COLLAPSING IN THE L2 CURVATURE FLOW JEFFREY STREETS Abstract. We show some results on collapsing phenomena for the L2 curvature flow, which exhibit a stark difference in the behavior of the flow in the subcritical dimensions n=2,3 and 2 supercritical dimensions n≥5. First we show long time existence and convergence of the flow 1 forSO(3)-invariantinitialdataonS3,aswellasanoptimallongtimeexistenceandconvergence 0 2 statementforthree-manifoldswithinitialL2 normofcurvaturechosensmallwithrespecttothe diameterandvolume. Inthecriticaldimensionn=4weshowarelatedlow-energyconvergence n statement with an additional hypothesis. Finally we exhibit some finite time singularities in a dimension n≥5, and show examples of finite time singularities in dimension n≥ 6 which are J collapsed on thescale of curvature. 5 ] G D 1. Introduction . h Let Mn be a smooth compact manifold. Consider the functional of Riemannian metrics t a m (1.1) (g) = Rm 2dV . F | g|g g [ ZM This is a natural analogue of the Yang-Mills energy for a Riemannian metric, and studying its 1 negative gradient flow, v 6 ∂ 6 g = grad (1.2) ∂t − F 2 1 g(0) = g0, . 1 is a natural approach to understanding the structure of this functional. For convenience below, 0 we will call this the L2 flow. This is a fourth-order, degenerate parabolic equation. Papers 2 1 on this flow and closely related topics include [4], [20], [26]. Certain obstructions to the long : time existence of this flow have by now been established. For instance, curvature blowup at v i the first singular time was established in [24]. As in the case of Ricci flow, one key difficulty is X to understand the possible collapsing behavior at a singular time. Interestingly, it is a simple r a matter to show that finite time singularities of the flow in dimensions n = 2,3 must becollapsed (see Proposition 3.6 below). The contrapositive statement of Proposition 3.6 is that if one were able to show a noncollapsing result analogous to Perelman’s estimate for Ricci flow [16], one immediately concludes the long time existence of solutions to theL2 flow in dimensions n= 2,3. Note that such a statement is plausible on PDE grounds due to the “supercriticality” of the functional in those dimensions, though perhaps counterintuitive due to the highly singular F nature of Ricci flow on three-manifolds. Given the discussion above, let us remark on two cases where the noncollapsing issue for solutions to the L2 flow is well understood. The first is the case of Riemann surfaces. A compactness/concentration criterion originally due to Chen [8] states roughly that sequences of conformal metrics on Riemann surfaces either converge or experience concentration volume and L1 concentration of curvature at a point. By fixing a special gauge to reduce the L2 flow Date: January 6, 2012. The authorwas partly supported by a grant from the National Science Foundation. 1 2 JEFFREYSTREETS to a conformal flow, and by exploiting some energy estimates, we were able to rule out the concentration behavior at finite time to establish long time existence of the L2 flow on compact Riemann surfaces ([23] Theorem 1). In higher dimensions the situation is more difficult since the flow cannot be reduced to a conformal flow, and no convenient compactness criteria are available to deal with the collapsing issue. One situation where this difficulty was overcome is related to a certain conformal sphere theorem in four dimensions. In [5], the authors show that a compact Riemannian four- manifoldwithpositiveYamabeconstant andsufficiently smallL2 normofWeyl curvaturetensor is smoothly deformable to a spherical space form. In particular, the result yields a classification of the possible diffeotypes satisfying the hypotheses. Moreover, the pinching condition on the Weyl tensor is sharp. Recently, we showed a weaker version of this theorem using the L2 flow. Specifically, we showed that given a compact Riemannian four-manifold with positive Yamabe constant and sufficiently small L2 norm of the traceless curvature tensor, the solution to the L2 flow exists for all time and converges to a spherical space form ([22] Theorem 1). Later this flow result was improved in [4], using a similar method, to yield an explicit value for the required pinching, though this value is still suboptimal. A critical feature of the flow proofs mentioned in the above paragraph is that the hypotheses ofsmallL2 normofWeyltensorandpositiveYamabeconstanttogether implyaboundontheL2 Sobolev constant. This is a beautiful argument which combines the Gauss-Bonnet theorem and the solution to the Yamabe problem, and is due independently to Gursky [10] and Tian. In the method of [22] and [4] the Sobolev constant bound is used to produce nonflat blowup limits of finite time singularities, and these limits are automatically critical metrics for the corresponding functionals. Then, via a Liouville’s Theorem argument one shows that noncompact critical metrics satisfying certain Sobolev constant and L2 curvature estimates are automatically flat, implying the singularity could not have occurred, thus yielding the long time existence of the flow. The argument of Gursky-Tian perfectly resolves the crucial Sobolev constant issue in the setting of positive Yamabe constant and pinched curvature. Outside of this regime however this remains a difficult problem. The purpose of this paper is to further flesh out and determine the nature of collapse and singularity formation in the L2 flow, and moreover to highlight the crucial role played by the dimensionofthemanifoldinunderstandingthisbehavior. Ourfirstresultisalong-timeexistence and convergence result for certain warped product 3-manifolds. Theorem 1.1. Let (M3,g) satisfy one of the following conditions: (1) M = S3 and g is an SO(3)-invariant metric. ∼ (2) M = Σ S1, where Σ is a compact Riemann surface, χ(Σ) = 0, and g = ds2 +ψ2g , ∼ Σ where g×is a constant curvature metric on Σ, and ψ C (S6 1,R ). Σ ∞ >0 ∈ The solution to the volume-normalized L2 flow with initial condition g exists on [0, ), and in ∞ the case of SO(3)-invariant metrics on S3 converges to a critical metric. Remark 1.2. The long time existence of SO(3)-invariant metrics on S3 is surprising when taken in contrast to the behavior of Ricci flow solutions in this context. Indeed, as shown in [13], neckpinches occur for metrics with such symmetry. Moreover, as established in the work of Perelman [16], [17], neckpinches (degenerate and nondegenerate) are the only finite time local singularities for Ricci flow on three-manifolds. From a PDE perspective though, this long time existence is not so surprising, since the functional is “supercritical” for n 3, and hence one F ≤ expects very good existence properties in these dimensions. Indeed, from this perspective it is not so unreasonable to think that solutions to the L2 flow on three-manifolds always exist for all time (see Conjecture 9.5). The fact that Theorem 1.1 applies to metrics with arbitrary initial energy is encouraging in this respect. COLLAPSING IN THE L2 CURVATURE FLOW 3 Remark 1.3. The proof of the long time existence statements of Theorem 1.1 will hold for the gradient flows of more general functionals. In particular, for any quadratic Riemannian functional satisfying as δ Rm 2, the result of Theorem 1.1 should hold. F ≥ M | | The next main result is a low Renergy convergence statement on three-manifolds. Theorem 1.4. Given V > 0,D > 0, there exists ǫ > 0 so that if (M3,g) is a compact Rie- mannian manifold satisfying Vol(g) V, ≥ (1.3) diam(g) D, ≤ (g) ǫ, F ≤ the solution to the volume-normalized L2 flow exists for all time and converges to a flat metric. Remark 1.5. Neither the hypothesis of a lower bound on volume nor the hypothesis of the initial bound on diameter can be removed while keeping convergence to a flat metric, making the statement optimal. In particular, on S2 S1, the metric Ag 1 g has unit volume × S2 ⊕ A2 S1 and diam = O(A12) and (g) = O(A−2) for A large. Since the universal cover of the manifold is S2 R, it cannot admFit a flat metric. Indeed, a direct calculation shows that the solution × to the L2 flow with this initial condition satisfies A as . Likewise, the metrics → ∞ → ∞ g = g ǫ2g have bounded diameter and (g )= O(ǫ), Vol(g ) = O(ǫ). ǫ S2 S1 ǫ ǫ ⊕ F The key estimate intervening in Theorem 1.4 controls the growth of the first Dirichlet eigen- valueinthepresenceofaSobolevconstantboundandsmallglobalenergy. Thisestimate bearsa structural similarity to Perelman’s κ-noncollapsing estimate for Ricci flow. Very roughly speak- ing, Perelman observes that if a solution to the Ricci flow becomes sufficiently collapsed, there are test functions forcing his quantity µ to approach . Since µ is monotonically increasing −∞ along the flow, one thereby derives a contradiction. Our approach is similar in that we control the Dirichlet energy (g,φ) of a test function φ along the flow. If at a certain time T the first E Dirichlet eigenvalue is very small, one has a test function φ for which yields a very small, E positive value. Taking a cue from Perelman’s conjugate heat equation, we push this function back to the initial metric by means of the bi-Laplacian heat flow of the time varying metric and derive a test function φ such that (g ,φ) is again a very small, positive value. Considering the 0 E initial first Dirichlet eigenvalue as known, we thus derive an estimate for how fast it can decay alongtheflow. Weemphasizethattheproofsreally areonlysimilar intheirgeneraloutline. The energy is not monotonicalong solutions to theL2 flow, andthemain difficulty is incontrolling E along the flow. Moreover, we emphasize that we do not show a κ-noncollapsing estimate for E solutions to the L2 flow akin to Perelman’s estimate for Ricci flow. As already mentioned, such an estimate immediately implies long time existence of the flow on surfaces and three-manifolds (see Proposition 3.6). By adding an extra hypothesis, we obtain a low energy convergence statement on four- manifolds as well. Theorem 1.6. Given constants A,B > 0 there exists ǫ(A,B) >0 so that if (M4,g) is a compact Riemannian manifold with unit volume satisfying grad A • || F||L2 ≤ C (g) B S • ≤ Rm 2 ǫ • M | | ≤ then the solution to the L2 flow with initial condition g exists for all time and converges to a R flat metric. 4 JEFFREYSTREETS Remark 1.7. ItfollowsfromTheorem1.6thatthedifficultyinprovingaversionofTheorem1.4 forfour-manifoldslies entirely inunderstandingtheshort-timebehavioroftheSobolevconstant. In particular, if one could obtain a doubling-time estimate for the Sobolev constant along the L2 flow, then by exploiting the gradient property, we conclude the existence of a metric at some time in that time interval with a uniform bound on the Sobolev constant and the L2 norm of grad . The analogue of Theorem 1.4 for four-manifolds would then follow, i.e. given energy F sufficiently small with respect to the Sobolev constant, the solution to the L2 flow with this initial condition would exist for all time and converge to a flat metric. Moving tohigherdimensions,weshowin 8thatin alldimensionsn 5, theL2 flow exhibits § ≥ finite time singularities, and in dimensions n 6, the L2 flow exhibits finite time singularities ≥ which do not satisfy an injectivity radius estimate on the scale of curvature. Thus we see quite strikingly the role the dimension plays in understanding the behavior of the L2 flow, and indeed it seems wise to restrict attention entirely to dimensions n 4. ≤ Here is an outline of the rest of the paper. In 2 we give some background on Sobolev and § isoperimetric constants, and in 3 we give background results on the L2 flow. Section 4 has § the proof of Theorem 1.1. In 5 we derive an estimate for the growth of the first Dirichlet § eigenvalue along solutions to the L2 flow, and we use this in 6 to prove Theorem 1.4. Section 7 § contains the proof of Theorem 1.6. In 8 we address the behavior of solutions to the L2 flow in § dimensionsn 5. We end in 9with aconjectural discussionof theoptimal long timeexistence ≥ § results for the L2 flow and their possible applications. Acknowledgments: The author would like to thank Aaron Naber, Peter Petersen, and Laurent Saloff-Coste for several helpful discussions. 2. Isoperimetric and Sobolev Constants In this section we recall some definitions and theorems related to isoperimetric and Sobolev constants on compact Riemannian manifolds. Definition 2.1. Let (Mn,g) be a compact Riemannian manifold, and let Ω denote a proper open subset of M The isoperimetric constant is Area(∂Ω) C (M,g) := inf . I n−1 Ω min Vol(Ω),Vol(M Ω) n { \ } Definition 2.2. Let (Mn,g) be a compact Riemannian manifold. The L1 Sobolev constant of M is the infimum of all C such that for any f C1(M), ∈ n−1 inf f α n−n1 dV n C f 2dV. α∈R(cid:18)ZM | − | (cid:19) ≤ ZM |∇ | Definition 2.3. Let (Mn,g) be a compact Riemannian manifold, n 3. The L2 Sobolev ≥ constant, denoted C (g), is the infimum of all C such that for any f C1(M), S ∈ n−2 fn2−n2dV n C f 2dV +V−n2 f2dV . ≤ |∇ | (cid:18)ZM (cid:19) (cid:18)ZM ZM (cid:19) Remark 2.4. The L1 Sobolev inequality is equivalent to the isoperimetric constant by [3] (see also [6]). Furthermore, by ([15] Lemma 2) an upper bound on the L1 Sobolev constant implies an upper bound for the L2 Sobolev constant. Theorem 2.5. Let (Mn,g) be a complete Riemannian manifold, n 3. Given q 2, there q ≥ ≥ exists C(q) such that for all u H (M), ∈ 1 ||u||Lp ≤ CCSα||u||1L−mα(||∇u||Lq +||u||Lq)α, COLLAPSING IN THE L2 CURVATURE FLOW 5 where 2 m p, ≤ ≤ 1 1 α = m − p , 1 1 + 1 m − q n and if q < n, then p nq and C = C(n,q), ≤ n q − if q = n, then p < and C = C(m,p),  ∞  if q > n, then p and C = C(n,m,q). ≤ ∞ Proof. See [14] or [4] fora more recent exposition. (cid:3)  Lemma 2.6. Given A> 0 there exists D = D(A) > 0 so that if (Mn,g) is a compact Riemann- ian manifold with Vol(g) = 1 and C (g) < A, then diam(g) < D. S Proof. It follows from the argument of ([12] Lemma 3.2) that for (M4,g) satisfying the hypotheses, for any x M one has ∈ C (2.1) Vol(B (1)) x ≥ An for a universal constant C. The result follows from a standard packing argument. (cid:3) Remark2.7. Amuchmorerobustargumentforthislemma,showingtherelationshipofSobolev inequalities and diameter, appears in [1]. Definition 2.8. Let (Mn,g) be a compact Riemannian manifold. Consider the functional φ 2 (φ,g) := ||∇ ||L2. E φ 2 || ||L2 The first Dirichlet eigenvalue of g is λ(g) = inf (φ,g). φ R φdV=0 E { | M } 3. Background on the L2 flow Inthissection wecollect somefacts aboutthefunctional andsolutions totheL2 flow. First F we recall ([2] Chapter 4.H) that 1 grad = δdRc Rˇ+ Rm 2g F − 4 | | where d is the exterior derivative induced by the Levi-Civita connection on Λ1 Λ1, δ is the ⊗ L2-adjoint of d, and Rˇ = RpqrR . ij i jpqr From this it follows that n (3.1) trgrad = ∆s+ 1 Rm 2. F − 4 − | | (cid:16) (cid:17) Nextwenotecertainellipticandparaboliccoercivityestimateswhichwewilluseintheestimates below. Lemma 3.1. ([21] Lemma 2.2) There exists a universal constant C such that if (M4,g) is a compact four-manifold, then 2Rm 2 C grad 2 + Rm2 Rm . ∇ L2 ≤ || F||L2 |∇ | | | (cid:20) ZM (cid:21) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) 6 JEFFREYSTREETS Proposition 3.2. ([21] Proposition 4.6) Given A > 0 there exists ǫ > 0 so that if (M4,g ) a t solution to the L2 flow on [0,T] such that sup C (g ) A, S t ≤ t [0,T] ∈ (g ) ǫ, 0 F ≤ then T sup||gradF||2L2 + ∇2gradF 2L2 ≤ 2||E||2L2(g0)+CA2ǫ41. [0,T] Z0 (cid:12)(cid:12) (cid:12)(cid:12) Next we recall a fundamental existe(cid:12)(cid:12)nce result (cid:12)o(cid:12)n the existence and long time behavior of solutions to the L2 flow. Theorem 3.3. ([24] Corollary 1.10) Let (Mn,g) be a compact Riemmanian manifold. The solution to the L2 flow with initial condition g exists on a maximal time interval [0,T). Furthermore, if T < then ∞ limsup Rm = . | |g(t) ∞ t T → Theorem 3.4. ([24] Theorem 1.9) Fix m,n 0. There exists a constant C = C(m,n) so that ≥ if (Mn,g(t)) is a complete solution to the L2 flow on [0,T] satisfying sup Rm K, | |≤ M [0,T] × then 1+m 1 2 (3.2) sup mRm C K + . M [0,T]|∇ |≤ (cid:18) t12(cid:19) × Corollary 3.5. ([24] Corollary 1.5) Let (Mn,g (t),p ) be a sequence of complete pointed { i i i } solutions of the L2 flow, where t (α,ω), α < ω . Suppose there exists K < and ∈ −∞ ≤ ≤ ∞ ∞ δ > 0 such that sup Rm(g ) K, inj (p ) δ. | i |gi ≤ gi(0) i ≥ Mi (α,ω) × Then there exists a subsequence (M ,g (t),p ) and a one parameter family of Riemannian { ij ij ij } manifolds (M ,g (t),p ) such that (M ,g ,p ) converges to (M ,g (t),p ) in the C ∞ ∞ ∞ { ij ij ij } ∞ ∞ ∞ ∞ Cheeger-Gromov topology. Next we make the observation mentioned in the introduction, namely that for solutions in dimensions n= 2,3 any finite time singularity must be collapsed on the scale of curvature. Proposition 3.6. Let (Mn,g(t)) be a solution to the L2 flow, n = 2,3. Suppose g(t) exists on a maximal time interval [0,T), T < . Let (x ,t ) be a sequence of points such that t T i i i ∞ { } → and Rm (x ,t ) = sup Rm . i i | | | | [0,ti] Then lim inj (x ) Rm (x ) = 0. g i i i | | →∞ COLLAPSING IN THE L2 CURVATURE FLOW 7 Proof. Supposetheclaim is false, and let (x ,t ) bethesequence of points as in thestatement. i i { } Let λ = Rm (x ,t ), and let i i i | | t gi = λ g t + . i i λ2 (cid:18) i (cid:19) A direct calculation shows that gi is aesolution to the L2 flow on [−tiλ2i,0] with bounded curvature. Moreover,sincewehaveassumedlim inj(x ) Rm (x ) > 0,weconcludelim inj (x ) > i i i i gi i 0. It follows fromCorollary 3.5 tehat theseq→u∞ence of so|lutio|ns M,gi,xi contains a s→ub∞sequeence { } converging to a smooth, nonflat solution to the L2 flow which we denote (M ,g ,x ). But ∞ scionncterandi0 and consider the Riemannian metric (4.1) g = φ(x)2dx2+ψ(x)2g . Σ It will frequently be useful to use the natural geometric coordinate of lateral distance from the slice Σ 0 . In particular, set ×{ } x (4.2) s(x)= φ(w)dw. Z0 The range of s is some interval we will always refer to as (a,b). Moreover, let 1 L(g) := φ(x)dx. Z−1 L represents the length of the base circle in the case of a product topology, or the distance from the north pole to the south pole in the case of SO(3) invariant metrics on S3. Of course L = b a. Using the parameter s, the metric (4.1) takes the form − (4.3) g = ds2+ψ(s)2g . Σ If we impose periodic boundary conditions, i.e. that φ and ψ extend to smooth functions on S1, then g defines a metric on S1 Σ. To produce SO(3)-invariant metrics on S3, we will impose × lim ψ = 0 x 1 (4.4) →± lim ψ = 1. s x 1 ∓ →± Next we establish some geometric estimates for the Riemannian manifolds described above. We will informally refer to these as warped products, with the assumption that the boundary conditions have been chosen appropriate to the given topology. Lemma 4.1. Let (M3,g) be a warped product. Let v and w denote vectors in π TΣ. Then ∗ ∂ ψ K ψ2 (4.5) K = K v = ss, K =K(v w) = Σ− s. 1 ∂s ∧ − ψ 2 ∧ ψ2 (cid:18) (cid:19) Lemma 4.2. Let (M3,g) be a warped product. Then b (4.6) Vol(g) = Vol(g ) ψ2dw, Σ Za b K ψ2 2 (4.7) Rm 2dV = Vol(g ) 4ψ2 +2 Σ− s dw. | | Σ ss ψ2 ZM Za (cid:0) (cid:1) ! Proof. We directly compute b b Vol(g) = dV = ψ2dΣdw = Vol(g ) ψ2dw. g Σ ZM Za ZΣ Za Next, using (4.5) we compute b ψ2 K ψ2 2 Rm 2 = 4 ss +2 Σ − s ψ2dΣdw | | ψ2 ψ4 ZM Za ZΣ (cid:0) (cid:1) ! b K ψ2 2 = Vol(g ) 4ψ2 +2 Σ − s dw. Σ ss ψ2 Za (cid:0) (cid:1) ! (cid:3) COLLAPSING IN THE L2 CURVATURE FLOW 9 Lemma 4.3. There is a constant δ > 0 so that if (S3,g) is an SO(3)-invariant metric with Rm 1, then L δ. | |≤ ≥ Proof. Recall that our metric satisfies (4.4). Let µ = sup s > a 1 ψ 2 . Clearly L µ, { |2 ≤ s ≤ } ≥ thus it suffices to bound µ from below. First note that on [0,µ] certainly ψ 2µ. Without loss ≤ of generality assume µ 1, so that ψ 1 on [0,µ]. Now observe that since Rm 1, ≤ 8 ≤ 4 | | ≤ 1 ψ = ψK ψ . ss 1 − ≥ − ≥ −4 It follows that, on [0,µ], ψ 1 µ. Likewise we can estimate on [0,µ] s ≥ − 4 1 ψ2 = K K ψ2 1+ψ2 1+(2µ)2 1+ . s Σ − 2 ≤ ≤ ≤ 4 This implies a lower bound for µ, and the lemma follows. (cid:3) Proposition 4.4. Let (M3,g) be a warped product with fiber Σ, χ(Σ) = 0, further satisfying 6 Rm 1. Given ǫ > 0, there exists δ > 0 so that if L ǫ then g | |≤ ≥ (4.8) inj δ. g ≥ Proof. We first consider the case of product topology, i.e. M = S1 Σ. Fix s (a,b) a ∼ 0 × ∈ minimum point for ψ. At this point we compute K ψ2 K 1 K = Σ − s = | Σ|. ≥| 2| ψ2 ψ2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) We conclude that for all s, (cid:12) (cid:12) ψ(s) ψ(s ) K = 1 0 Σ ≥ ≥ | | since χ(Σ)= 0. We now show a lower volume growth estimate for sufficiently small balls. Fix a 6 constant r > 0sothatr inj(g ), andalso r ǫ L. Now fixan arbitrary(p ,s ) Σ S1 0 0 ≤ Σ 0 ≤ 2 ≤ 2 0 0 ∈ × and consider B (p ,s ). We want to show that there is a uniform constant µ > 0 so that, for all r 0 0 r r , 0 ≤ Vol(B (p ,s )) r 0 0 µ. r3 ≥ Without loss of generality wecan reparameterize s sothat s = 0, andtherange of sis L,L . 0 −2 2 First we claim the inclusion (cid:0) (cid:1) r r (4.9) U := Br2,ψ2(s0)gΣ(p0)× s0− 2,s0+ 2 ⊂ Br,g(p0,s0). h i To show this let (q,t) U and let γ denote the curve which is the concatenation of the shortest ∈ geodesic in Σ connecting p and q, in the metric ψ2(s )g , with the lateral curve connecting 0 0 Σ (q,s ) to (q,t). One has 0 r r d ((p ,s ),(q,t)) Length(γ) + = r. g 0 0 ≤ ≤ 2 2 10 JEFFREYSTREETS Therefore the inclusion (4.9) holds. One can then compute Vol(B (p ,s )) Vol(U) r 0 0 ≥ = s0+r2 ψ2(w)dΣdw Zs0−2r ZBr2,ψ2(s0)gΣ(p0,s0) infψ2 s0+r2 dΣdw ≥ Zs0−r2 ZB2r,ψ2(s0)gΣ(p0,s0) ν r3. ≥ 2 where ν is a lower bound on the volume growth of g . The lower bound on volume growth Σ follows, and by Cheeger’s lemma the proposition follows. This argument can be repeated for SO(3)-invariant metrics on S3. (cid:3) Proof of Theorem 1.1. Let (M3,g(t)) be a solution to the volume normalized L2 flow as in the statement. The first step is to show long time existence. We know from Theorem 3.3 that if the maximal existence time is T < then ∞ lim Rm = . t T | |g(t) ∞ → Choose a sequence of points (x ,t ) such that i i λ := Rm(x ) = sup Rm . i | i |g(ti) | | M [0,ti] × Let t g (t):= λ g t + i i i λ2 (cid:18) i (cid:19) By construction one notes that the solution g (t) exists on [ λ2t ,0], and moreover i − i i (4.10) sup Rm = Rm(x ) = 1. | | | i |gi(0) M×[−λ2iti,0] We want to take a convergent subsequence of these solutions g , and to obtain a manifold in the i limit we require a lower bound on the injectivity radius. By Proposition 4.4 it suffices to show a lower bound on the lateral distance L. e Here we break into cases. In the case of SO(3)-invariant metrics on S3, Lemma 4.3 provides the required lower bound. For the remaining cases we argue by contradiction. Assume L 0. i → Note that the solutions (M,g (t)) exist, for sufficiently large i, on [ 1,0]. It follows from (4.10) i − and Theorem 3.4 that there is a uniform constant C such that sup Rm C. |∇ |gi(0) ≤ M Say the point x is given by (p ,s ) Σ (a,b). By integrating along a lateral geodesic one i i i ∈ × concludes Rm (p ,s τ) Rm (p ,s ) τC 1 CL . | |gi(0) i i± ≥ | |gi(0) i i − ≥ − i Since the curvatures are functions of the parameter s only, and since L 0 we may conclude i → that for sufficiently large i one has 1 (4.11) inf Rm . M | |gi(0) ≥ 2

Collapsing in the $L^2$ curvature flow

Similar Collapsing in the $L^2$ curvature flow