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LAPLACIAN FLOW FOR CLOSED G2 STRUCTURES: REAL ANALYTICITY JASON D.LOTAYAND YONGWEI Abstract. Let ϕ(t),t∈[0,T ] be a solution to the Laplacian flow for 6 0 closed G structures on a compact 7-manifold M. We show that for 1 2 0 each fixed time t∈(0,T0], (M,ϕ(t),g(t)) is real analytic, where g(t) is 2 themetric inducedby ϕ(t). Consequently,any Laplacian soliton is real analytic and we obtain uniquecontinuation results for the flow. n a J 7 1 Contents ] 1. Introduction 1 G 2. Preliminaries 4 D 3. Criterion for a G structure to be real analytic 6 . 2 h 4. Laplacian flow and evolution equations 9 t a 5. Global real analyticity 12 m 6. Local real analyticity 24 [ References 28 1 v 8 5 1. Introduction 2 4 Let M be a compact 7-manifold and let ϕ be a closed G structure on 0 2 0 M. We consider solutions ϕ(t),t [0,T ], to the Laplacian flow for closed . ∈ 0 1 G structures: 2 0 ∂ ϕ = ∆ ϕ, 6 ∂t ϕ 1 dϕ = 0, (1.1)  : ϕ(0) = ϕ , v  0 i where ∆ ϕ = dd ϕ+d dϕ is the Hodge Laplacian of ϕ(t) with respect to X ϕ ∗ ∗  the metric g(t) determined by ϕ(t). r a The flow (1.1) was introduced by Bryant (see [2, 5]) as a potential way § to study the challenging problem of existence of torsion-free G structures 2 and thus Ricci-flat metrics with exceptional holonomy G , since stationary 2 points of the flow are the G structures ϕ satisfying dϕ = d ϕ = 0, which 2 ∗ is the torsion-free condition. (Although this statement about the stationary points is true for compact manifolds by integration by parts, we gave an alternative argument in [13] which shows that stationary points of the flow arealwaystorsion-free,eveninthenon-compactsetting.) Moreover, theflow moveswithinthecohomology classofϕ andhasavariationalinterpretation 0 2010 Mathematics Subject Classification. 53C44, 53C25, 53C10. Key words and phrases. Laplacian flow, G structure,real analyticity. 2 This research was supported by EPSRCgrant EP/K010980/1. 1 2 JASOND.LOTAYANDYONGWEI due to Hitchin [4,9]. The primary goal in the field is to find conditions on an initial closed G structure ϕ such that the flow (1.1) will exist for all 2 0 time and converge to a torsion-free G structure. Situations under which 2 this occurs were proved by the authors in [14]. AsM iscompact,theLaplacianflowstartingfromanyclosedG structure 2 ϕ is guaranteed to have a unique solution ϕ(t) for a short time t [0,ǫ), 0 ∈ whereǫdependsonϕ (see[2,4]). Inourpreviouspapers[13,14], westudied 0 various foundational analytical and geometric properties of the flow (1.1), including Shi-type derivative estimates, uniqueness theorems, compactness results, soliton solutions, long-time existence results and stability of torsion- free G structures along the flow. 2 On the face of it these analytic results are somewhat surprising because the velocity of the flow (1.1) is defined by the Hodge Laplacian, which we would usually think of as a positive operator, and thus the flow appears to look like a backwards heat equation. In spite of this, the Laplacian flow is actually weakly parabolic in a certain non-standard sense: it is parabolic in the direction of closed forms, modulo the action of diffeomorphisms. It is this fact that enables the analysis of the flow to proceed. The reader is referred to [13,14] for more detailed information about the Laplacian flow. In this paper, we continue to analyze the Laplacian flow (1.1) and inves- tigate the regularity of the solution ϕ(t) for each positive time t. Our main result is the following. Theorem 1.1. If ϕ(t),t [0,T ] is a smooth solution to the Laplacian flow 0 ∈ (1.1) for closed G structures on a compact 7-manifold M, then for each 2 time t (0,T ], (M,ϕ(t),g(t)) is real analytic. 0 ∈ Readersarereferredto 3forthedefinitionandcriterionforaG structure 2 § to be real analytic. Real analyticity for positive times is well known for linearparabolicPDE(suchastheheatequation)andsomeweakly parabolic nonlinear PDE (such as Ricci flow [1]). However, as we have indicated, the Laplacian flow is not weakly parabolic in a standard manner, and so one should not immediately expect such a regularity result. By using a cut-off function to localize the discussion, we also have the following local version of real analyticity. Theorem 1.2. If ϕ(t),t [0,T ] is a smooth solution to the Laplacian flow 0 ∈ (1.1) for closed G structures on an open set U M, then for each time 2 ⊂ t (0,T ], (U,ϕ(t),g(t)) is real analytic. 0 ∈ Since any Laplacian soliton corresponds to a local self-similar solution to the Laplacian flow (1.1), we have the following corollary. Corollary 1.3. Suppose (M,ϕ,X,λ) is a Laplacian soliton (not necessarily compact), i.e., dϕ = 0 and ∆ ϕ = λϕ+ ϕ (1.2) ϕ X L for some smooth vector field X and constant λ. Then (M,ϕ) is real analytic. The real analyticity of a torsion-free G structure, i.e., the case X = 0, 2 λ = 0 in (1.2), is already well-known (see [3] for example). Moreover, real analyticity plays a significant role in G geometry, as can be seen in [3]. 2 LAPLACIAN FLOW FOR CLOSED G2 STRUCTURES 3 Forconvenience wesayaG structureϕonM iscompleteifitsassociated 2 metric is complete. By modifying the argument in the proof of [12, Corol- lary 6.4, p.256], Theorems 1.1–1.2 immediately imply the following unique continuation results. Corollary 1.4. Suppose that M7 is compact, connected and simply connected, and ϕ(t), ϕ˜(t) are smooth complete solutions to the Laplacian flow (1.1) on M [0,T ]. Then, for any t (0,T ], the following hold. 0 0 × ∈ (a) If ϕ(t) = ϕ˜(t) on some open set U M, then there exists a diffeo- ⊂ morphism F of M such that F ϕ˜(t) ϕ(t). ∗ ≡ (b) Any local diffeomorphism F : U V between connected open sets → U,V M such that F (ϕ ) = ϕ can be uniquely extended to a ∗ V U ⊂ | | global diffeomorphism F of M with F ϕ = ϕ. ∗ Corollary 1.5. Suppose that M7 is connected and simply-connected and (ϕ,X,λ) and (ϕ˜,X˜,λ˜) are complete Laplacian solitons on M. If ϕ = ϕ˜ on some connected open set U M, then there exists a diffeomorphism F of ⊂ M such that F ϕ˜ ϕ. ∗ ≡ SinceaG structureϕdeterminesauniquemetricg ,anydiffeomorphism 2 ϕ F : (M,ϕ) (M,ϕ˜) such that F ϕ˜ = ϕ is an isometry between (M,g ) ∗ ϕ → and (M,g ). The converse is clearly not always true, since the G structure ϕ˜ 2 encodes strictly more information than the metric. Our approach to prove Theorems 1.1 and 1.2 is similar to Bando’s [1] proof of the real analyticity of Ricci flow, namely to use derivative estimates for the Riemann curvature tensor Rm, the torsion tensor T and ϕ along the flow. In our previous paper [13], we derived Shi-type derivative estimates along the Laplacian flow, which take the form tk2 kRm(x,t) + k+1T(x,t) CkK, x M, t [0,1/K], (1.3) |∇ | |∇ | ≤ ∈ ∈ (cid:16) (cid:17) where C is a constant depending on the order k and K is the bound on k Λ(x,t) = (Rm 2(x,t)+ T 2(x,t))1/2. (1.4) | | |∇ | However, in [13], we do not analyze how C depends on k, which is partic- k ularly relevant when k is large. Whenoneappliestheheatoperatorto kRm(x,t) + k+1T(x,t),lower |∇ | |∇ | order terms are generated duringthe computation, and the number of these terms grows with the order k of differentiation, which then contributes to the growth of the constants C . By showing that the C are of sufficiently k k slow growth in the order k, we may deduce that the G structure ϕ(t) and 2 associated metric g(t) are real analytic at each fixed time t > 0. The key step is to revisit the derivation of the derivative estimates (1.3) from [13] and obtain the following much more refined estimates: n tk kRm 2(x,t)+ k+2ϕ2(x,t) C(T ,K ) (1.5) (k+1)!2 |∇ | |∇ | ≤ 0 0 Xk=0 (cid:16) (cid:17) on M [0,α/K ] for all n N (we assume N to include 0), where K = 0 0 × ∈ sup Λ(x,0), α,C(T ,K )areconstants. As wewillseein 3, theestimate M | | 0 0 § (1.5) leads to the real analyticity of (M,ϕ(t),g(t)) for each time t > 0. 4 JASOND.LOTAYANDYONGWEI 2. Preliminaries We collect some facts on closed G structures, mainly based on [2,10,13]. 2 Let e ,e , ,e be the standard basis of R7 and let e1,e2, ,e7 be 1 2 7 { ··· } { ··· } its dual basis. For simplicity we write eijk = ei ej ek and define a 3-form ∧ ∧ φ by: φ = e123+e145 +e167+e246 e257 e347 e356. − − − The subgroup of GL(7,R) fixing φ is the exceptional Lie group G , which 2 is a compact, connected, simple Lie subgroup of SO(7) of dimension 14. It is well-known that G acts irreducibly on R7 and preserves the metric and 2 orientation for which e ,e , ,e is an oriented orthonormal basis. 1 2 7 { ··· } Let M be a 7-manifold. For x M we let ∈ Λ3(M) = ϕ Λ3T M u Hom(T M,R7),u φ= ϕ . + x { x ∈ x∗ |∃ ∈ x ∗ x} The bundle Λ3(M) = Λ3(M) is an open subbundle of Λ3T M. We + x + x ∗ call a section ϕ of Λ3(M) a G structure on M and denote the space of G + F 2 2 structuresonM byΩ3(M). Thenotationismotivatedbythefactthatthere + is a 1-1 correspondence between G structures in the sense of subbundles 2 of the frame bundle and Ω3(M). The bundle Λ3(M) has sections, which + + means that G structures exist, if and only if M is oriented and spin. 2 A G structure ϕ induces a unique metric g and orientation (given by a 2 volume form vol of g) which satisfy g 1 g(u,v)vol = (uyϕ) (vyϕ) ϕ. g 6 ∧ ∧ Themetric andorientation determinetheHodge staroperator , sowecan ϕ ∗ define ψ = ϕ. Notice that the relationship between g and ϕ, and hence ϕ ∗ between ψ and ϕ, is nonlinear. Although G acts irreduciblyon R7 (and hence on Λ1(R7) and Λ6(R7) ), 2 ∗ ∗ it acts reducibly on Λk(R7) for 2 k 5. Hence a G structure ϕ induces ∗ 2 ≤ ≤ splittings of the bundles ΛkT M (2 k 5), which we denote by Λk(T M) ∗ ≤ ≤ l ∗ so that l indicates the rank of the bundle, and we let the space of sections of Λk(T M) be Ωk(M). Explicitly, we have that l ∗ l Ω2(M) =Ω2(M) Ω2 (M) and Ω3(M) = Ω3(M) Ω3(M) Ω3 (M), 7 ⊕ 14 1 ⊕ 7 ⊕ 27 where (using the orientation in [2] rather than [10]) Ω27(M) = {β ∈ Ω2(M)|β ∧ϕ = 2∗ϕ β} ={Xyϕ|X ∈C∞(TM)}, Ω2 (M) = β Ω2(M) β ϕ = β = β Ω2(M) β ψ = 0 , 14 { ∈ | ∧ −∗ϕ } { ∈ | ∧ } and Ω31(M) = {fϕ|f ∈ C∞(M)}, Ω37(M) = {Xyψ|X ∈ C∞(TM)}, Ω3 (M) = γ Ω3(M) γ ϕ = 0 = γ ψ . 27 { ∈ | ∧ ∧ } Hodge duality gives corresponding decompositions of Ω4(M) and Ω5(M). In our study it is convenient to write key quantities with respect to local coordinates x1, ,x7 on M. We write a k-form α locally as { ··· } 1 α = α dxi1 dxik, k! i1i2···ik ∧···∧ LAPLACIAN FLOW FOR CLOSED G2 STRUCTURES 5 where α is totally skew-symmetric in its indices. In particular, we i1i2···ik write ϕ and ψ locally as 1 1 ϕ = ϕ dxi dxj dxk, ψ = ψ dxi dxj dxk dxl. ijk ijkl 6 ∧ ∧ 24 ∧ ∧ ∧ As in [2] (up to a constant factor), we define an operator i : S2T M ϕ ∗ → Λ3T M locally by ∗ (i (h)) = hlϕ hlϕ hlϕ ϕ ijk i ljk − j lik − k lji where h = h dxidxj. Then Λ3 (T M) = i (S2T M), where S2T M de- ij 27 ∗ ϕ 0 ∗ 0 ∗ notes the bundle of trace-free symmetric 2-tensors on M, and i (g) = 3ϕ. ϕ We have contraction identities for ϕ and ψ in index notation (see [2,10]): ϕ ϕ giagjb = 6g , (2.1) ijk abl kl ϕ ψ giagjb = 4ϕ , (2.2) ijq abkl qkl ϕ ϕ gia = g g g g +ψ , (2.3) ipq ajk pj qk pk qj pqjk − ϕ ψ gia = g ϕ g ϕ +g ϕ ipq ajkl pj qkl jq pkl pk jql − g ϕ +g ϕ g ϕ . (2.4) kq jpl pl jkq lq jkp − − Given any G structure ϕ Ω3(M), there exist unique differential forms 2 ∈ + (called the intrinsic torsion forms) τ Ω0(M),τ Ω1(M),τ Ω2 (M) 0 ∈ 1 ∈ 2 ∈ 14 and τ Ω3 (M) such that dϕ and dψ can be expressed as follows (see [2]): 3 ∈ 27 dϕ = τ ψ+3τ ϕ+ τ and dψ = 4τ ψ+τ ϕ. 0 1 ϕ 3 1 2 ∧ ∗ ∧ ∧ We shall only consider closed G structures ϕ in this article. In this case 2 dϕ = 0 forces τ = τ = τ = 0, and hence the only non-zero torsion form is 0 1 3 τ . We therefore from now on set τ = τ Ω2 (M) and reiterate that 2 2 ∈ 14 dϕ = 0 and dψ = τ ϕ= τ. ϕ ∧ −∗ We see immediately that d τ = d τ = 0, ∗ ϕ ϕ ∗ ∗ which is given in local coordinates by gmi τ = 0. m ij ∇ The full torsion tensor is a 2-tensor T satisfying (see [10]) 1 ϕ = T mψ , T j = ϕ ψjlmn, (2.5) ∇i jkl i mjkl i 24∇i lmn ψ = T ϕ T ϕ T ϕ T ϕ , (2.6) m ijkl mi jkl mj ikl mk jil ml jki ∇ − − − − (cid:16) (cid:17) whereT =T(∂ ,∂ ) andT j = T gjk. Inour setting we may computethat ij i j i ik 1 T = τ, −2 so T is divergence-free as d τ = 0. ∗ Given these formulae we can compute the Hodge Laplacian of ϕ, which is the velocity of the Laplacian flow, as in [2,13]. Proposition 2.1. For a closed G structure ϕ, the Hodge Laplacian of ϕ 2 satisfies ∆ ϕ= dτ = i (h) Ω3(M) Ω3 (M), ϕ ϕ ∈ 1 ⊕ 27 6 JASOND.LOTAYANDYONGWEI where h is a symmetric 2-tensor on M, locally given by 1 h = T ϕmn T 2g T lT . (2.7) ij −∇m ni j − 3| | ij − i lj Since ϕ determines a unique metric g on M, we then have the Riemann curvature tensor Rm of g on M, which in our convention is given by Rm(X,Y,Z,W) = g( Z Z Z,W) X Y Y X [X,Y] ∇ ∇ −∇ ∇ −∇ for vector fields X,Y,Z,W on M. In local coordinates, we denote the components of Rm by R = Rm(∂ ,∂ ,∂ ,∂ ). The Ricci curvature Rc and ijkl i j k l scalar curvature R are given locally by R = gjlR and R = gijR , and ik ijkl ij may be computed in terms of the torsion tensor as follows (see e.g. [13]). Proposition 2.2. The Ricci tensor and the scalar curvature of the associated metric g of a closed G structure ϕ are given as 2 R = T ϕ jl T jT and R = T 2 = T T gijgkl. (2.8) ik ∇j li k − i jk −| | − ik jl Notice that Rm and T are both second order in ϕ, and T is essentially ∇ ϕ, so we might expect Rm and T to be related. The next proposition ∇ ∇ from [13] says that T can be expressed using T and Rm. ∇ Proposition 2.3. For a closed G structure ϕ, we have 2 1 1 1 2 T = R ϕ mn + R ϕ mn R ϕ mn ∇i jk 2 ijmn k 2 kjmn i − 2 ikmn j T T ϕ mn T T ϕ mn +T T ϕ mn. − im jn k − km jn i im kn j 3. Criterion for a G structure to be real analytic 2 Given a 7-manifold M, a real analytic structure on M is an atlas (U , xi 7 ) , j { j}i=1 j J ∈ where J is some indexing s(cid:8)et, such that(cid:9)the transition functions are real analytic. A Riemannian metric g on a real analytic manifold M is then real analytic if the components g of g are real analytic functions with respect ij to a subatlas of real analytic coordinates. We can provide an alternative criterion for when a Riemannian manifold is real analytic as follows (see [6, Lemma 13.20], [7]). Lemma 3.1. Let (M,g) be a Riemannian manifold. If there exists a subatlas of normal coordinate systems such that the components g of g are ij real analytic functions with respect to each normal coordinate system in the subatlas, then (M,g) is a real analytic Riemannian manifold with respect to this subatlas. Let M be an orientable and spinnable 7-manifold, let ϕ be a G structure 2 on M and let g be its associated Riemannian metric. Suppose further that there is a subatlas of normal coordinate systems on M such that the components of g are real analytic functions in each of these coordinate systems. From Lemma 3.1, (M,g) is then a real analytic Riemannian manifold with respect to this subatlas and, in particular, an atlas for M can be found with real analytic transition functions. In fact, by [7, Lemma 1.2 & Theorem 2.1], for such (M,g) there exists an atlas of harmonic coordinates which are LAPLACIAN FLOW FOR CLOSED G2 STRUCTURES 7 real analytic functions of the normal coordinates and so that the metric g is real analytic in these harmonic coordinates. Real analyticity of the transition functions for the atlas of harmonic coordinates then follows from the fact that the coordinates are harmonic and the metric is real analytic. If in addition the components ϕ of ϕ are real analytic with respect to the nor- ijk mal coordinates, which implies that ϕ is also real analytic in the harmonic coordinates by [7, Corollary 1.4], then we say that (M,ϕ) is real analytic. Let inj (p) denote the injectivity radius of g at p M, and if xi 7 are g ∈ { }i=1 coordinatescentredatpwedenotetheChristoffelsymbolsoftheLevi-Civita connection of g by Γl as usual and let ∂k = ∂k . ij k1+...+kn=k (∂x1)k1 (∂xn)kn With this notation in hand, we have the following derivative estim·a··tes of ϕ P and g in normal coordinates. Lemma 3.2. Let ϕ be a G structure on M and let g be its associated 2 metric. Let p M and suppose there exist constants C and r > 0 such that 1 ∈ kRm (x)+ k+2ϕ(x) C k!r k 2 (3.1) 1 − − |∇ | |∇ | ≤ in a geodesic ball B(p,r) for all k N. ∈ There exist constants C ,C ,C ,r = r (r),r = r (r) such that if we 2 3 4 1 1 2 2 set ρ = min C2 r,inj (p) then, for all x B(p,ρ) and k N we have in {√C1 g } ∈ ∈ normal coordinates centred at p: 1 δ g (x) 2δ , ∂kg (x) C k!r k (3.2) 2 ij ≤ ij ≤ ij | ij| ≤ 3 1− ∂kΓl (x) C k!r k 1, ∂kϕ (x) C k!r k. (3.3) | ij| ≤ 3 1− − | ijl| ≤ 4 2− Proof. The assumption (3.1) implies kRm (x) C k!r k 2 in B(p,r). 1 − − |∇ | ≤ The proof of [8, Corollary 4.12] (see also [6, Lemma 13.31]) gives the existence of constants C ,C ,r = r (r) > 0 such that for any x B(p,ρ), 2 3 1 1 ∈ whereρisas stated, wehavethederivative estimates forg andΓl in(3.2)- ij ij (3.3) for all k N. Thus it remains to show that, under the assumption (3.1), there are∈constants C and r (r) > 0 such that for all k N we have 4 2 ∈ ∂kϕ (x) C k!r k. | ijl| ≤ 4 2− In the following, we will prove a slightly stronger estimate: ∂l k lϕ(x) C k!2 (k l)r k (3.4) | ∇ − | ≤ 4 − − 2− for all 0 l k m, where k,l,m N. We prove (3.4) by induction on ≤ ≤ ≤ ∈ m. The case m = 0 of (3.4) is trivial as ϕ2 = 7. Suppose now that m > 1 | | and (3.4) holds for all 0 l k m 1. We therefore only need to deal ≤ ≤ ≤ − with the case where k = m and we can perform an induction on l. Again, the case k = m,l = 0 is trivial if we take r r/2, as the condition (3.1) 2 ≤ gives that mϕ C (m 2)!r m C m!r m C m!2 mr m. |∇ | ≤ 1 − − ≤ 1 − ≤ 1 − 2− 8 JASOND.LOTAYANDYONGWEI So we now suppose that (3.4) holds for all 0 l < s for some s k = m ≤ ≤ and consider the case l = s. Since (k s)ϕ is a (k s+3)-tensor, we have − ∇ − ∂s (k s)ϕ(x) = ∂(s 1) (k s+1)ϕ(x)+(k s+3)Γ(x) (k s)ϕ(x) − − − − | ∇ | ∇ − ∗∇ (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) ∂(s 1) (k s+1(cid:12))ϕ(x) (cid:12) ≤ − ∇ − (cid:12) (cid:12) (cid:12) s 1 (cid:12) (cid:12) − (cid:12)s 1 (cid:12) +(k s+3) (cid:12) − ∂iΓ(x) ∂(s−1−i) (k−s)ϕ(x) − i ∇ Xi=0(cid:18) (cid:19)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) ≤ C4k!2−(k−s+1)r2−k (cid:12) (cid:12)(cid:12) (cid:12) s 1 +(k−s+3)2−(k−s)C3C4 − s−i 1 i!(k−1−i)!r1−i−1r2−(k−i−1) i=0(cid:18) (cid:19) X 1 s−1 (s 1)! (k 1 i)!r1+i ≤ C4k!2−(k−s)r2−k 2 +(k−s+3)C3 (s −1 i)! −k!− r21+i (cid:18) i=0 − − 1 (cid:19) X I To estimate the term|I in the bracket above,{bzy choosing r2 r1 we h}ave ≤ s 1 I 1 +C k−s+3 −i r2 1 +4C r2, ≤ 2 3 k k 1 r ≤ 2 3r (cid:0) −i (cid:1) 1 1 as k 1. Thus we can choose (cid:0) (cid:1) ≥ r r 1 r = r (r)= min ,r , > 0 2 2 1 8C 2 (cid:26) 3 (cid:27) such that I 1 and then ≤ ∂s (k s)ϕ(x) C k!2 (k s)r k. | ∇ − | ≤ 4 − − 2− This completes the induction. (cid:3) For any smooth function f on M, to show f is real analytic in B(p,r) with respect to normal coordinates xi centered at p, we need to show that { } for any a B(p,r), the Taylor series of f at a ∈ Taf(x)= ∞ (x1−a1)k1···(x7−a7)k7 ∂kf(a) ∞ k1! k7! ∂xk1 ∂xk7 Xk=0k1+·X··+k7=k ··· 1 ··· 7 converges to f(x) for x sufficiently close to a. Note that the remainder for the m-th truncated Taylor series is (x a )k1 (x a )k7 ∂m+1f Ra f(x)= 1− 1 ··· 7− 7 (a+s (x a)) m k1! k7! ∂xk1 ∂xk7 0 − k1+···+Xk7=m+1 ··· 1 ··· 7 for some s (0,1). So it also suffices to show Ra f(x) converges to 0 0 ∈ m uniformly as m for x sufficiently close to a. If the G structure ϕ and 2 → ∞ its associated metric g satisfies the following derivative estimates ∂kg (x) Ck!r k, ∂kϕ (x) Ck!r k ij − ijl − | | ≤ | | ≤ LAPLACIAN FLOW FOR CLOSED G2 STRUCTURES 9 for x B(p,r), with respect to the normal coordinates xi centered at p, ∈ { } then for any x sufficiently close to a (for example, for x B(a, r )), we have ∈ 14 r m+1 (m+1)! Ra g (x) C r (m+1) | m ij | ≤ 14 k ! k ! − 1 7 k1+···+Xk7=m+1(cid:16) (cid:17) ··· m+1 1 (m+1)! C = C = , 14 k ! k ! 2m+1 (cid:18) (cid:19) k1+···+Xk7=m+1 1 ··· 7 which converges uniformly to 0 as m . Thus the coefficients g of g are ij → ∞ real analytic in B(p,r) with respect to normal coordinates. Similarly, the coefficents ϕ of ϕ are also real analytic in B(p,r) with respect to normal ijl coordinates. In summary, if we have the derivative estimates (3.1) for Rm and ϕ, then we can conclude that (M,ϕ,g) is real analytic. 4. Laplacian flow and evolution equations The goal of this paper is to prove the real analyticity of the solution to the Laplacian flow (1.1). From Proposition 2.1, (1.1) is equivalent to ∂ ϕ(t) = i (h(t)), (4.1) ϕ ∂t where h(t) is the symmetric 2-tensor on M given locally in (2.7). By (2.8), we can also write h locally as 1 h = R T 2g 2T kT . (4.2) ij − ij − 3| | ij − i kj Notice that T k = T gkl and T = T . i il il − li Throughout the remainder of the article we will use the symbol ∆ to denote the “analyst’s Laplacian”, which is a non-positive operator given in local coordinates as i , in contrast to the Hodge Laplacian ∆ . i ϕ ∇ ∇ Under (4.1), the associated metric g(t) of ϕ(t) evolves by ∂ g(t) = 2h(t). (4.3) ∂t Substituting (4.2) into this equation, we have that ∂ 1 g = 2(R + T 2g +2T kT ). (4.4) ∂t ij − ij 3| | ij i kj Moreover, by (4.4), the inverse of the metric evolves by ∂ 1 gij = 2hij = 2gikgjl(R + T 2g +2T mT ). (4.5) ∂t − kl 3| | kl k ml The next lemma describes the evolution equations of the torsion tensor T, ϕ and the curvature tensor Rm along the Laplacian flow. Here, and for∇the rest of the article, if A,B are tensors and k N, then A B denotes ∈ ∗ a contraction of tensors A,B using only the metric g (which is covariant constant) and we write a tensor S / kA B1 if S is equal to the sum of at ∗ most k terms of the form A B. ∗ 1Note that theinequality “/” can be differentiated, i.e., ∇S /k∇A∗B+kA∗∇B, unliketheusual inequality “≤” case. 10 JASOND.LOTAYANDYONGWEI Lemma 4.1. Suppose that ϕ(t),t [0,T ] is a solution to the Laplacian 0 ∈ flow (1.1) on a compact manifold M. The evolution equations of the torsion tensor T, ϕ and the curvature tensor Rm satisfy the following estimates: ∇ ∂ ∆ T / 8Rm T +Rm ϕ+11 T T ϕ+4T T T; ∂t − ∗ ∗∇ ∇ ∗ ∗ ∗ ∗ (cid:18) (cid:19) (4.6) ∂ ∆ ϕ / 62 T T ϕ+6 T ϕ ϕ+29Rm ϕ+Rm T ∂t − ∇ ∇ ∗ ∗ ∇ ∗∇ ∗ ∗∇ ∗ (cid:18) (cid:19) +Rm ϕ (T ϕ+ ϕ ϕ)+24T T ϕ (4.7) ∗ ∗ ∗ ∇ ∗ ∗ ∗∇ and ∂ ∆ Rm / 33Rm Rm+4Rm T T +35 ( T T). (4.8) ∂t − ∗ ∗ ∗ ∇ ∇ ∗ (cid:18) (cid:19) Proof. Theestimates (4.6)and(4.8)follow directly fromtheevolution equations of T and Rm along the Laplacian flow, which have been derived in [13, 3]. To show (4.7), recall that ϕ and T are related by § ∇ ϕ = T gmnψ . i jkl im njkl ∇ Then we have ∂ ∂ ∂ ∂ ϕ = T gmnψ +T gmn ψ +T gmn ψ i jkl im njkl im njkl im njkl ∂t∇ ∂t ∂t ∂t (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) = I +II +III. (4.9) For the first term I, recall that from [13, 3.2], we have § ∂ 1 1 T =∆T +RkT + R Tmk + R k ϕ pm ∂t ij ij i kj 2 ijmk 2 mpi ∇k j 1 + T ϕ pqϕmn+ T 2ϕ m ∇p qi∇m n j 3∇m| | ij 1 + (T kT )ϕmn T k T ϕpq T 2T T kT mT . ∇m i kn j − i ∇p qk j − 3| | ij − i k mj Then ∂ I = T gmnψ im njkl ∂t (cid:18) (cid:19) 1 = ∆T gmnψ +R pT gmnψ + R Tpqgmnψ im njkl i pm njkl 2 impq njkl 1 + R s ϕ qpgmnψ + T ϕ pqϕstgmnψ 2 pqi ∇s m njkl ∇p qi∇s t m njkl 1 + T 2ϕ pgmnψ + (T sT )ϕ pqgmnψ 3∇p| | im njkl ∇p i sq m njkl 1 T s T ϕ pqgmnψ T 2T gmnψ − i ∇p qs m njkl− 3| | im njkl T pT qT gmnψ . (4.10) − i p qm njkl Using (2.6), ∆ ϕ = ∆(T gmnψ ) i jkl im njkl ∇ =(∆T )gmnψ +T gmn∆ψ +2( T )gmngpq ψ im njkl im njkl p im q njkl ∇ ∇