Baixe Ricci Flow with Surgery: Extinction Time and Area Estimates e outras Provas em PDF para Física, somente na Docsity!
arXiv:math/0307245v1 [math.DG] 17 Jul 2003
Finite extinction time for the solutions to the
Ricci flow on certain three-manifolds
Grisha Perelman∗
February 1, 2008
In our previous paper we constructed complete solutions to the Ricci flow with surgery for arbitrary initial riemannian metric on a (closed, oriented) three-manifold [P,6.1], and used the behavior of such solutions to classify three- manifolds into three types [P,8.2]. In particular, the first type consisted of those manifolds, whose prime factors are diffeomorphic copies of spherical space forms and S^2 × S^1 ; they were characterized by the property that they admit metrics, that give rise to solutions to the Ricci flow with surgery, which become extinct in finite time. While this classification was sufficient to answer topological ques- tions, an analytical question of significant independent interest remained open, namely, whether the solution becomes extinct in finite time for every initial metric on a manifold of this type. In this note we prove that this is indeed the case. Our argument (in con- junction with [P,§1-5]) also gives a direct proof of the so called ”elliptization conjecture”. It turns out that it does not require any substantially new ideas: we use only a version of the least area disk argument from [H,§11] and a regu- larization of the curve shortening flow from [A-G].
1 Finite time extinction
1.1 Theorem. Let M be a closed oriented three-manifold, whose prime decom- position contains no aspherical factors. Then for any initial metric on M the solution to the Ricci flow with surgery becomes extinct in finite time. Proof for irreducible M. Let ΛM denote the space of all contractible loops in C^1 (S^1 → M ). Given a riemannian metric g on M and c ∈ ΛM, define A(c, g) to be the infimum of the areas of all lipschitz maps from D^2 to M, whose restriction to ∂D^2 = S^1 is c. For a family Γ ⊂ ΛM let A(Γ, g) be the supremum of A(c, g) over all c ∈ Γ. Finally, for a nontrivial homotopy class α ∈ π∗(ΛM, M ) let A(α, g) be the infimum of A(Γ, g) over all Γ ∈ α. Since M is not aspherical, it follows from a classical (and elementary) result of Serre that such a nontrivial homotopy class exists. ∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191023, Russia. Email: [email protected] or [email protected]
1.2 Lemma. (cf. [H,§11]) If gt^ is a smooth solution to the Ricci flow, then for any α the rate of change of the function At^ = A(α, gt) satisfies the estimate
d dt At^ ≤ − 2 π −
Rt minAt
(in the sense of the lim sup of the forward difference quotients), where Rt min denotes the minimum of the scalar curvature of the metric gt. A rigorous proof of this lemma will be given in §3, but the idea is simple and can be explained here. Let us assume that at time t the value At^ is attained by the family Γ, such that the loops c ∈ Γ where A(c, gt) is close to At^ are embedded and sufficiently smooth. For each such c consider the minimal disk Dc with boundary c and with area A(c, gt). Now let the metric evolve by the Ricci flow and let the curves c evolve by the curve shortening flow (which moves every point of the curve in the direction of its curvature vector at this point) with the same time parameter. Then the rate of change of the area of Dc can be computed as (^) ∫
Dc
(−Tr(RicT)) +
c
(−kg )
where RicT^ is the Ricci tensor of M restricted to the tangent plane of Dc, and kg is the geodesic curvature of c with respect to Dc (cf. [A-G, Lemma 3.2]). In three dimensions the first integrand equals − 12 R − (K − det II), where K is the intrinsic curvature of Dc and det II, the determinant of the second fundamental form, is nonpositive, because Dc is minimal. Thus, the rate of change of the area of Dc can be estimated from above by ∫
Dc
R − K) +
c
(−kg ) =
Dc
R) − 2 π
by the Gauss-Bonnet theorem, and the statement of the lemma follows. The problem with this argument is that if Γ contains curves, which are not immersed (for instance, a curve could pass an arc once in one direction and then make an about turn and pass the same arc in the opposite direction), then it is not clear how to define curve shortening flow so that it would be continuous both in the time parameter and in the family parameter. In §3 we’ll explain how to circumvent this difficulty, essentially by adding one dimension to the ambient manifold. This regularization of the curve shortening flow has been worked out by Altschuler and Grayson [A-G] (who were interested in approximating the singular curve shortening flow on the plane and obtained for that case more precise results than what we need). 1.3 Now consider the solution to the Ricci flow with surgery. Since M is assumed irreducible, the surgeries are topologically trivial, that is one of the components of the post-surgery manifold is diffeomorphic to the pre-surgery manifold, and all the others are spheres. Moreover, by the construction of the surgery [P,4.4], the diffeomorphism from the pre-surgery manifold to the post-surgery one can be chosen to be distance non-increasing ( more precisely, (1 + ξ)-lipschitz, where ξ > 0 can be made as small as we like). It follows that
Denote by Xt^ the tangent vector field to ct, and let St^ = gt(Xt, Xt)−^ (^12) Xt be the unit tangent vector field; then H = ∇S S (from now on we drop the superscript t except where this omission can cause confusion). We compute
d dt
g(X, X) = −2Ric(X, X) − 2 g(X, X)k^2 , (1)
which implies [H, S] = (k^2 + Ric(S, S))S (2)
Now we can compute
d dt
k^2 = (k^2 )′′^ − 2 g((∇S H)⊥, (∇S H)⊥) + 2k^4 + ..., (3)
where primes denote differentiation with respect to the arclength parameter s, and where dots stand for the terms containing the curvature tensor of g, which can be estimated in absolute value by const · (k^2 + k). Thus the curvature k satisfies d dt
k ≤ k′′^ + k^3 + const · (k + 1) (4)
Now it follows from (1) and (4) that the length L and the total curvature Θ =
kds satisfy d dt
L ≤
(const − k^2 )ds, (5)
d dt
const · (k + 1)ds (6)
In particular, both quantities can grow at most exponentially in t (they would be non-increasing in a flat manifold). 2.2 In general the curvature of ct^ may concentrate near certain points, cre- ating singularities. However, if we know that this does not happen at some time t∗, then we can estimate the curvature and higher derivatives at times shortly thereafter. More precisely, there exist constants ǫ, C 1 , C 2 , ... (which may depend on the curvatures of the ambient space and their derivatives, but are independent of ct), such that if at time t∗^ for some r > 0 the length of ct^ is at least r and the total curvature of each arc of length r does not exceed ǫ, then for every t ∈ (t∗, t∗^ + ǫr^2 ) the curvature k and higher derivatives satisfy the estimates k^2 = g(H, H) ≤ C 0 (t − t∗)−^1 , g(∇S H, ∇S H) ≤ C 1 (t − t∗)−^2 , ... This can be proved by adapting the arguments of Ecker and Huisken [E-Hu]; see also [A-G,§4]. 2.3 Now suppose that our manifold (M, gt) is a metric product ( M ,¯ g¯t) × S^1 λ, where the second factor is the circle of constant length λ; let U denote the unit tangent vector field to this factor. Then u = g(S, U ) satisfies the evolution equation d dt u = u′′^ + (k^2 + Ric(S, S))u (7)
Assume that u was strictly positive everywhere at time t 0 (in this case the curve is called a ramp). Then it will remain positive and bounded away from zero as long as the solution exists. Now combining (4) and (7) we can estimate the right hand side of the evolution equation for the ratio k u and conclude that this ratio, and hence the curvature k, stays bounded (see [A-G,§2]). It follows that ct^ is defined on the whole interval [t 0 , t 1 ]. 2.4 Assume now that we have two ramp solutions ct 1 , ct 2 , each winding once around the S^1 λ factor. Let μt^ be the infimum of the areas of the annuli with boundary ct 1 ∪ ct 2. Then
d dt μt^ ≤ (2n − 1)|Rmt|μt, (8)
where |Rmt| denotes a bound on the absolute value of sectional curvatures of gt. Indeed, the curves ct 1 and ct 2 , being ramps, are embedded and without substantial loss of generality we may assume them to be disjoint. In this case the results of Morrey [M] and Hildebrandt [Hi] yield an analytic minimal annulus A, immersed, except at most finitely many branch points, with prescribed boundary and with area μ. The rate of change of the area of A can be computed as ∫
A
(−Tr(RicT^ )) +
∂A
(−kg ) ≤
A
(−Tr(RicT^ ) + K)
A
(−Tr(RicT^ ) + RmT^ ) ≤ (2n − 1)|Rm|μ,
where the first inequality comes from the Gauss-Bonnet theorem, with possible contribution of the branch points, and the second one is due to the fact that a minimal surface has nonpositive extrinsic curvature with respect to any normal vector. 2.5 The estimate (8) implies that μt^ can grow at most exponentially; in particular, if ct 1 and ct 2 were very close at time t 0 , then they would be close for all t ∈ [t 0 , t 1 ] in the sense of minimal annulus area. In general this does not imply that the lengths of the curves are also close. However, an elementary argument shows that if ǫ > 0 is small then, given any r > 0 , one can find ¯μ, depending only on r and on upper bound for sectional curvatures of the ambient space, such that if the length of ct 1 is at least r, each arc of ct 1 with length r has total curvature at most ǫ, and μt^ ≤ μ,¯ then L(ct 2 ) ≥ (1 − 100 ǫ)L(ct 1 ).
3 Proof of lemma 1.
3.1 In this section we prove the following statement Let M be a closed three-manifold, and let (M, gt) be a smooth solution to the Ricci flow on a finite time interval [t 0 , t 1 ]. Suppose that Γ ⊂ ΛM is a compact family. Then for any ξ > 0 one can construct a continuous deformation Γt, t ∈ [t 0 , t 1 ], Γt^0 = Γ, such that for each curve c ∈ Γ either the value A(ct^1 , gt^1 ) is bounded from above by ξ plus the value at t = t 1 of the solution to the ODE
disk argument as in 1.2, and this implies the corresponding estimate for p 1 ctλ if λ ∈ Λc is small enough, whereas for the intervals of the complement of JB (c) we can use the estimate in 3.3. On the other hand, if our assumption on the lower bound for lengths does not hold, then it follows from (5) that L(ct λ^2 ) ≤ CB−^1 ≤ 12 ξ. 3.5 Now apply the previous argument to all elements of some finite μ-net ˆΓ ⊂ Γ for small μ > 0 to be determined later. We get a λ > 0 such that for each
ˆc ∈ Γ eitherˆ A(p 1 ˆct λ^1 , gt^1 ) ≤ wˆc(t 1 ) + 12 ξ or L(ˆct λ^2 ) ≤ 12 ξ. Now for any curve c ∈ Γ
pick a curve ˆc ∈ Γˆ, μ-close to c, and apply the result of 2.4. It follows that if A(p 1 ˆct λ^1 , gt^1 ) ≤ wˆc(t 1 ) + 12 ξ and μ ≤ C−^1 ξ, then A(p 1 ct λ^1 , gt^1 ) ≤ wc(t 1 ) + ξ. On the other hand, if L(ˆct λ^2 ) ≤ 12 ξ, then we can conclude that L(ct λ^1 ) ≤ ξ provided
that μ > 0 is small enough in comparison with ξ and B−^1. Indeed, if L(ct λ^1 ) > ξ, then L(ctλ) > 34 ξ for all t ∈ [t 2 , t 1 ]; on the other hand, using (5) we can find a t ∈ [t 2 , t 1 ], such that
k^2 ds ≤ CB for ctλ; hence, applying 2.5, we get L(ˆctλ) > 23 ξ
for this t, which is incompatible with L(ˆct λ^2 ) ≤ 12 ξ. The proof of the statement 3.1 is complete.
References
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