resultfor the Plateau problem is proven by taking a minimizing sequence for area and reparametriz-ing to make these maps conformal, i.e., choosing isothermal coordinates. There are a fewtechnical difficulties in carrying this out since the pull-back metric may be degenerate and isonly in L , while the existence of isothermal coordinates requires that the induced metric bepositive and bounded; see, e.g., proposition 5 . t .D.1. Density of smooth mappings.
The next lemma observes that the regularizationusing convolution of Schoen-Uhlenbeck in the proposition in section 4 of [SU2] is continuous.
Lemma D.1.
Given γ ∈ Ω and ǫ >
0, there exists a regularization ˜ γ ∈ Ω γ so that(D.2) max t || ˜ γ ( · , t ) − γ ( · , t ) || W , ≤ ǫ , each slice ˜ γ ( · , t ) is C , and the map t → ˜ γ ( · , t ) is continuous from [0 ,
1] to C ( S , M ). Proof.
Since M is smooth, compact and embedded, there exists a δ > x inthe δ -tubular neighborhood M δ of M in R N , there is a unique closest point Π( x ) ∈ M andso the map x → Π( x ) is smooth. Π is called nearest point projection .Given y in the open ball B (0) ⊂ R , define T y : S → S by T y ( x ) = x − y | x − y | . Sinceeach T y is smooth and these maps depend smoothly on y , it follows that the map ( y, f ) → f ◦ T y is continuous from B (0) × C ∩ W , ( S , R N ) → C ∩ W , ( S , R N ) (this is clearfor f ∈ C and follows for C ∩ W , by density). Therefore, since T is the identity, given f ∈ C ∩ W , ( S , R N ) and µ >
0, there exists r > | y |≤ r || f ◦ T y − f || C ∩ W , < µ .Applying this to γ ( · , t ) for each t and using that t → γ ( · , t ) is continuous to C ∩ W , and[0 ,
1] is compact, we get ¯ r > t ∈ [0 , sup | y |≤ ¯ r || T y γ ( · , t ) − γ ( · , t ) || C ∩ W , < µ . Next fix a smooth radial mollifier φ ≥ R . For each r ∈ (0 , φ r ( x ) = r − φ ( x/r ) and set(D.4) γ r ( x, t ) = Z B r (0) φ r ( y ) γ ( T y ( x ) , t ) dy = Z B r ( x ) φ r ( x − y ) γ ( y | y | , t ) dy . We have the following standard properties of convolution with a mollifier (see, e.g., section5 . γ r ( · , t ) is smooth and for each k the map t → γ r ( · , t ) is continuous from [0 ,
1] to C k ( S , R N ). Second, || γ r ( · , t ) − γ ( · , t ) || C ≤ sup | y |≤ r || T y γ ( · , t ) − γ ( · , t ) || C , (D.5) ||∇ γ r ( · , t ) − ∇ γ ( · , t ) || L ≤ sup | y |≤ r || T y γ ( · , t ) − γ ( · , t ) || L . It follows from (D.5) and (D.3) that for r ≤ ¯ r and all t we have(D.6) || γ r ( · , t ) − γ ( · , t ) || C ∩ W , < µ . The map γ r ( · , t ) may not land in M , but it is in M δ when µ is small by (D.6). Hence, themap ˜ γ ( x, t ) = Π ◦ γ r ( x, t ) satisfies (D.2), each slice ˜ γ ( · , t ) is C , and t → ˜ γ ( · , t ) is continuousfrom [0 ,
1] to C ( S , M ). Finally, s → ˜ γ sr is an explicit homotopy connecting ˜ γ and γ . (cid:3) D.2.
Equivalence of energy and area.
We will also need the existence of isothermalcoordinates, taking special care on the dependence on the metric. Let S g denote the roundmetric on S with constant curvature one. Lemma D.7.
Given a C metric ˜ g on S , there is a unique orientation preserving C , / conformal diffeomorphism h ˜ g : S g → S g that fixes 3 given points.Moreover, if ˜ g and ˜ g are two C metrics that are both ≥ ǫ g for some ǫ >
0, then(D.8) || h ˜ g − h ˜ g || C ∩ W , ≤ C || ˜ g − ˜ g || C , where the constant C depends on ǫ and the maximum of the C norms of the ˜ g i ’s. Proof.
The Riemann mapping theorem for variable metrics (see theorem 3 . . . . h ˜ g : S g → S g .We will separately bound the C and W , norms. First, lemma 17 in [ABe] gives(D.9) || h ˜ g − h ˜ g || C ≤ C || ˜ g − ˜ g || C , where C depends on ǫ and the C norms of the metrics. Second, theorem 8 in [ABe] givesa uniform L p bound for ∇ ( h ˜ g − h ˜ g ) on any unit ball in S where p > ||∇ ( h ˜ g − h ˜ g ) || L p ( B ) ≤ C || ˜ g − ˜ g || C ( S ) , where C depends on ǫ and the C norms of the metrics. Covering S by a finite collectionof unit balls and applying H¨older’s inequality gives the desired energy bound. (cid:3) We can now prove the equivalence of the two widths.
Proof. (of Proposition 1.5). By (1.4), we have that W A ≤ W E . To prove that W E ≤ W A ,given ǫ >
0, let γ ∈ Ω β be a sweepout with max t ∈ [0 , Area ( γ ( · , t )) < W A + ǫ/
2. By LemmaD.1, there is a regularization ˜ γ ∈ Ω β so that each slice ˜ γ ( · , t ) is C , the map t → ˜ γ ( · , t ) iscontinuous from [0 ,
1] to C ( S , M ), and (also by (A.4))(D.11) max t Area (˜ γ ( · , t )) < W A + ǫ . The maps ˜ γ ( · , t ) induce a continuous one-parameter family of pull-back (possibly degener-ate) C metrics g ( t ) on S . Lemma D.7 requires that the metrics be non-degenerate, sodefine perturbed metrics ˜ g ( t ) = g ( t ) + δ g . For each t , Lemma D.7 gives C , / conformaldiffeomorphisms h t : S g → S g ( t ) that vary continuously in C ∩ W , ( S , S ). The continuity IDTH AND FINITE EXTINCTION TIME OF RICCI FLOW 33 of t → ˜ γ ( · , t ) ◦ h t as a map from [0 ,
1] to C ∩ W , ( S , M ) follows from this, the continuityof t → ˜ γ ( · , t ) in C , and the chain rule.We will now use the conformality of the map h t to control the energy of the compositionas in proposition 5 . γ ( · , t ) ◦ h t ) = E ( h t : S g → S g ( t ) ) ≤ E ( h t : S g → S g ( t ) )= Area ( S g ( t ) ) = Z S [det( g − g ( t )) + δ Tr( g − g ( t )) + δ ] / dvol g (D.12) ≤ Area ( S g ( t ) ) + 4 π [ δ + 2 δ sup t | g − g ( t ) | ] / . Choose δ > π [ δ + 2 δ sup t | g − g ( t ) | ] / < ǫ .We would be done if ˜ γ ( · , t ) ◦ h t was homotopic to ˜ γ . However, the space of orientationpreserving diffeomorphisms of S is homotopic to RP by Smale’s theorem. To get aroundthis, note that t → || ˜ γ ( · , t ) || C is continuous and zero when t = 1, thus for some τ < t ≥ τ || ˜ γ ( · , t ) || C ≤ ǫ sup t ∈ [0 , || h t || W , . Consequently, if we set ˜ h t equal to h t ≡ h ( t ) on [0 , τ ] and equal to h ( τ (1 − t ) / (1 − τ )) on[ τ, t ∈ [0 , E (˜ γ ( · , t ) ◦ ˜ h t ) ≤ W A + 2 ǫ . Moreover, themap ˜ γ ( · , t ) ◦ ˜ h t is also in Ω. Finally, replacing τ by sτ and taking s → γ ( · , t ) ◦ ˜ h t to ˜ γ ( · , t ). (cid:3) References [ABe] L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics,
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