Shi-type estimates and finite time singularities of flows of G2 structures

Shi-type estimates and finite time singularities of flows of G2 structures Abstract In this paper, we extend Lotay–Wei’s Shi-type estimate from Laplacian flow to more general flows of G2 structures including the modified Laplacian co-flow. Then we prove a version of κ-non-collapsing theorem. We will use both of them to study finite-time singularities of general flows of G2 structures. 1. Introduction Let M be a compact 7-manifold. A G2 structure on M is defined by a 3-form ϕ such that at each point, there exists an element in GL(7,R) which maps ϕ into e123+e145+e167+e246−e257−e347−e356, (1.1) where eijk=ei∧ej∧ek and {ei} are the standard coframe of R7. It induces a metric g by g(u,v)Volg=16(u⌟ϕ)∧(v⌟ϕ)∧ϕ. (1.2) If ϕ is closed, it is called a closed G2 structure. If ψ=∗ϕ is closed, it is called a co-closed G2 structure. For a G2 structure, the torsion tensor T is defined by ∇aϕbcd=Taψebcde. (1.3) If the torsion tensor T vanishes, then it is called a torsion-free G2 structure. The holonomy group of the metric induced by a G2 structure is contained in G2 if it is torsion-free. In order to get general existence results for the torsion-free G2 structures, many versions of flows have been introduced. For example, Bryant [1] proposed the Laplacian flow of closed G2 structures: ∂∂tϕ=Δϕϕ. (1.4) Karigiannis et al. [6] proposed an analogy for the co-closed form ψ. Later, Grigorian [4] proposed a modified version and proved the short-time existence: ∂∂tψ=Δψψ+2d((A−TrT)ϕ), (1.5) where A is a suitable constant. There may be other important flows of G2 structures. In general, they should satisfy the equation ∂∂tϕijk=12hilϕljkdxi∧dxj∧dxk+16Xlψlijkdxi∧dxj∧dxk, (1.6) where X is a vector field and h is a symmetric tensor. According to Karigiannis [5], the equivalent equation for ψ is ∂∂tψijkl=himψmjkl+hjmψimkl+hkmψijml+hlmψijkm−Xiϕjkl+Xjϕikl−Xkϕijl+Xlϕijk. (1.7) The induced equations for the metric and torsion tensor are [5] ∂∂tgij=2hij, (1.8) and ∂∂tTij=Tijhmj+TijXlϕlmj+(∇khil)ϕklj+∇iXj. (1.9) In this paper, we require that ∂∂tgij=2hij=−2Rij+C+L(T)+T*T, (1.10) X=C+L(T)+L(Rm)+L(∇T)+T*T, (1.11) and ∂∂tTij=ΔTij+L(T)+L(∇T)+Rm*T+∇T*T+T*T+T*T*T, (1.12) where L and * denote linear maps and multi-linear maps in variables other than ϕ, ψ, g, respectively. Their meanings may vary in different lines. For example, both Tklψijkl and Tklgkl are considered as L(T). Therefore, we have formulas like ∇(L(T))=L(∇T)+T*T. (1.13) Definition 1.1. In this paper, we call a flow of G2 structures reasonable if it satisfies Equations (1.6), (1.10), (1.11), (1.12), the short-time existence and the uniqueness. For example, for Laplacian flow [8], X=0, and ∂∂tgij=−2Rij−23∣T∣2gij−4TiTkjk. (1.14) The condition for the torsion is also satisfied. For the modified Laplacian co-flow [4], X=∇TrT, and ∂∂tgij=−2Rij+TkmTlnϕiklϕjmn+(4A−2TrT)Tij. (1.15) The condition for the torsion is also satisfied. The short-time existence and uniqueness of the Laplacian flow were proved by Bryant–Xu [2]. The analogous results for the modified Laplacian co-flow were proved by Grigorian [4]. In the case of Laplacian flow, Lotay and Wei [8] proved a global version of Shi-type estimate with respect to (∣Rm(p,t)∣g(t)2+∣∇T(p,t)∣g(t)2)12. It is equivalent to (∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12 in that case because the scalar curvature is equal to −∣T(p,t)∣g(t)2 in that case. The first goal of this paper is to show a local version of Shi-type estimate with respect to (∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12 for all reasonable flows of G2 structures including both the Laplacian flow and the modified Laplacian co-flow. Using the global Shi-type estimate, Lotay–Wei proved that supp∈M(∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12≥CT−t, (1.16) if T is the maximal existence time for the Laplacian flow. For a reasonable flow of G2 structures, using our Shi-type estimate, (1.16) is also true. One may ask whether there are any estimates for the Ricci curvature, scalar curvature and torsion torsion at maximal existence time. The answer is yes. Using the Shi-type estimate and the method of Lotay and Wei, it is easy to see that ∫0TsupM(∣Ric∣+∣T∣2)dt=∞. (1.17) In order to get better estimates using the method of Wang in [11], we need a κ-non-collapsing theorem. We will show that the κ-non-collapsing theorem is true if ∫0T(T−t)supM∣T∣4dt<∞. (1.18) In that case, we will prove that limsupt→T[(T−t)supM(∣Ric∣+∣T∣2)]>0, (1.19) and limsupt0→T[(T−t0)2supt≤t0(1+∣R∣+∣T∣2)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]>0. (1.20) In particular, if in addition to (1.18), supM(∣R∣+∣T∣2)=o(1T−t), (1.21) then the singularity cannot be type-I. In other words, supM(∣Rm∣+∣T∣2+∣∇T∣)=O(1(T−t)) (1.22) cannot be true. Moreover, using our κ-non-collapsing theorem, we can also show that any blow-up limit near finite-time singularity satisfying (1.18) and (1.21) must be a manifold with holonomy contained in G2 and has maximal volume growth rate. In Section 2, we prove the Shi-type estimate. In Section 3, we derive the evolution equation for Perelman’s W-functional. In Section 4, we prove the κ-non-collapsing theorem. In Section 5, we discuss the finite-time singularity. 2. Shi-type estimate Theorem 2.1. Let Br(p)be the ball of radius rwith respect to g(0)for a reasonable flow of G2structures. Assume the coefficients in Eqs. (1.6), (1.10), (1.11), (1.12) are bounded by Λ. For example, in the modified Laplacian co-flow case, we assume ∣A∣≤Λ. If ∣Rm∣+∣T∣2+∣∇T∣<Λ (2.1)on Br(p)×[0,T], then ∣∇kRm∣+∣∇k+1T∣<C(k,r,Λ,T) (2.2)on Br/2(p)×[T/2,T]for all k=1,2,3,… Proof We will use the method proposed by Shi in [10]. We start from the evolution equations for the Riemannian curvature, the torsion tensor and their higher order derivatives. It is well known [3] that if ∂∂tgij=2hij, then ∂∂tRijkl=glp(∇i∇jhkp+∇i∇khjp−∇i∇phjk−∇j∇ihkp−∇j∇khip+∇j∇phik), (2.3) ∂∂tRjk=gpq(∇q∇jhkp+∇q∇khjp−∇q∇phjk−∇j∇khqp) (2.4) and ∂∂tR=−2ΔTrh+2∇i∇jhij−2hijRij. (2.5) When h=−Ric, then ∂∂tRm−ΔRm=Rm*Rm, (2.6) ∂∂tRic−ΔRic=Ric*Rm, (2.7) and ∂∂tR−ΔR=2∣Ric∣2. (2.8) For general h satisfying (1.10), we can treat the Ricci part of h as in Ricci flow and then compute the rest terms. Since our equation is more complicated than the Lotay–Wei case [8], we would like to use a more concise notation. Therefore, we define the degree of T and ∇ be 1 and the degree of Rm be 2, and use it to compute the degree of polynomials in them. For example ∇Rm*T has degree 4=3+1 and Rm+L(T) has degree 2=max{2,1}. Using such notation, the degree of (∂∂t−Δ)Rm is 4, but it contains no ∇2Rm or ∇3T term. The degree of (∂∂t−Δ)T is 3, but it contains no ∇Rm or ∇2T term. The term ∂∂tR−ΔR−2∣Ric∣2 is a degree 4 polynomial of Ric, ∇2T, ∇T and T, but contains no Ric*Ric term. On the other hand ∂∂tΓijk=gkl(∇jhil+∇ihjl−∇lhij). (2.9) So the degree of (∂∂t−Δ)∇T is degree 4, but it contains no ∇2Rm or ∇3T term. Therefore, all the terms (∂∂t−Δ)∣Rm∣2+2∣∇Rm∣2, (∂∂t−Δ)∣T∣4 and (∂∂t−Δ)∣∇T∣2+2∣∇2T∣2 can be bounded by ϵ(∣∇Rm∣2+∣∇2T∣2)+Cϵ(∣Rm∣2+∣T∣4+∣∇T∣2+1)3/2. (2.10) In fact, all of them have degree 6. When we consider a monomial with degree at most 6, it contains a degree at least 4 part like L(∇2Rm), is the product of a degree 3 part and another degree 3 part like ∇Rm*∇2T, the product of a degree 3 part with several order 2, 1 or 0 parts like ∇Rm*Rm*T, or the product of parts with order 2, 1 and 0 like Rm*∇T*T. In the expressions of (∂∂t−Δ)∣Rm∣2+2∣∇Rm∣2, (∂∂t−Δ)∣T∣4 and (∂∂t−Δ)∣∇T∣2+2∣∇2T∣2, the first two possibilities do not happen. The third possibility can be bounded by ϵ times degree three part squared plus Cϵ times a monomial in the final possibility. The sixth root of the norm of the final possibility can be bounded by the sum of square root of the norm of degree 2 part, the norm of degree 1 part and 1. That is how we get (2.10). Choose ϵ=1, then (∂∂t−Δ)(∣Rm∣2+∣T∣4+∣∇T∣2+1)≤−(∣∇Rm∣2+∣∇2T∣2)+C(∣Rm∣2+∣T∣4+∣∇T∣2+1)3/2. (2.11) Similarly, for all k=1,2,3…, both the degree of (∂∂t−Δ)∇kRm and the degree of (∂∂t−Δ)∇k+1T are k+4, but they contain no ∇k+2Rm or ∇k+3T term. So (∂∂t−Δ)(∣∇kRm∣2+∣∇k+1T∣2)≤−(∣∇k+1Rm∣2+∣∇k+2T∣2)+C(k)(∑j=0k(∣∇jRm∣2(k+3)j+2+∣∇j+1T∣2(k+3)j+2))+∣T∣2(k+3)+1). (2.12) As Shi did in [10, Section 7], the next step is to consider Q=(μ+∣Rm∣2+∣T∣4+∣∇T∣2)(∣∇Rm∣2+∣∇2T∣2), (2.13) where μ is a constant to be determined later. Then using (2.1), (∂∂t−Δ)Q=[(∂∂t−Δ)(μ+∣Rm∣2+∣T∣4+∣∇T∣2)](∣∇Rm∣2+∣∇2T∣2)+(μ+∣Rm∣2+∣T∣4+∣∇T∣2)(∂∂t−Δ)(∣∇Rm∣2+∣∇2T∣2)−[∇(μ+∣Rm∣2+∣T∣4+∣∇T∣2)][∇(∣∇Rm∣2+∣∇2T∣2)]≤(−∣∇Rm∣2−∣∇2T∣2+C(Λ))(∣∇Rm∣2+∣∇2T∣2)+(μ+∣Rm∣2+∣T∣4+∣∇T∣2)[−∣∇2Rm∣2−∣∇3T∣2]+C(∣Rm∣4+∣∇Rm∣83+∣∇T∣4+∣∇2T∣83+∣T∣8+1)+(Rm*∇Rm+∣T∣2T*∇T+∇T*∇2T)*(∇Rm*∇2Rm+∇2T*∇3T)≤(−∣∇Rm∣2−∣∇2T∣2+C(Λ))(∣∇Rm∣2+∣∇2T∣2)+μ(−∣∇2Rm∣2−∣∇3T∣2)+C(μ+C(Λ))(C(Λ)+∣∇Rm∣83+∣∇2T∣83)+C(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣)≤−(∣∇Rm∣2+∣∇2T∣2)2−μ(∣∇2Rm∣2+∣∇3T∣2)+C(Λ,μ)(∣∇Rm∣2+∣∇2T∣2)43+C(Λ,μ)+C0(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣). (2.14) Choose μ=C(Λ) large enough so that C0(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣)≤14(∣∇Rm∣2+∣∇2T∣2+1)2+μ(∣∇2Rm∣2+∣∇3T∣2), (2.15) then (∂∂t−Δ)Q≤−34(∣∇Rm∣2+∣∇2T∣2)2+C(Λ)(∣∇Rm∣2+∣∇2T∣2+1)43≤−12(∣∇Rm∣2+∣∇2T∣2)2+C(Λ)≤−C(Λ)Q2+C(Λ)=−C1(Λ)Q2+C2(Λ). (2.16) Between the second and the third line, we used (2.1) and (2.13). Let χ be a cut-off function which is 0 outside Br(p), and is 1 inside Br/2(p). We are done if we can find a constant ν>0 such that H=νχ2+1C1(Λ)t+C2(Λ)C1(Λ) (2.17) satisfies (∂∂t−Δ)H>−C1(Λ)H2+C2(Λ) (2.18) at the first time t0∈(0,T] when supM(Q−H)=0 and at the point p0 where the maximal is achieved. In fact, before t0, Q−H<0. So (∂∂t−Δ)(Q−H)(p0,t0)≥0. This will provide a contradiction. The non-existence of t0 will provide the desired control Q<H for all t∈[0,T]. However ∂∂tH=−1C1(Λ)t2, (2.19) H2≥ν2χ4+1C1(Λ)2t2+C2(Λ)C1(Λ), (2.20) and ΔH=νΔ1χ2=ν∇·(−2∇χχ3)=νχ4(−2ϕΔχ+6∣∇χ∣2). (2.21) So if C1(Λ)ν>−2χΔχ+6∣∇χ∣2 at (p0,t0), we are done. Let g˜ be the metric at time 0, let γ be the distance to p with respect to g˜. Pick a non-increasing cut-off function η which is 0 on [r2,∞) and is 1 on [0,r2/4]. Let χ=η(γ2). Then for the ordinary derivatives ∂iχ=2η′(γ2)γ∂iγ, (2.22) ∂i∂jχ=2η′(γ2)γ∂i∂jγ+(4η″(γ2)γ2+2η′(γ2))∂iγ∂jγ. (2.23) By Hessian comparison theorem, ∇˜ij2γ=∂i∂jγ−Γ˜ijp∂pγ≤C(Λ)g˜ij/γ. (2.24) So Δγ=gij(∂i∂jγ−Γijp∂pγ)≤C(Λ)gijg˜ij/γ+gij(Γ˜ijp−Γijp)∂pγ. (2.25) Since ∣∂∂tgij∣≤C(Λ), we see that C(Λ,T)−1g˜ij≤gij≤C(Λ,T)g˜ij. (2.26) On the other hand, the degree of ∂∂tΓijk=gkl(∇jhil+∇ihjl−∇lhij) (2.27) is 3, so it is bounded by C(Λ,T)(∣∇Rm∣+∣∇2T∣+1). Using Q≤H and (2.13), we see that before t0, ∣∂∂tΓijk∣≤C(Λ,T)(νχ+1t+1). (2.28) So at t0, Δγ≤C(Λ,T)γ+C(Λ,T)(νχ+1). (2.29) Using (2.23) and (2.26), Δχ≥2η′(γ2)γΔγ−Cgij∂iγ∂jγ≥−C(Λ,T,r)(νχ+1). (2.30) Therefore −2χΔχ+6∣∇χ∣2≤C(Λ,T,r)(ν+1). (2.31) So if we choose ν=C(Λ,T,r) large enough, then C(Λ,T,r)(ν+1)<C1(Λ)ν (2.32) can be achieved. We are done for the bound of (∣∇Rm∣2+∣∇2T∣2). Using Qk=(μk+∣∇kRm∣2+∣∇k+1T∣2)(∣∇k+1Rm∣2+∣∇k+2T∣2), (2.33) we can get higher derivative bounds.□ 3. Perelman’s W functional In [9], Perelman introduced the W functional W(g,f,τ)=∫M[τ(Rg+∣∇f∣2)+f−n](4πτ)−n/2e−fdg. (3.1) By routine calculations [7], if δgij=vij, δf=h, v=gijvij, δτ=σ, then the variation of W is δW=∫M[(v2−h−nσ2τ)(τ(R+2Δf−∣∇f∣2)+f−n)]+σ(R+∣∇f∣2)+h−τ(Rij+fij)vij(4πτ)−n/2e−fdg, (3.2) where fij means the second covariant derivative of f. For a general geometric flow ∂∂tgij=−2Rij+Eij. (3.3) Let f(t,p) solve the backwards heat equation: {∂∂tf=−Δf−R+12gijEij+n2τ+∣∇f∣2τ=T−t, (3.4) where T is any given real number. Let φt be the diffeomorphism generated by the time-dependent vector fields −∇f, define g˜(t)=φt*g(t) and f˜(t)=φt*f(t), then {∂∂tg˜ij=−2R˜ij+E˜ij−2f˜ij:=v˜ij∂∂tf˜=−Δ˜f˜−R˜+12g˜ijE˜ij+n2τ=v˜2+n2τ:=h˜, (3.5) where the quantities with ∼ sign are just the original quantities pulled back under φt. Since W(g(t),f(t),τ(t))=W(g˜(t),f˜(t),τ(t)), (3.6) we could use the variation formula to obtain ddtW(g(t),f(t),τ(t))=ddtW(g˜(t),f˜(t),τ(t))=∫{−τ(R˜ij+f˜ij)v˜ij+∂τ∂t(R˜+∣∇˜f˜∣2)+h˜+(v˜2−h˜−n2τ∂τ∂t)(τ(R˜+2Δ˜f˜−∣∇˜f˜∣2)+f˜−n)}(4πτ)−n2e−f˜dg˜=∫{2τ(R˜ij+f˜ij)(R˜ij+f˜ij−12E˜ij)−(R˜+∣∇˜f˜∣2)−Δ˜f˜−R˜+12g˜ijE˜ij+n2τ}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij∣2−2(R˜+Δ˜f˜)+n2τ−τ(R˜ij+f˜ij−g˜ij2τ)E˜ij}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij−g˜ij2τ ∣2−τ(R˜ij+f˜ij−g˜ij2τ)E˜ij}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij−g˜ij2τ−E˜ij4 ∣2−τ8∣E˜∣2}(4πτ)−n2e−f˜dg˜=∫{2τ∣Rij+fij−gij2τ−Eij4 ∣2−τ8∣E∣2}(4πτ)−n2e−fdg≥−τ8(supM∣E∣)2∫M(4πτ)−n/2e−fdg. (3.7) In the fourth equation, we used the fact that ∫(∣∇˜f˜∣2−Δ˜f˜)e−f˜dg˜=∫Δ˜(e−f˜)dg˜=0. (3.8) Now we are interested in the infimum μ(g,τ)=inf∫(4πτ)−n/2e−fdg=1W(g,f,τ). (3.9) Suppose τ1<τ2 and f achieves the infimum at T−τ1. Then if the equation ∂∂tf=−Δf−R+12gijEij+n2τ+∣∇f∣2 (3.10) can be solved backwards, ∫(4πτ)−n/2e−fdg=1 (3.11) is still true for all τ∈[τ1,τ2] because ∂∂tf˜=g˜ij2∂∂tg˜ij+n2τ. (3.12) This will imply that μ(g(T−τ2),τ2)≤μ(g(T−τ1),τ1)+18∫τ1τ2τsupt=T−τ∣E∣2dτ. (3.13) In fact, the nonlinear equation (3.10) can be solved by rewriting it as a linear parabolic equation ∂∂t(e−f)=−Δ(e−f)−(−R+12gijEij+n2τ)(e−f). (3.14) 4. κ-Non-collapsing theorem The original κ-non-collapsing theorem of Perelman for Ricci flow in [9] requires the Riemannian curvature bound. However, the definition can be modified to the following version: Definition 4.1. The Riemannian metric gon Mnis said to be κ-non-collapsing relative to upper bound of scalar curvature on the scale ρif for any Bg(p,r)⊂Mwith r<ρsuch that supBg(p,r)Rg≤r−2, we have VolgBg(p,r)≥κrn. The κ-non-collapsing theorem relative to upper bound of scalar curvature for Ricci flow was proved by Perelman ([7, Section 13]). The proof can be modified to get the following theorem using the quasi-monotonicity formula (3.13) in the previous section: Theorem 4.2. Let ∂∂tgij=−2Rij+Eijbe a geometric flow on a compact manifold Mn. Then there exists a positive function κwith four variables such that if 0<ρ≤ρ0<∞, 0<T2≤t0≤T<∞and ∫0t0(t0+ρ2−t)supM∣E∣2dt<∞, (4.1)then g(t0)is κ(g(0),T,ρ0,∫0t0(t0+ρ2−t)supM∣E∣2dt)-non-collapsing relative to upper bound of scalar curvature on scale ρ. Proof Fix a cut-off function χ(s) such that χ(s)=1 when ∣s∣≤12, and χ(s)=0 when ∣s∣≥1. For any g(t0)-metric ball B(p,r) of radius r<ρ which satisfies R(x)≤r−2 for every x∈B(p,r), we can define u(x)=eL/2χ(d(x,p)r), (4.2) where L is chosen so that (4πr2)−n/2∫Mu2=1. (4.3) In particular, Vol(B(p,r))≥e−L(4π)n/2rn, (4.4) Vol(B(p,r2))≤e−L(4π)n/2rn. (4.5) By (3.13) applied to T=t0+r2, W(g(t0),−2lnu,r2)≥μ(g(t0),r2)≥μ(g(0),t0+r2)−18∫t0+r2r2τsupt=t0+r2−τ∣E∣2dτ=μ(g(0),t0+r2)−18∫0t0(t0+r2−t)(supM∣E∣2)dt≥μ0−18∫0t0(t0+ρ2−t)(supM∣E∣2)dt:=μ1, (4.6) where μ0 is the lower bound of μ(g(0),τ) when τ∈[T2,T+ρ02]. So μ1≤W(g(t0),−2lnu,r2)=∫M(4πr2)−n/2[r2(R+∣−2∇lnu∣2)−2lnu−n]u2dg(t0)=∫M(4πr2)−n/2[r2(Ru2+4∣∇u∣2)+u2(−2lnu−n)]dg(t0). (4.7) R<Cr−2 in B(p,r), −2u2lnu=−2u2(L/2+lnχ) and ∣∇u∣≤eL/2r∣χ′(d(x,p)r)∣≤CeL/2r. (4.8) So μ1≤C−L+CeLVolB(p,r)rn≤C−L+CVol(B(p,r))Vol(B(p,r2)). (4.9) Thus if Vol(B(p,r))Vol(B(p,r2))<2n, then −L≥C, so Vol(B(p,r))≥C1rn. Let ω be the volume of unit ball in the Euclidean space Rn and let κ=min(C1,ω2), then we claim that Vol(B(p,r))≥κrn. Otherwise, Vol(B(p,r))<κrn, so Vol(B(p,r))Vol(B(p,r2))≥2n, so Vol(B(p,r2))<2−nκrn=κ(r2)n. (4.10) We can apply the same thing for r2k and obtain that Vol(B(p,r2k))<κ(r2k)n, (4.11) which is a contradiction for large enough k.□ 5. Finite-time singularity Now we are ready to study the finite-time singularities of reasonable flows of G2 structures. First of all, using the method of Lotay–Wei and our Shi-type estimate, we can prove the following theorem: Theorem 5.1. If ϕ(t)is a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T), then supM(∣Rm∣2+∣T∣4+∣∇T∣2)12≥CT−t (5.1)for some constant C>0. Proof As Lotay–Wei did in [8], if supM(∣Rm∣2+∣T∣4+∣∇T∣2)12 is bounded, then all the higher order derivatives are also bounded. So ∂∂tg and ∂∂tϕ are all bounded. So they and their higher order derivatives are all bounded using the background metric g(0). So we can take the smooth limit. This will violate the short-time existence assumption. Still as Lotay–Wei, we can use Eq. (2.11) to get the required blow-up rate.□ Then we can get the following estimate: Theorem 5.2. If ϕ(t)is a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T), then ∫0TsupM(∣Ric∣+∣T∣2)dt=∞. (5.2) Proof If it is not true, then using the evolution equation, we can see that ∫0Tsup∣∂∂tg∣ is finite. So the metric is uniformly continuous, in other words, for any ϵ>0, there exists δ>0 such that for all t1,t2∈(T−δ,T), (1+ϵ)−1g(t2)≤g(t1)≤(1+ϵ)g(t2). (5.3) Now the proof of [8, Theorem 8.1] can be applied directly. In fact, according to Theorem 5.1, there exist ti→T and a sequence of points xi∈M such that Λi:=supx∈M,t∈[0,ti](∣Rm∣2+∣T∣4+∣∇T∣2) is achieved at (xi,ti). Now we can rescale the flow so that this quantity becomes 1. Then using Shi-type estimate, all of the higher order derivatives of the curvature are also bounded in any finite scale. Using the uniform continuity of the metric, the harmonic radius has a lower bound and we can take the limit of metric for a subsequence. The higher order derivatives of ϕ is also bounded, so we can also take the limit of the 3-form in a subsequence. Remark that the quantity ∫T00sup(∣Ric∣+∣T∣2)dt after rescaling is the same as the quantity ∫T−(Λi)−1/2T0TsupM(∣Ric∣+∣T∣2)dt before rescaling for any fixed T0<0. By Fatou’s lemma, in the limit, this quantity is 0. In other words, Ric=T=0 in the limit. Now the uniform continuity of the metric implies that the limit has the same volume of balls as the Euclidean space, so it must be the Euclidean space. This is a contradiction.□ As for the better estimates of Ricci curvature, scalar curvature and torsion tensor, we can prove the following theorem using the method in [11] Theorem 5.3. Let ϕ(t)be a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T). Assume that ∫0T(T−t)supM∣T∣4dt<∞, (5.4)then limsupt→T(T−t)supM(∣Ric∣+∣T∣2)>0, (5.5)and limsupt0→T[(T−t0)2supt≤t0(1+∣R∣+∣T∣2)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]>0. (5.6) Proof In this case, the flow is κ-non-collapsing on the scale T−t. Define Q(t0)=supt≤t0(∣Rm∣+∣T∣2+∣∇T∣), (5.7) and O(t0)=supt≤t0(1+∣R∣+∣T∣2)≤1+100Q(t0), (5.8) then according to Theorem 5.1, Q(t)→∞ as t→T. Therefore, there exist an increasing sequence tk→T such that Q(tk)=2k and the maximum is achieved at (pk,tk). By Theorem 5.1 again, there exists a subsequence of tk such that (T−tk)≥T−tk−13. Now if (5.5) is not true, for the subsequence ∫tk−1tk∣∂∂tg∣dt≤C(T−tk)supt≤tk(∣Ric∣+∣T∣2+1)→0. (5.9) Now using Theorems 2.1 and 4.2, we can show that as before, both (M,pk,23k2ϕ,2kg,2k(t−tk)) and (M,pk,23k2ϕ,2kg,2k(t−tk−1)) converge to the trivial product of a torsion-free G2 structure times (−∞,0]. The two limits must be isometric using the uniform continuity of metric. It is a contradiction. As for the second estimate, we need to show that when T−t0<1, supt≤t0(∣Ric∣+∣T∣2)≤CO(t0)Q(t0), (5.10) We will still follow the method in [11]. By Theorem 5.1, we see that Q(t0)≥C(T−t)−1. So the flow is κ-non-collapsing on the scale Q(t0)−12. Now we rescale the flow so that Q(t0)=1. Then the harmonic radius has a lower bound. So inside a finite size of ball, the metric and all of its higher derivatives are uniformly bounded. So (∂∂t−D)T=0 for some elliptic operator D with bounded coefficients and higher derivatives of coefficients. So after rescaling, sup∣∇kT∣≤C(k)sup∣T∣. (5.11) Now ∂∂tR=−2ΔTrh+2∇i∇jhij−2hijRij=2ΔR−2∇i∇jRij+2∣Ric∣2+L(Ric)Q(t0)+T*RicQ(t0)+T*T*Ric+L(∇2(CQ(t0)+TQ(t0)+T*T)). (5.12) The terms 2ΔR−2∇i∇jRij+2∣Ric∣2 are equal to ΔR+2∣Ric∣2 by Bianchi identity as in Ricci flow case. The terms L(Ric)Q(t0)+T*RicQ(t0)+T*T*Ric are bounded by CO(t0)Q(t0)∣Ric∣≤∣Ric∣2+CO(t0)Q(t0). The rest terms are bounded by CO(t0)Q(t0). In conclusion, ∣(∂∂t−Δ)R−2∣Ric∣2∣≤∣Ric∣2+CO(t0)Q(t0). (5.13) For any p, we can pick a cut-off function χ such that it is 0 outside Bg(t0)(p,Q(t0)−1/2)×[t0−1Q(t0),t0], (5.14) and is 1 inside Bg(t0)(p,12Q(t0)−1/2)×[t0−12Q(t0),t0]. (5.15) After rescaling, it vanishes outside Bg(0)(p,1)×[−1,0] and is 1 inside Bg(0)(p,12)×[−12,0]. Thus ∫t=0χR=∫−10(∫M∂∂t(χR))dt=∫−10∫M[R(∂∂t−Δ)χ+χ(∂∂t−Δ)R]dt. (5.16) Since the geometry is bounded, (5.13) and (5.16) imply that ∫Bg(0)(p,12)×[−12,0]∣Ric∣2≤CO(t0)Q(t0). (5.17) Now the Ricci curvature satisfies the equation ∂∂tRjk=gpq(∇q∇jhkp+∇q∇khjp−∇q∇phjk−∇j∇khqp). (5.18) As in the Ricci flow case, −gpq(∇q∇jRkp+∇q∇kRjp−∇q∇pRjk−∇j∇kRqp)=ΔRjk+Ric*Rm. (5.19) The rest term gpq(∇q∇j(h+R)kp+∇q∇k(h+R)jp−∇q∇p(h+R)jk−∇j∇k(h+R)qp) (5.20) is bounded by CO(t0)Q(t0) by (1.10). Therefore, ∣(∂∂t−D)Ric∣≤CO(t0)Q(t0) (5.21) for some elliptic operator D with bounded coefficients and higher derivatives of coefficients. Therefore, we have ∣Ric∣2≤CO(t0)Q(t0). Before rescaling, it is exactly ∣Ric∣≤CO(t0)Q(t0).□ If in addition supM(∣R∣+∣T∣2)=o(1T−t), (5.22) we can also show that any blow-up limit at finite time must be a manifold with maximal volume growth rate whose holonomy is contained in G2. Theorem 5.4. Let ϕ(t)be a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T). If ∫0T(T−t)supM∣T∣4dt<∞, (5.22)and supM(∣R∣+∣T∣2)=o(1T−t), (5.24)then there exists a sequence tk→T,pk∈Msuch that Qk=(∣Rm∣2+∣T∣4+∣∇T∣2)12(pk,tk)→∞, (5.23)and (M,Qk3/2ϕ(tk),Qkg(tk),pk)converges to a complete manifold M∞with a torsion-free G2structure (ϕ∞,g∞,p∞)such that Volg∞(Bg∞(p∞,r))≥κr7 (5.24)for some κ>0and all r>0. Proof First of all, we can see that limsupt0→T[(T−t0)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]=∞. (5.25) So we can choose a sequence such that (T−tk)Qk→∞. After rescaling, (∣Rm∣2+∣T∣4+∣∇T∣2)12 is bounded. Moreover, sup(∣R∣+∣T∣2) converges to 0, and the manifold is κ-non-collapsing on a scale going to infinity. In particular, we get a uniform volume lower bound in any finite scale. Therefore, by our Shi-type estimate, the G2 structures converge in C∞ sense to a limit G2 structure. In the limit, both the scalar curvature and the torsion tensor are everywhere 0. In other words, the limit is torsion-free. Moreover, it has maximal volume growth rate.□ Funding The author is supported by the National Science Foundation under Grant no. 1638352, as well as support from the S.S. Chern Foundation for Mathematics Research Fund when this article is been revised. Acknowledgement The author is grateful to the valuable comments of the referee. The author also thanks the helpful discussions with Xiuxiong Chen, Jason Lotay and Chengjian Yao. References 1 R. L. Bryant , Some remarks on G2-structures. Proceedings of Gökova Geometry-Topology Conference 2005, 75-109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006. 2 R. L. Bryant and F. Xu , Laplacian Flow for Closed G2-Structures: Short Time Behavior. Preprint, arxiv:1101.2004. 3 B. Chow and D. Knopf , The Ricci flow: an introduction. Mathematical Surveys and Monographs Vol. 110 , American Mathematical Society , Providence, RI , 2004 . 4 S. Grigorian , Short-time behaviour of a modified Laplacian coflow of G2-structures , Adv. Math. 248 ( 2013 ), 378 – 415 . Google Scholar Crossref Search ADS 5 S. Karigiannis , Flows of G2-structures , I. Q. J. Math. 60 ( 2009 ), 487 – 522 . Google Scholar Crossref Search ADS 6 S. Karigiannis , B. McKay and M.-P. Tsui , Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry , Differential Geom. Appl. 30 ( 2012 ), 318 – 333 . Google Scholar Crossref Search ADS 7 B. Kleiner and J. Lott , Notes on Perelman’s papers , Geom. Topol. 12 ( 2008 ), 2587 – 2855 . Google Scholar Crossref Search ADS 8 J. D. Lotay and Y. Wei , Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness , Geom. Funct. Anal. 27 ( 2017 ), 165 – 233 . Google Scholar Crossref Search ADS 9 G. Perelman , The entropy formula for the Ricci flow and its geometric applications, Preprint. arXiv:math/0211159. 10 W.-X. Shi , Deforming the metric on complete Riemannian manifolds , J. Differential Geom. 30 ( 1989 ), 223 – 301 . Google Scholar Crossref Search ADS 11 B. Wang , On the conditions to extend Ricci flow (II) , Int. Math. Res. Not. IMRN 2012 ( 2012 ), 3192 – 3223 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

Shi-type estimates and finite time singularities of flows of G2 structures

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Abstract

Abstract In this paper, we extend Lotay–Wei’s Shi-type estimate from Laplacian flow to more general flows of G2 structures including the modified Laplacian co-flow. Then we prove a version of κ-non-collapsing theorem. We will use both of them to study finite-time singularities of general flows of G2 structures. 1. Introduction Let M be a compact 7-manifold. A G2 structure on M is defined by a 3-form ϕ such that at each point, there exists an element in GL(7,R) which maps ϕ into e123+e145+e167+e246−e257−e347−e356, (1.1) where eijk=ei∧ej∧ek and {ei} are the standard coframe of R7. It induces a metric g by g(u,v)Volg=16(u⌟ϕ)∧(v⌟ϕ)∧ϕ. (1.2) If ϕ is closed, it is called a closed G2 structure. If ψ=∗ϕ is closed, it is called a co-closed G2 structure. For a G2 structure, the torsion tensor T is defined by ∇aϕbcd=Taψebcde. (1.3) If the torsion tensor T vanishes, then it is called a torsion-free G2 structure. The holonomy group of the metric induced by a G2 structure is contained in G2 if it is torsion-free. In order to get general existence results for the torsion-free G2 structures, many versions of flows have been introduced. For example, Bryant [1] proposed the Laplacian flow of closed G2 structures: ∂∂tϕ=Δϕϕ. (1.4) Karigiannis et al. [6] proposed an analogy for the co-closed form ψ. Later, Grigorian [4] proposed a modified version and proved the short-time existence: ∂∂tψ=Δψψ+2d((A−TrT)ϕ), (1.5) where A is a suitable constant. There may be other important flows of G2 structures. In general, they should satisfy the equation ∂∂tϕijk=12hilϕljkdxi∧dxj∧dxk+16Xlψlijkdxi∧dxj∧dxk, (1.6) where X is a vector field and h is a symmetric tensor. According to Karigiannis [5], the equivalent equation for ψ is ∂∂tψijkl=himψmjkl+hjmψimkl+hkmψijml+hlmψijkm−Xiϕjkl+Xjϕikl−Xkϕijl+Xlϕijk. (1.7) The induced equations for the metric and torsion tensor are [5] ∂∂tgij=2hij, (1.8) and ∂∂tTij=Tijhmj+TijXlϕlmj+(∇khil)ϕklj+∇iXj. (1.9) In this paper, we require that ∂∂tgij=2hij=−2Rij+C+L(T)+T*T, (1.10) X=C+L(T)+L(Rm)+L(∇T)+T*T, (1.11) and ∂∂tTij=ΔTij+L(T)+L(∇T)+Rm*T+∇T*T+T*T+T*T*T, (1.12) where L and * denote linear maps and multi-linear maps in variables other than ϕ, ψ, g, respectively. Their meanings may vary in different lines. For example, both Tklψijkl and Tklgkl are considered as L(T). Therefore, we have formulas like ∇(L(T))=L(∇T)+T*T. (1.13) Definition 1.1. In this paper, we call a flow of G2 structures reasonable if it satisfies Equations (1.6), (1.10), (1.11), (1.12), the short-time existence and the uniqueness. For example, for Laplacian flow [8], X=0, and ∂∂tgij=−2Rij−23∣T∣2gij−4TiTkjk. (1.14) The condition for the torsion is also satisfied. For the modified Laplacian co-flow [4], X=∇TrT, and ∂∂tgij=−2Rij+TkmTlnϕiklϕjmn+(4A−2TrT)Tij. (1.15) The condition for the torsion is also satisfied. The short-time existence and uniqueness of the Laplacian flow were proved by Bryant–Xu [2]. The analogous results for the modified Laplacian co-flow were proved by Grigorian [4]. In the case of Laplacian flow, Lotay and Wei [8] proved a global version of Shi-type estimate with respect to (∣Rm(p,t)∣g(t)2+∣∇T(p,t)∣g(t)2)12. It is equivalent to (∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12 in that case because the scalar curvature is equal to −∣T(p,t)∣g(t)2 in that case. The first goal of this paper is to show a local version of Shi-type estimate with respect to (∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12 for all reasonable flows of G2 structures including both the Laplacian flow and the modified Laplacian co-flow. Using the global Shi-type estimate, Lotay–Wei proved that supp∈M(∣Rm(p,t)∣g(t)2+∣T(p,t)∣g(t)4+∣∇T(p,t)∣g(t)2)12≥CT−t, (1.16) if T is the maximal existence time for the Laplacian flow. For a reasonable flow of G2 structures, using our Shi-type estimate, (1.16) is also true. One may ask whether there are any estimates for the Ricci curvature, scalar curvature and torsion torsion at maximal existence time. The answer is yes. Using the Shi-type estimate and the method of Lotay and Wei, it is easy to see that ∫0TsupM(∣Ric∣+∣T∣2)dt=∞. (1.17) In order to get better estimates using the method of Wang in [11], we need a κ-non-collapsing theorem. We will show that the κ-non-collapsing theorem is true if ∫0T(T−t)supM∣T∣4dt<∞. (1.18) In that case, we will prove that limsupt→T[(T−t)supM(∣Ric∣+∣T∣2)]>0, (1.19) and limsupt0→T[(T−t0)2supt≤t0(1+∣R∣+∣T∣2)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]>0. (1.20) In particular, if in addition to (1.18), supM(∣R∣+∣T∣2)=o(1T−t), (1.21) then the singularity cannot be type-I. In other words, supM(∣Rm∣+∣T∣2+∣∇T∣)=O(1(T−t)) (1.22) cannot be true. Moreover, using our κ-non-collapsing theorem, we can also show that any blow-up limit near finite-time singularity satisfying (1.18) and (1.21) must be a manifold with holonomy contained in G2 and has maximal volume growth rate. In Section 2, we prove the Shi-type estimate. In Section 3, we derive the evolution equation for Perelman’s W-functional. In Section 4, we prove the κ-non-collapsing theorem. In Section 5, we discuss the finite-time singularity. 2. Shi-type estimate Theorem 2.1. Let Br(p)be the ball of radius rwith respect to g(0)for a reasonable flow of G2structures. Assume the coefficients in Eqs. (1.6), (1.10), (1.11), (1.12) are bounded by Λ. For example, in the modified Laplacian co-flow case, we assume ∣A∣≤Λ. If ∣Rm∣+∣T∣2+∣∇T∣<Λ (2.1)on Br(p)×[0,T], then ∣∇kRm∣+∣∇k+1T∣<C(k,r,Λ,T) (2.2)on Br/2(p)×[T/2,T]for all k=1,2,3,… Proof We will use the method proposed by Shi in [10]. We start from the evolution equations for the Riemannian curvature, the torsion tensor and their higher order derivatives. It is well known [3] that if ∂∂tgij=2hij, then ∂∂tRijkl=glp(∇i∇jhkp+∇i∇khjp−∇i∇phjk−∇j∇ihkp−∇j∇khip+∇j∇phik), (2.3) ∂∂tRjk=gpq(∇q∇jhkp+∇q∇khjp−∇q∇phjk−∇j∇khqp) (2.4) and ∂∂tR=−2ΔTrh+2∇i∇jhij−2hijRij. (2.5) When h=−Ric, then ∂∂tRm−ΔRm=Rm*Rm, (2.6) ∂∂tRic−ΔRic=Ric*Rm, (2.7) and ∂∂tR−ΔR=2∣Ric∣2. (2.8) For general h satisfying (1.10), we can treat the Ricci part of h as in Ricci flow and then compute the rest terms. Since our equation is more complicated than the Lotay–Wei case [8], we would like to use a more concise notation. Therefore, we define the degree of T and ∇ be 1 and the degree of Rm be 2, and use it to compute the degree of polynomials in them. For example ∇Rm*T has degree 4=3+1 and Rm+L(T) has degree 2=max{2,1}. Using such notation, the degree of (∂∂t−Δ)Rm is 4, but it contains no ∇2Rm or ∇3T term. The degree of (∂∂t−Δ)T is 3, but it contains no ∇Rm or ∇2T term. The term ∂∂tR−ΔR−2∣Ric∣2 is a degree 4 polynomial of Ric, ∇2T, ∇T and T, but contains no Ric*Ric term. On the other hand ∂∂tΓijk=gkl(∇jhil+∇ihjl−∇lhij). (2.9) So the degree of (∂∂t−Δ)∇T is degree 4, but it contains no ∇2Rm or ∇3T term. Therefore, all the terms (∂∂t−Δ)∣Rm∣2+2∣∇Rm∣2, (∂∂t−Δ)∣T∣4 and (∂∂t−Δ)∣∇T∣2+2∣∇2T∣2 can be bounded by ϵ(∣∇Rm∣2+∣∇2T∣2)+Cϵ(∣Rm∣2+∣T∣4+∣∇T∣2+1)3/2. (2.10) In fact, all of them have degree 6. When we consider a monomial with degree at most 6, it contains a degree at least 4 part like L(∇2Rm), is the product of a degree 3 part and another degree 3 part like ∇Rm*∇2T, the product of a degree 3 part with several order 2, 1 or 0 parts like ∇Rm*Rm*T, or the product of parts with order 2, 1 and 0 like Rm*∇T*T. In the expressions of (∂∂t−Δ)∣Rm∣2+2∣∇Rm∣2, (∂∂t−Δ)∣T∣4 and (∂∂t−Δ)∣∇T∣2+2∣∇2T∣2, the first two possibilities do not happen. The third possibility can be bounded by ϵ times degree three part squared plus Cϵ times a monomial in the final possibility. The sixth root of the norm of the final possibility can be bounded by the sum of square root of the norm of degree 2 part, the norm of degree 1 part and 1. That is how we get (2.10). Choose ϵ=1, then (∂∂t−Δ)(∣Rm∣2+∣T∣4+∣∇T∣2+1)≤−(∣∇Rm∣2+∣∇2T∣2)+C(∣Rm∣2+∣T∣4+∣∇T∣2+1)3/2. (2.11) Similarly, for all k=1,2,3…, both the degree of (∂∂t−Δ)∇kRm and the degree of (∂∂t−Δ)∇k+1T are k+4, but they contain no ∇k+2Rm or ∇k+3T term. So (∂∂t−Δ)(∣∇kRm∣2+∣∇k+1T∣2)≤−(∣∇k+1Rm∣2+∣∇k+2T∣2)+C(k)(∑j=0k(∣∇jRm∣2(k+3)j+2+∣∇j+1T∣2(k+3)j+2))+∣T∣2(k+3)+1). (2.12) As Shi did in [10, Section 7], the next step is to consider Q=(μ+∣Rm∣2+∣T∣4+∣∇T∣2)(∣∇Rm∣2+∣∇2T∣2), (2.13) where μ is a constant to be determined later. Then using (2.1), (∂∂t−Δ)Q=[(∂∂t−Δ)(μ+∣Rm∣2+∣T∣4+∣∇T∣2)](∣∇Rm∣2+∣∇2T∣2)+(μ+∣Rm∣2+∣T∣4+∣∇T∣2)(∂∂t−Δ)(∣∇Rm∣2+∣∇2T∣2)−[∇(μ+∣Rm∣2+∣T∣4+∣∇T∣2)][∇(∣∇Rm∣2+∣∇2T∣2)]≤(−∣∇Rm∣2−∣∇2T∣2+C(Λ))(∣∇Rm∣2+∣∇2T∣2)+(μ+∣Rm∣2+∣T∣4+∣∇T∣2)[−∣∇2Rm∣2−∣∇3T∣2]+C(∣Rm∣4+∣∇Rm∣83+∣∇T∣4+∣∇2T∣83+∣T∣8+1)+(Rm*∇Rm+∣T∣2T*∇T+∇T*∇2T)*(∇Rm*∇2Rm+∇2T*∇3T)≤(−∣∇Rm∣2−∣∇2T∣2+C(Λ))(∣∇Rm∣2+∣∇2T∣2)+μ(−∣∇2Rm∣2−∣∇3T∣2)+C(μ+C(Λ))(C(Λ)+∣∇Rm∣83+∣∇2T∣83)+C(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣)≤−(∣∇Rm∣2+∣∇2T∣2)2−μ(∣∇2Rm∣2+∣∇3T∣2)+C(Λ,μ)(∣∇Rm∣2+∣∇2T∣2)43+C(Λ,μ)+C0(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣). (2.14) Choose μ=C(Λ) large enough so that C0(Λ)(∣∇Rm∣+∣∇2T∣+1)2(∣∇2Rm∣+∣∇3T∣)≤14(∣∇Rm∣2+∣∇2T∣2+1)2+μ(∣∇2Rm∣2+∣∇3T∣2), (2.15) then (∂∂t−Δ)Q≤−34(∣∇Rm∣2+∣∇2T∣2)2+C(Λ)(∣∇Rm∣2+∣∇2T∣2+1)43≤−12(∣∇Rm∣2+∣∇2T∣2)2+C(Λ)≤−C(Λ)Q2+C(Λ)=−C1(Λ)Q2+C2(Λ). (2.16) Between the second and the third line, we used (2.1) and (2.13). Let χ be a cut-off function which is 0 outside Br(p), and is 1 inside Br/2(p). We are done if we can find a constant ν>0 such that H=νχ2+1C1(Λ)t+C2(Λ)C1(Λ) (2.17) satisfies (∂∂t−Δ)H>−C1(Λ)H2+C2(Λ) (2.18) at the first time t0∈(0,T] when supM(Q−H)=0 and at the point p0 where the maximal is achieved. In fact, before t0, Q−H<0. So (∂∂t−Δ)(Q−H)(p0,t0)≥0. This will provide a contradiction. The non-existence of t0 will provide the desired control Q<H for all t∈[0,T]. However ∂∂tH=−1C1(Λ)t2, (2.19) H2≥ν2χ4+1C1(Λ)2t2+C2(Λ)C1(Λ), (2.20) and ΔH=νΔ1χ2=ν∇·(−2∇χχ3)=νχ4(−2ϕΔχ+6∣∇χ∣2). (2.21) So if C1(Λ)ν>−2χΔχ+6∣∇χ∣2 at (p0,t0), we are done. Let g˜ be the metric at time 0, let γ be the distance to p with respect to g˜. Pick a non-increasing cut-off function η which is 0 on [r2,∞) and is 1 on [0,r2/4]. Let χ=η(γ2). Then for the ordinary derivatives ∂iχ=2η′(γ2)γ∂iγ, (2.22) ∂i∂jχ=2η′(γ2)γ∂i∂jγ+(4η″(γ2)γ2+2η′(γ2))∂iγ∂jγ. (2.23) By Hessian comparison theorem, ∇˜ij2γ=∂i∂jγ−Γ˜ijp∂pγ≤C(Λ)g˜ij/γ. (2.24) So Δγ=gij(∂i∂jγ−Γijp∂pγ)≤C(Λ)gijg˜ij/γ+gij(Γ˜ijp−Γijp)∂pγ. (2.25) Since ∣∂∂tgij∣≤C(Λ), we see that C(Λ,T)−1g˜ij≤gij≤C(Λ,T)g˜ij. (2.26) On the other hand, the degree of ∂∂tΓijk=gkl(∇jhil+∇ihjl−∇lhij) (2.27) is 3, so it is bounded by C(Λ,T)(∣∇Rm∣+∣∇2T∣+1). Using Q≤H and (2.13), we see that before t0, ∣∂∂tΓijk∣≤C(Λ,T)(νχ+1t+1). (2.28) So at t0, Δγ≤C(Λ,T)γ+C(Λ,T)(νχ+1). (2.29) Using (2.23) and (2.26), Δχ≥2η′(γ2)γΔγ−Cgij∂iγ∂jγ≥−C(Λ,T,r)(νχ+1). (2.30) Therefore −2χΔχ+6∣∇χ∣2≤C(Λ,T,r)(ν+1). (2.31) So if we choose ν=C(Λ,T,r) large enough, then C(Λ,T,r)(ν+1)<C1(Λ)ν (2.32) can be achieved. We are done for the bound of (∣∇Rm∣2+∣∇2T∣2). Using Qk=(μk+∣∇kRm∣2+∣∇k+1T∣2)(∣∇k+1Rm∣2+∣∇k+2T∣2), (2.33) we can get higher derivative bounds.□ 3. Perelman’s W functional In [9], Perelman introduced the W functional W(g,f,τ)=∫M[τ(Rg+∣∇f∣2)+f−n](4πτ)−n/2e−fdg. (3.1) By routine calculations [7], if δgij=vij, δf=h, v=gijvij, δτ=σ, then the variation of W is δW=∫M[(v2−h−nσ2τ)(τ(R+2Δf−∣∇f∣2)+f−n)]+σ(R+∣∇f∣2)+h−τ(Rij+fij)vij(4πτ)−n/2e−fdg, (3.2) where fij means the second covariant derivative of f. For a general geometric flow ∂∂tgij=−2Rij+Eij. (3.3) Let f(t,p) solve the backwards heat equation: {∂∂tf=−Δf−R+12gijEij+n2τ+∣∇f∣2τ=T−t, (3.4) where T is any given real number. Let φt be the diffeomorphism generated by the time-dependent vector fields −∇f, define g˜(t)=φt*g(t) and f˜(t)=φt*f(t), then {∂∂tg˜ij=−2R˜ij+E˜ij−2f˜ij:=v˜ij∂∂tf˜=−Δ˜f˜−R˜+12g˜ijE˜ij+n2τ=v˜2+n2τ:=h˜, (3.5) where the quantities with ∼ sign are just the original quantities pulled back under φt. Since W(g(t),f(t),τ(t))=W(g˜(t),f˜(t),τ(t)), (3.6) we could use the variation formula to obtain ddtW(g(t),f(t),τ(t))=ddtW(g˜(t),f˜(t),τ(t))=∫{−τ(R˜ij+f˜ij)v˜ij+∂τ∂t(R˜+∣∇˜f˜∣2)+h˜+(v˜2−h˜−n2τ∂τ∂t)(τ(R˜+2Δ˜f˜−∣∇˜f˜∣2)+f˜−n)}(4πτ)−n2e−f˜dg˜=∫{2τ(R˜ij+f˜ij)(R˜ij+f˜ij−12E˜ij)−(R˜+∣∇˜f˜∣2)−Δ˜f˜−R˜+12g˜ijE˜ij+n2τ}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij∣2−2(R˜+Δ˜f˜)+n2τ−τ(R˜ij+f˜ij−g˜ij2τ)E˜ij}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij−g˜ij2τ ∣2−τ(R˜ij+f˜ij−g˜ij2τ)E˜ij}(4πτ)−n2e−f˜dg˜=∫{2τ∣R˜ij+f˜ij−g˜ij2τ−E˜ij4 ∣2−τ8∣E˜∣2}(4πτ)−n2e−f˜dg˜=∫{2τ∣Rij+fij−gij2τ−Eij4 ∣2−τ8∣E∣2}(4πτ)−n2e−fdg≥−τ8(supM∣E∣)2∫M(4πτ)−n/2e−fdg. (3.7) In the fourth equation, we used the fact that ∫(∣∇˜f˜∣2−Δ˜f˜)e−f˜dg˜=∫Δ˜(e−f˜)dg˜=0. (3.8) Now we are interested in the infimum μ(g,τ)=inf∫(4πτ)−n/2e−fdg=1W(g,f,τ). (3.9) Suppose τ1<τ2 and f achieves the infimum at T−τ1. Then if the equation ∂∂tf=−Δf−R+12gijEij+n2τ+∣∇f∣2 (3.10) can be solved backwards, ∫(4πτ)−n/2e−fdg=1 (3.11) is still true for all τ∈[τ1,τ2] because ∂∂tf˜=g˜ij2∂∂tg˜ij+n2τ. (3.12) This will imply that μ(g(T−τ2),τ2)≤μ(g(T−τ1),τ1)+18∫τ1τ2τsupt=T−τ∣E∣2dτ. (3.13) In fact, the nonlinear equation (3.10) can be solved by rewriting it as a linear parabolic equation ∂∂t(e−f)=−Δ(e−f)−(−R+12gijEij+n2τ)(e−f). (3.14) 4. κ-Non-collapsing theorem The original κ-non-collapsing theorem of Perelman for Ricci flow in [9] requires the Riemannian curvature bound. However, the definition can be modified to the following version: Definition 4.1. The Riemannian metric gon Mnis said to be κ-non-collapsing relative to upper bound of scalar curvature on the scale ρif for any Bg(p,r)⊂Mwith r<ρsuch that supBg(p,r)Rg≤r−2, we have VolgBg(p,r)≥κrn. The κ-non-collapsing theorem relative to upper bound of scalar curvature for Ricci flow was proved by Perelman ([7, Section 13]). The proof can be modified to get the following theorem using the quasi-monotonicity formula (3.13) in the previous section: Theorem 4.2. Let ∂∂tgij=−2Rij+Eijbe a geometric flow on a compact manifold Mn. Then there exists a positive function κwith four variables such that if 0<ρ≤ρ0<∞, 0<T2≤t0≤T<∞and ∫0t0(t0+ρ2−t)supM∣E∣2dt<∞, (4.1)then g(t0)is κ(g(0),T,ρ0,∫0t0(t0+ρ2−t)supM∣E∣2dt)-non-collapsing relative to upper bound of scalar curvature on scale ρ. Proof Fix a cut-off function χ(s) such that χ(s)=1 when ∣s∣≤12, and χ(s)=0 when ∣s∣≥1. For any g(t0)-metric ball B(p,r) of radius r<ρ which satisfies R(x)≤r−2 for every x∈B(p,r), we can define u(x)=eL/2χ(d(x,p)r), (4.2) where L is chosen so that (4πr2)−n/2∫Mu2=1. (4.3) In particular, Vol(B(p,r))≥e−L(4π)n/2rn, (4.4) Vol(B(p,r2))≤e−L(4π)n/2rn. (4.5) By (3.13) applied to T=t0+r2, W(g(t0),−2lnu,r2)≥μ(g(t0),r2)≥μ(g(0),t0+r2)−18∫t0+r2r2τsupt=t0+r2−τ∣E∣2dτ=μ(g(0),t0+r2)−18∫0t0(t0+r2−t)(supM∣E∣2)dt≥μ0−18∫0t0(t0+ρ2−t)(supM∣E∣2)dt:=μ1, (4.6) where μ0 is the lower bound of μ(g(0),τ) when τ∈[T2,T+ρ02]. So μ1≤W(g(t0),−2lnu,r2)=∫M(4πr2)−n/2[r2(R+∣−2∇lnu∣2)−2lnu−n]u2dg(t0)=∫M(4πr2)−n/2[r2(Ru2+4∣∇u∣2)+u2(−2lnu−n)]dg(t0). (4.7) R<Cr−2 in B(p,r), −2u2lnu=−2u2(L/2+lnχ) and ∣∇u∣≤eL/2r∣χ′(d(x,p)r)∣≤CeL/2r. (4.8) So μ1≤C−L+CeLVolB(p,r)rn≤C−L+CVol(B(p,r))Vol(B(p,r2)). (4.9) Thus if Vol(B(p,r))Vol(B(p,r2))<2n, then −L≥C, so Vol(B(p,r))≥C1rn. Let ω be the volume of unit ball in the Euclidean space Rn and let κ=min(C1,ω2), then we claim that Vol(B(p,r))≥κrn. Otherwise, Vol(B(p,r))<κrn, so Vol(B(p,r))Vol(B(p,r2))≥2n, so Vol(B(p,r2))<2−nκrn=κ(r2)n. (4.10) We can apply the same thing for r2k and obtain that Vol(B(p,r2k))<κ(r2k)n, (4.11) which is a contradiction for large enough k.□ 5. Finite-time singularity Now we are ready to study the finite-time singularities of reasonable flows of G2 structures. First of all, using the method of Lotay–Wei and our Shi-type estimate, we can prove the following theorem: Theorem 5.1. If ϕ(t)is a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T), then supM(∣Rm∣2+∣T∣4+∣∇T∣2)12≥CT−t (5.1)for some constant C>0. Proof As Lotay–Wei did in [8], if supM(∣Rm∣2+∣T∣4+∣∇T∣2)12 is bounded, then all the higher order derivatives are also bounded. So ∂∂tg and ∂∂tϕ are all bounded. So they and their higher order derivatives are all bounded using the background metric g(0). So we can take the smooth limit. This will violate the short-time existence assumption. Still as Lotay–Wei, we can use Eq. (2.11) to get the required blow-up rate.□ Then we can get the following estimate: Theorem 5.2. If ϕ(t)is a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T), then ∫0TsupM(∣Ric∣+∣T∣2)dt=∞. (5.2) Proof If it is not true, then using the evolution equation, we can see that ∫0Tsup∣∂∂tg∣ is finite. So the metric is uniformly continuous, in other words, for any ϵ>0, there exists δ>0 such that for all t1,t2∈(T−δ,T), (1+ϵ)−1g(t2)≤g(t1)≤(1+ϵ)g(t2). (5.3) Now the proof of [8, Theorem 8.1] can be applied directly. In fact, according to Theorem 5.1, there exist ti→T and a sequence of points xi∈M such that Λi:=supx∈M,t∈[0,ti](∣Rm∣2+∣T∣4+∣∇T∣2) is achieved at (xi,ti). Now we can rescale the flow so that this quantity becomes 1. Then using Shi-type estimate, all of the higher order derivatives of the curvature are also bounded in any finite scale. Using the uniform continuity of the metric, the harmonic radius has a lower bound and we can take the limit of metric for a subsequence. The higher order derivatives of ϕ is also bounded, so we can also take the limit of the 3-form in a subsequence. Remark that the quantity ∫T00sup(∣Ric∣+∣T∣2)dt after rescaling is the same as the quantity ∫T−(Λi)−1/2T0TsupM(∣Ric∣+∣T∣2)dt before rescaling for any fixed T0<0. By Fatou’s lemma, in the limit, this quantity is 0. In other words, Ric=T=0 in the limit. Now the uniform continuity of the metric implies that the limit has the same volume of balls as the Euclidean space, so it must be the Euclidean space. This is a contradiction.□ As for the better estimates of Ricci curvature, scalar curvature and torsion tensor, we can prove the following theorem using the method in [11] Theorem 5.3. Let ϕ(t)be a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T). Assume that ∫0T(T−t)supM∣T∣4dt<∞, (5.4)then limsupt→T(T−t)supM(∣Ric∣+∣T∣2)>0, (5.5)and limsupt0→T[(T−t0)2supt≤t0(1+∣R∣+∣T∣2)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]>0. (5.6) Proof In this case, the flow is κ-non-collapsing on the scale T−t. Define Q(t0)=supt≤t0(∣Rm∣+∣T∣2+∣∇T∣), (5.7) and O(t0)=supt≤t0(1+∣R∣+∣T∣2)≤1+100Q(t0), (5.8) then according to Theorem 5.1, Q(t)→∞ as t→T. Therefore, there exist an increasing sequence tk→T such that Q(tk)=2k and the maximum is achieved at (pk,tk). By Theorem 5.1 again, there exists a subsequence of tk such that (T−tk)≥T−tk−13. Now if (5.5) is not true, for the subsequence ∫tk−1tk∣∂∂tg∣dt≤C(T−tk)supt≤tk(∣Ric∣+∣T∣2+1)→0. (5.9) Now using Theorems 2.1 and 4.2, we can show that as before, both (M,pk,23k2ϕ,2kg,2k(t−tk)) and (M,pk,23k2ϕ,2kg,2k(t−tk−1)) converge to the trivial product of a torsion-free G2 structure times (−∞,0]. The two limits must be isometric using the uniform continuity of metric. It is a contradiction. As for the second estimate, we need to show that when T−t0<1, supt≤t0(∣Ric∣+∣T∣2)≤CO(t0)Q(t0), (5.10) We will still follow the method in [11]. By Theorem 5.1, we see that Q(t0)≥C(T−t)−1. So the flow is κ-non-collapsing on the scale Q(t0)−12. Now we rescale the flow so that Q(t0)=1. Then the harmonic radius has a lower bound. So inside a finite size of ball, the metric and all of its higher derivatives are uniformly bounded. So (∂∂t−D)T=0 for some elliptic operator D with bounded coefficients and higher derivatives of coefficients. So after rescaling, sup∣∇kT∣≤C(k)sup∣T∣. (5.11) Now ∂∂tR=−2ΔTrh+2∇i∇jhij−2hijRij=2ΔR−2∇i∇jRij+2∣Ric∣2+L(Ric)Q(t0)+T*RicQ(t0)+T*T*Ric+L(∇2(CQ(t0)+TQ(t0)+T*T)). (5.12) The terms 2ΔR−2∇i∇jRij+2∣Ric∣2 are equal to ΔR+2∣Ric∣2 by Bianchi identity as in Ricci flow case. The terms L(Ric)Q(t0)+T*RicQ(t0)+T*T*Ric are bounded by CO(t0)Q(t0)∣Ric∣≤∣Ric∣2+CO(t0)Q(t0). The rest terms are bounded by CO(t0)Q(t0). In conclusion, ∣(∂∂t−Δ)R−2∣Ric∣2∣≤∣Ric∣2+CO(t0)Q(t0). (5.13) For any p, we can pick a cut-off function χ such that it is 0 outside Bg(t0)(p,Q(t0)−1/2)×[t0−1Q(t0),t0], (5.14) and is 1 inside Bg(t0)(p,12Q(t0)−1/2)×[t0−12Q(t0),t0]. (5.15) After rescaling, it vanishes outside Bg(0)(p,1)×[−1,0] and is 1 inside Bg(0)(p,12)×[−12,0]. Thus ∫t=0χR=∫−10(∫M∂∂t(χR))dt=∫−10∫M[R(∂∂t−Δ)χ+χ(∂∂t−Δ)R]dt. (5.16) Since the geometry is bounded, (5.13) and (5.16) imply that ∫Bg(0)(p,12)×[−12,0]∣Ric∣2≤CO(t0)Q(t0). (5.17) Now the Ricci curvature satisfies the equation ∂∂tRjk=gpq(∇q∇jhkp+∇q∇khjp−∇q∇phjk−∇j∇khqp). (5.18) As in the Ricci flow case, −gpq(∇q∇jRkp+∇q∇kRjp−∇q∇pRjk−∇j∇kRqp)=ΔRjk+Ric*Rm. (5.19) The rest term gpq(∇q∇j(h+R)kp+∇q∇k(h+R)jp−∇q∇p(h+R)jk−∇j∇k(h+R)qp) (5.20) is bounded by CO(t0)Q(t0) by (1.10). Therefore, ∣(∂∂t−D)Ric∣≤CO(t0)Q(t0) (5.21) for some elliptic operator D with bounded coefficients and higher derivatives of coefficients. Therefore, we have ∣Ric∣2≤CO(t0)Q(t0). Before rescaling, it is exactly ∣Ric∣≤CO(t0)Q(t0).□ If in addition supM(∣R∣+∣T∣2)=o(1T−t), (5.22) we can also show that any blow-up limit at finite time must be a manifold with maximal volume growth rate whose holonomy is contained in G2. Theorem 5.4. Let ϕ(t)be a solution to a reasonable flow of G2structures on a compact manifold M7in a finite maximal time interval [0,T). If ∫0T(T−t)supM∣T∣4dt<∞, (5.22)and supM(∣R∣+∣T∣2)=o(1T−t), (5.24)then there exists a sequence tk→T,pk∈Msuch that Qk=(∣Rm∣2+∣T∣4+∣∇T∣2)12(pk,tk)→∞, (5.23)and (M,Qk3/2ϕ(tk),Qkg(tk),pk)converges to a complete manifold M∞with a torsion-free G2structure (ϕ∞,g∞,p∞)such that Volg∞(Bg∞(p∞,r))≥κr7 (5.24)for some κ>0and all r>0. Proof First of all, we can see that limsupt0→T[(T−t0)supt≤t0(∣Rm∣+∣T∣2+∣∇T∣)]=∞. (5.25) So we can choose a sequence such that (T−tk)Qk→∞. After rescaling, (∣Rm∣2+∣T∣4+∣∇T∣2)12 is bounded. Moreover, sup(∣R∣+∣T∣2) converges to 0, and the manifold is κ-non-collapsing on a scale going to infinity. In particular, we get a uniform volume lower bound in any finite scale. Therefore, by our Shi-type estimate, the G2 structures converge in C∞ sense to a limit G2 structure. In the limit, both the scalar curvature and the torsion tensor are everywhere 0. In other words, the limit is torsion-free. Moreover, it has maximal volume growth rate.□ Funding The author is supported by the National Science Foundation under Grant no. 1638352, as well as support from the S.S. Chern Foundation for Mathematics Research Fund when this article is been revised. Acknowledgement The author is grateful to the valuable comments of the referee. The author also thanks the helpful discussions with Xiuxiong Chen, Jason Lotay and Chengjian Yao. References 1 R. L. Bryant , Some remarks on G2-structures. Proceedings of Gökova Geometry-Topology Conference 2005, 75-109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006. 2 R. L. Bryant and F. Xu , Laplacian Flow for Closed G2-Structures: Short Time Behavior. Preprint, arxiv:1101.2004. 3 B. Chow and D. Knopf , The Ricci flow: an introduction. Mathematical Surveys and Monographs Vol. 110 , American Mathematical Society , Providence, RI , 2004 . 4 S. Grigorian , Short-time behaviour of a modified Laplacian coflow of G2-structures , Adv. Math. 248 ( 2013 ), 378 – 415 . Google Scholar Crossref Search ADS 5 S. Karigiannis , Flows of G2-structures , I. Q. J. Math. 60 ( 2009 ), 487 – 522 . Google Scholar Crossref Search ADS 6 S. Karigiannis , B. McKay and M.-P. Tsui , Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry , Differential Geom. Appl. 30 ( 2012 ), 318 – 333 . Google Scholar Crossref Search ADS 7 B. Kleiner and J. Lott , Notes on Perelman’s papers , Geom. Topol. 12 ( 2008 ), 2587 – 2855 . Google Scholar Crossref Search ADS 8 J. D. Lotay and Y. Wei , Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness , Geom. Funct. Anal. 27 ( 2017 ), 165 – 233 . Google Scholar Crossref Search ADS 9 G. Perelman , The entropy formula for the Ricci flow and its geometric applications, Preprint. arXiv:math/0211159. 10 W.-X. Shi , Deforming the metric on complete Riemannian manifolds , J. Differential Geom. 30 ( 1989 ), 223 – 301 . Google Scholar Crossref Search ADS 11 B. Wang , On the conditions to extend Ricci flow (II) , Int. Math. Res. Not. IMRN 2012 ( 2012 ), 3192 – 3223 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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The Quarterly Journal of MathematicsOxford University Press

Published: Sep 1, 2018

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