On short time existence of Lagrangian mean curvature flow

We consider a short time existence problem motivated by a conjecture of Joyce. Specifically we prove that given any compact Lagrangian $L\subset \mathbb{C}^n$ with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains $L$ as $t \searrow 0$ as varifolds, and smoothly locally away from the singularities.


Introduction
A long-standing open problem in the study of Calabi-Yau manifolds is whether given a Lagrangian submanifold, one can find a special Lagrangian in its homology or Hamiltonian isotopy class. Special Lagrangians are always area minimising, so one way to approach the existence problem is to try to minimise area among all Lagrangians in a given class. This minimisation problem turns out to be very subtle and fraught with difficulties. Indeed Schoen-Wolfson [13] showed that when the real dimension is 4, given a particular class one can find a Lagrangian minimising area among Lagrangians in that class, but that the minimiser need not be a special Lagrangian. Later Wolfson [18] found a K3 surface and a Lagrangian sphere in this surface such that the area minimiser among Lagrangians in the homology class of the sphere, is not special Lagrangian, and the area minimiser in the class is not Lagrangian. An alternative way of approaching the problem is to consider mean curvature flow. Mean curvature flow is a geometric evolution of submanifolds where the velocity at any point is given by the mean curvature vector. This can also be seen as the gradient descent for the area functional. Smoczyk showed in [14] that the Lagrangian condition is preserved by mean curvature flow if the ambient space is Kähler-Einstein, and consequently mean curvature flow has been proposed as a means of constructing special Lagrangians. In order to flow to a special Lagrangian, one would need to show that the flow exists for all time. This however can't be expected in general, as finite time singularities abound. See for example Neves [12]. For a nice overview on what is known about singularities of Lagrangian mean curvature flow, we refer the reader to the survey paper of Neves [11]. A natural question is whether it might be possible to continue the flow in a weaker sense once a singularity develops and, in doing so, to push through the singularity. Since all special Lagrangians are zero-Maslov class, and the Maslov class is preserved by Lagrangian mean curvature flow, of particular interest is the mean curvature flow of zero-Maslov class Lagrangians. In this case, the structure of singularities is relatively well understood. Indeed Neves [10] has shown that a singularity of zero-Maslov class Lagrangian mean curvature flow must be asymptotic to a union of special Lagrangian cones. We note that in C 2 every such union is simply a union of Lagrangian planes, and so the case we consider in the below theorem is not necessarily overly restrictive. In this paper we consider the simplest such singularity, namely that where the singularities are each asymptotic to the union of two nonarea-minimising, transversally intersecting Lagrangian planes. Specifically we prove the following theorem which serves as a partial answer to Problem 3.14 in [8].
Theorem (Short-time Existence). Suppose that L ⊂ C n is a compact Lagrangian submanifold of C n with a finite number of singularities, each of which is asymptotic to a pair of transversally intersecting planes P 1 + P 2 where neither P 1 + P 2 nor P 1 − P 2 are area minimizing. Then there exists T > 0 and a Lagrangian mean curvature flow (L t ) 0<t<T such that as t ց 0, L t → L as varifolds and in C ∞ loc away from the singularities.
We remark that the assumptions L ⊂ C n and L compact are made to simplify the analysis in the sequel, however since the analysis is all of an entirely local nature we may relax this to L ⊂ M for some Calabi-Yau manifold M, and to L non-compact provided, in the latter case, that we impose suitable conditions at infinity. In the one-dimensional case all curves are Lagrangian. Ilmanen-Neves-Schulze considered the flow of planar networks, that is finite unions of embedded line segments of non-zero length meeting only at their endpoints, in [5]. They showed that there exists a flow of regular networks, that is networks where at any meeting point exactly three line segments come together at angles of 2π/3, starting at any initial non-regular network. To do so they performed a gluing procedure to get an approximating family of regular initial conditions, and proved uniform estimates on the corresponding flows, allowing them to pass to a limit of flows to prove the result. The proof here is based heavily on their arguments, and many of the calculations we do are similar to those in that paper. To prove the short-time existence, we construct a smooth approximating family L s of initial conditions via a surgery procedure. Specifically we take a singularity asymptotic to some non-area-minimising pair of planes P 1 + P 2 , cut it out and glue in a piece of the Lagrangian self-expander asymptotic to those planes at a scale determined by s. For full details see Section 7. Each of these approximating Lagrangians is smooth, and hence standard short time existence theory gives a smooth Lagrangian mean curvature flow L s t corresponding to each s. As s → 0 the curvature of L s blows up so the existence time of the flows L s t guaranteed by the standard short time existence theory goes to zero. Instead we are able to prove uniform estimates on the Gaussian density ratios of L s t , which combined with the local regularity result of Brian White [17] provides uniform curvature estimates, interior in time, on the flows L s t , from which we obtain a uniform time of existence allowing us to pass to a limit of flows and prove the main result. There are two key components in the proof of the estimates on the Gaussian density ratios. The first is a stability result for self-expanding solutions to Lagrangian mean curvature flow. More specifically we show that if a Lagrangian is weakly close to a Lagrangian self expander in an L 2 sense, then it is close in a stronger C 1,α sense. The proof of this stability result depends crucially on a uniqueness result for zero-Maslov smooth self-expanders asymptotic to transverse pairs of planes due to Lotay-Neves [9] and Imagi-Joyce-Oliveira dos Santos [6]. The second component is a monotonicity formula for the self-expander equation, which allows us to show that the approximating family of initial conditions that we construct in the proof, which are self-expanders in a ball, remain weakly close to the self-expander for a short time. The combination of these results tells us that the evolution of the approximating flows is close to the evolution of the self-expander near the singularity. Since self-expanders move by dilation, we have good curvature control on the selfexpander, and hence estimates on the Gaussian densities of the approximating flow. Organisation. The paper is organised as follows. In Section 2 we recall key definitions and results. In Section 3 we derive evolution equations and monotonicity formulas for geometric quantities under the flow. In Section 4 we prove the Stability result mentioned above. Section 5 contains the proof of the main theorem which gives uniform estimates on the Gaussian density ratios of the approximating family near the singularity. From this we get uniform estimates, interior in time, on the curvature of the approximating family which allows us to appeal to a compactness argument. Section 6 contains the proof of the short time existence result itself. Section 7 details the construction of the approximating family used in the proof of the main theorem. Finally the appendix, Section 8, contains miscellaneous technical results, including Ecker-Huisken style curvature estimates for high-codimension mean curvature flow. Acknowledgements. Both authors would like to thank Jason Lotay and Felix Schulze for their help, guidance and feedback.

Preliminaries
2.1. Mean Curvature Flow. Let M n ⊂ R n+k be an n-dimensional embedded submanifold of R n+k . A mean curvature flow is a one parameter family of immersions F : M × [0, T ) → R n+k such that the normal velocity at any point is given by the mean curvature vector, that is dF dt = H.
Of particular interest to us are self-expanders. These are submanifolds M ⊂ R n+k satisfying the elliptic equation where (·) ⊥ is the projection to the normal space. In this case one can show that M t = √ 2tM is a solution of mean curvature flow. A fundamental tool in the analysis of mean curvature flow is the Gaussian density. We first define the backwards heat kernel ρ (x 0 ,t 0 ) as follows Next, for a mean curvature flow (M t ) 0≤t<T ) we define the Gaussian density ratio centred at (x 0 , t 0 ) and at scale r by this is defined for 0 < t 0 ≤ T , 0 < r ≤ √ t 0 and any x 0 ∈ R n+k . Huisken in [4] proved the following monotonicity formula.
In particular, it follows that Θ(x 0 , t 0 , r) is non-decreasing in r. Consequently we can define the Gaussian density as One can show that (x 0 , t 0 ) is a regular point of the flow if and only if Θ(x 0 , t 0 ) = 1. The following local regularity theorem of White [17] says that if the density ratios are close to 1, then that is enough to get curvature estimates.
Finally, we introduce what it means for two manifolds to be ε-close in C 1,α . Given an open set U and two n-dimensional manifolds Σ and L defined in U, we say that Σ and L are 1-close in C 1,α (W ) for any W with dist(W, ∂U) ≥ 1 if for all x ∈ W , B 1 (x) ∩Σ and B 1 (x) ∩L are both graphical over some common n-dimensional plane, and if u and v denote the respective graph functions then u − v 1,α ≤ 1. We then say that Σ and L are ε-close in W if after rescaling by a factor 1/ε, Σ and L are 1-close in ε −1 W for any W with dist(ε −1 W, ε −1 ∂U) ≥ 1.

2.2.
Lagrangian Submanifolds and Lagrangian Mean Curvature Flow. We consider C n with the standard complex coordinates z j = x j + iy j . We will often identify C n with R 2n . We let J denote the standard complex structure on C n and ω the standard symplectic form on C n . We say that a smooth n-dimensional submanifold of C n is Lagrangian if ω| L = 0. We also consider the closed n-form Ω, called the holomorphic volume form, defined by Ω := dz 1 ∧ · · · ∧ dz n .
On any oriented Lagrangian a simple computation shows that Ω| L = e iθ L vol L , where vol L is the volume form on L. e iθ L : L → S 1 is called the Lagrangian phase. θ L is called the Lagrangian angle, and may be a multi-valued function. We henceforth suppress the subscript L. In the case that θ is a single valued function, we say that the Lagrangian L is zero-Maslov. An equivalent condition is [dθ] = 0, that is, dθ is cohomologous to 0. If θ ≡ θ 0 is constant, then we say that L is special Lagrangian. In this case L is calibrated by Re(e −iθ 0 vol L ), and hence is area-minimising in its homology class. We also consider the Liouville form λ on C n defined by A simple calculation verifies that dλ = 2ω. If there exists some function β such that λ| L = dβ then we say that L is exact. In this paper we will be more interested in local exactness, that is when the Liouville form λ only has a primitive in some open set.
The following remarkable property of smooth Lagrangians relates the Lagrangian angle and mean curvature vector (see for example [15]) Consequently we see that the smooth minimal Lagrangians are exactly the smooth special Lagrangians. A Lagrangian mean curvature flow is a mean curvature flow (L t ) 0≤t<T with L 0 Lagrangian. As proved by Smoczyk [14], it turns out that the Lagrangian condition is preserved by the mean curvature flow.

Evolution Equations and Monotonicity Formulas
In this section we compute evolution equations for different geometric quantities under the flow, and then use these to prove a local monotonicity formula for a primitive of the expander equation.
Proof. (i) Differentiating the holomorphic volume form Ω and using Cartan's formula we have On the other hand Comparing real and imaginary parts we have (i).
(ii) Using Cartan's formula again and denoting λ t : Hence By possibly adding a time-dependent constant to β t this implies Hence it only remains to show that H λ t = ∆β t . We first show that ∇β t = (Jx) T . Indeed we have dβ t = λ t , thus for a tangent vector τ With this in hand we now choose normal coordinates at a point x, and denote the coordinate tangent vectors by {∂ 1 , . . . , ∂ n }. Then we calculate where h ij is the second fundamental form. Taking the trace of each side we have (iii) We may assume without loss of generality that x 0 = 0 and t 0 = 0, and we will suppress the subscripts of ρ. We first calculate Then we have To compute the Laplacian term, we once again fix a point in L and take normal coordinates at that point, with {∂ 1 , . . . , ∂ n } denoting the coordinate tangent vectors. Then So we find that , combining this with the previous calculation yields (iii).
Remark. From the above evolution equations we see that local exactness is preserved by the flow, indeed so by the fundamental theorem of calculus we have where the right hand side is exact if λ 0 is.
Let φ be a cut-off function supported on B 3 with 0 ≤ φ ≤ 1, φ ≡ 1 on B 2 and the estimates |Dφ| ≤ 2 and |D 2 φ| ≤ C. We then have the following lemma.
Using Young's inequality we estimate the last term as follows which is true of any compactly supported smooth (or even C 2 ) function. Thus we arrive at We now just differentiate under the integral to get The first integral is zero by Green's identities, so we are left with precisely the desired inequality since ∇α t = ∇β t + 2t∇θ t = Jx ⊥ − 2tJ H.

Stability of Self-Expanders
In this section we prove a dynamic stability result for Lagrangian self-expanders. More specifically we show that if a Lagrangian submanifold is asymptotic to some pair of planes and is almost a self-expander in a weak sense, then the submanifold is actually close in a stronger topology to some self-expander. Let P 1 , P 2 ⊂ C n be Lagrangian planes intersecting transversally such that neither P 1 + P 2 or P 1 − P 2 are area minimising. We denote by P := P 1 + P 2 . We will need the following uniqueness result, proved by Lotay-Neves [9] in dimension 2 and Imagi-Joyce-Oliveira dos Santos [6] in dimensions 3 and higher.
Theorem 4.1. There exists a unique smooth, zero-Maslov class Lagrangian selfexpander asymptotic to P . Theorem 4.2. Fix R, r, τ > 0, α,ε 0 < 1, and C, M < ∞. Let Σ be the unique smooth zero-Maslov Lagrangian self-expander asymptotic to P . Then for all ε > 0 there existsR ≥ R, η, ν > 0 each dependent on ε 0 , ε, r, R, τ , α, C, M and P such that if L is a smooth Lagrangian submanifold which is zero-Maslov in BR (iv) The connected components of L ∩ A(r,R) are in one to one correspondence with the connected components of P ∩ A(r,R) and for all x ∈ L ∩ A(r,R); Proof. Seeking a contradiction, suppose that the result were not true. Then there would exist sequences ν i ց 0, η i ց 0, R i → ∞ and L i such that each L i is a smooth Lagrangian submanifold of C n that is zero-Maslov in B R i , satisfying By virtue of (1), (4), and a suitable interpolation inequality, it follows that for some ρ > 0, outside of B ρ , L i and Σ are both ε/4-close to P in C 1,α . Hence, in order that On the other hand, by (1) and (2) we may extract a subsequence of L i that converges in C 1,α loc for all α < 1 to some limit L ∞ , a C 1,1 zero-Maslov Lagrangian submanifold. The estimate (2) passes to the limit and tells us that L ∞ has unit multiplicity everywhere, and bounded area ratios. Since L ∞ is C 1,1 we can define mean curvature in a weak sense, and (3) implies By standard Schauder theory for elliptic PDE, this immediately implies that L ∞ is in fact smooth and satisfies the expander equation in the classical sense. Consequently L ∞ is a smooth, zero-Maslov class Lagrangian submanifold, and (4) implies that L ∞ is asymptotic to P . Theorem 4.1 then implies that L ∞ = Σ, which contradicts (5).

Main Theorem
Suppose, as in the previous section, that P := P 1 + P 2 is a pair of transversely intersecting Lagrangian planes such that neither P 1 + P 2 nor P 1 − P 2 are minimising, and that Σ is a Lagrangian self-expander asymptotic to P . We assume the existence of a family (L s ) 0<s≤c of compact Lagrangians, each exact and zero-Maslov in B 4 satisfying the following properties. The existence of such a family will be established in section 7 (H1) The area ratios are uniformly bounded, i.e. there exists a constant D 1 such that (H2) There is a constant D 2 such that for every s and x ∈ L s ∩ B 4 where θ s and β s are, respectively, the Lagrangian angle of L s and a primitive for the Liouville form on L s . (H3) The rescaled manifoldsL s := (2s) −1/2 L s converge in C 1,α loc to Σ. Moreover the second fundamental form ofL s is bounded uniformly in s and without loss of generality we can assume that lim s→0 (θ s +β s ) = 0 locally onL s . (Note thatL s is exact in the ball B 4(2s) −1/2 so we can make sense ofβ s in the limit.) (H4) The connected components of P ∩ A(r 0 √ s, 4) are in one to one correspondence with the connected components of L s ∩ A(r 0 √ s, 4), and each component can be parametrised as a graph over the corresponding plane P i where the function u s : P ∩ A(r 0 √ s, 3) → P ⊥ is normal to P and satisfies the estimate where ∇ denotes the covariant derivative on P . We will denote by (L s t ) t∈[0,Ts) a smooth solution of Lagrangian mean curvature flow with initial condition L s . For x 0 ∈ R 2n and t > 0 we define We introduce a slightly modified notion of the Gaussian density ratios, which we will continue to refer to as the Gaussian density ratios, of L s t at x 0 , denoted Θ s t (x 0 , r) and defined as defined for t < T s . The monotonicity formula of Huisken tells us that We also defineL s t = L s t 2(s + t) .
We will denote byΘ s t (x 0 , r) the Gaussian density ratios of (L s t ), that is One of the primary reasons for modifying the Gaussian density ratios is that our new ratios behave well under the above rescaling. Indeed we can calculate The primary goal of this section is now to prove the following result.
We start by proving estimates like the one in the above theorem hold for a short time or far from the origin.
Proof. We first claim that there is a K 0 such that if y 0 ∈ R 2n has |y 0 | ≥ K 0 then for any λ > 0 and s we have Indeed suppose that this were not the case, then there would exist sequences y i , λ i and s i with |y i | → ∞ such that First we note that λ i must be unbounded since so it is easily seen that if λ i were bounded then (5.2) would fail for large i. Next from the estimate (H4) we have that for every x ∈ A(r 0 √ 2s, 4) and hence where C is a curvature bound for Σ. We rescale and defineL Consequently |A| → 0 uniformly on compact sets centred at y i , so it follows that locallyL i − y i converges to a plane, but this contradicts (5.2). We next observe that (H1) ensures that we may choose δ 1 > 0 small enough such that for any By the monotonicity formula we have that for any r 2 , t ≤ δ 1 so imposing the additional requirement that r 2 ≤ t this gives precisely the desired result.
Proof. By lemma 5.2 we need only prove the estimate for x ∈ B K 0 √ 2t . We apply lemma 8.2 with R = K 0 √ q 1 + 1 where q 1 = q 1 (Σ, ε 0 , α) and the rescaled floŵ This is a mean curvature flow with initial conditionL s . By (H3) we know thatL s → Σ in C 1,α loc , so in particular for s small enough, we can ensure thatL s is ε(ε 0 , Σ, α)-close to Σ in C 1,α . Hence for r 2 , t ≤ q 1 and The next lemma shows that in an annular region, and for short times, we retain control on both the distance to P and the Gaussian density ratios that is uniform in s.
Lemma 5.4 (Proximity to P = P 1 + P 2 ). There are constants C 1 , and r 1 such that for any ν > 0 there are s 2 , δ 2 > 0 such that the following holds. If s ≤ s 2 , t ≤ δ 2 and r ≤ 2 then we have the estimates Proof. We consider t ≤ δ 2 and s ≤ s 2 (both δ 2 and s 2 to be chosen) and define .
Clearly l ≤ 1/2 and also from (H4) we have that if s 2 , δ 2 are chosen small enough, then is the function arising from this graphical decomposition then we have by scaling the estimate of (H4) that Let c > 0 be a constant that will be chosen later. If s 2 (D 3 , r 0 , c) and δ 2 (D 3 , r 0 , c) > 0 are small enough and r 1 (P, c) ≥ max{r 0 , 1} is chosen to be large enough then we can ensure that From now on we fix some y 0 ∈L s t ∩ A 3r 1 + 1, (s + t) −1/8 , then y 0 2(s + t) is a regular point of (L s t ) so by the monotonicity formula Therefore by choosing since l is bounded independent of s and t, and the estimate (H1) is scale invariant, so in particular is satisfied by (3l) −1/2 Σ (s,t) . Next we estimate B. Similarly as before we find that for |x| ≤ r 1 ≤ |y 0 |/3 we have Finally we deal with C. We denote by a i the orthogonal projection of y 0 onto P i and by b i the orthogonal projection of y 0 onto P ⊥ i . We suppose without loss of generality that dist(y 0 , P ) = |b 1 |.
We will also denote by Σ (s,t) i the component of Σ (s,t) ∩ A(r 1 , 3(s + t) −1/8 ) that is graphical over Π i := P i ∩A(r 1 , 3(s + t) −1/8 ), and by v i (s,t) the corresponding function. Since we have that P 1 ∩ P 2 = {0} it follows that for some c = c(P ) > 0 we have that |b 2 | ≥ c|y 0 |. Notice that since |b 2 | ≤ |y 0 | we have that c ≤ 1. Suppose that x ∈ Σ (s,t) 2 , and denote by x ′ the orthogonal projection onto P 2 . Then we have Moreover by (5.4), if r 1 is chosen large enough (and in particular larger than 1), where we used (5.4) to estimate the gradient terms arising in the surface measure. Combining this with the estimates for A and B we have that Increasing r 1 for the last time if necessary, we can ensure that Therefore we have that .
The following two Lemmas show that we have additional control in annular regions, specifically on normal deviation, curvature, Lagrange angle and the primitive for the Liouville form.
Lemma 5.6. There are δ 4 > 0 and s 4 > 0 such that for 0 < s ≤ s 4 and t < δ 4 The estimate is clearly true for t = 0 by assumption (H2). Moreover, by (H4) we can assume that for s sufficiently small, each of the L s is the graph of a function with small gradient in the region A(1/4, 4). Applying Lemma 8.1 we find that L s remains graphical with small gradient in A(2/7, 7/2) for some short time, which implies that |θ s t | ≤ C for δ 4 chosen small enough. That |A s t | is bounded follows from Lemma 8.1 and Corollary 8.4, since Lemma 8.1 implies small gradient for a short time, which allows use to apply Corollary 8.4 to get uniform curvature bounds for some short time in A(1/3, 3).
Since |θ s t | and |A s t | are both bounded, we have from the evolution equations of β s t that dβ s t dt ≤ Jx, H + 2|θ s t | ≤ C. Hence for some suitable short time, |β s t | also remains bounded in A(1/3, 3). The last of the technical lemmas in this section uses the monotonicity formula of Section 3 to show that after waiting for a short time dependent on s, we can find times at which the scaled flowL s t is close to a self-expander in an L 2 sense. We later use this in the proof of the main theorem to get estimates on the density ratios via the stability result.
Lemma 5.7. Let a > 1. Let q 1 be as given by Lemma 5.3, and set q := q 1 /a. Then for all η > 0 and R > 0 there exist δ 5 > 0, s 5 > 0 such that for all s ≤ s 5 and qs ≤ T ≤ δ 5 we have Proof. Fix R > 0, η > 0. Suppose s ≤ s 5 and qs ≤ T ≤ δ 5 , with δ 5 and s 5 yet to be determined. Furthermore, we set T 0 := R 2 (s + aT ) + aT . Throughout the proof, we denote by C a constant which depends on a, R and q, but not on T or s. We estimate Now supposing that s 5 and δ 5 are small enough we can ensure that R 2(s + t) ≤ 2. Moreover on B R √ 2(s+t) we have hence we continue estimating (5.7) using the localized monotonicity formula of Lemma 3.2 (φ denotes the cut-off function given in that lemma which is 1 on B 2 and 0 outside of B 3 ) Now using the localized monotonicity a second time we have the estimate |2(s + t)θ s t + β s t | 2 ρ 0,T 0 dH n dt.

(5.9)
Now T 0 − T ≤ C(s + T ), with C depending only on R and a, so We first estimate B, for which we make use of the estimate of Lemma 5.6 We note that T 0 ≤ (R 2 (1/q + a) + a)T = CT , T 0 ≤ C(s + T ) and T 0 ≥ R 2 (s + aT ) so we can estimate where we can estimate the supremum by a uniform constant because L s t all have bounded area ratios with a uniform constant. Moreover T 0 ≤ R 2 δ 5 (1/q + a) + aδ 5 so that by possibly decreasing δ 5 we can ensure that B ≤ η/2. We next estimate A, First recall that if β s is primitive for the Liouville form on some L s , then β s l := l −2 β s is primitive for the Liouville form on l −1 L s . From here on we surpress the subscript 0 of the β s and θ s since we only ever integrate over the manifolds L s 0 , and we instead use a subscript l to denote the rescaling factor of the β s . We define |σθ s + β s l | 2 ρ 0,l −2 T 0 dH n since T ≥ qs, so we can absorb (s + T )/T into the constant. Define Notice that from the definition of T 0 we can find C > 0 independent of T and s such that l −2 T 0 ∈ [C −1 , C]. We want to show that by possibly again decreasing s 5 and δ 5 , we can ensure Seeking a contradiction, suppose that this is not the case. Then we can find sequences s i and T i both converging to 0 with qs i ≤ T i and such that After possibly extracting a subsequence which we don't relabel, we may assume that l −2 i T 0 → T 1 . We split the rest of the proof into two cases. Case 1: Suppose that (after possibly extracting a further subsequence) we have that σ i → σ > 0. Then by (H3) we have because |θ s i +β s i | is bounded by D 2 (1 + |x| 2 ) on B 3(2s i ) −1/2 , which means that since l −2 i σ −1 i T 0 → σ −1 T 1 > 0 the contribution to the integral outside some fixed large ball is small uniformly in i. Moreover by (H3) we have lim i→∞ |θ s i +β s i | 2 = 0 locally, so inside this large ball the integral can be made as small as desired.
Case 2: Suppose now that, again after possibly passing to a not relabelled subsequence, σ i → 0. Then because |θ s i +β s i | 2 → 0 locally, and ρ is bounded. So to estimate lim i→∞ F (T i , s i ) we need only control the integral in the annulus A(r 0 σ i /2, 3l −1 i ). We first notice that by (H4), provided i is large enough, l −1 i L s i ∩ A(r 0 σ i /2, 3l −1 i ) is graphical over P , and if v i is the function arising from this decomposition we have the estimate In the graphical region, the normal space to the graph is spanned by the vectors n j := (−∇v j i , e j ) for j = 1, . . . , n where e j denotes the vector in R n whose jth entry is 1, and all other entries are 0, and v j i is the jth coordinate of v i . Then given an orthonormal basis for the normal space ν 1 , . . . , ν n we have ν j = n k=1 α jk n k so it follows that where C depends only on the α jk . Now Using this estimate we can control β s i l i independently of i on the annular region Note that x i of course depends on the original choice of x as well as i. We may now define a curve inL s i by setting By the fundamental theorem of calculus we can write uniformly in x asθ s i is bounded and σ i → 0. Therefore we may bound the term where we again used the fact that l −2 i T 0 → T 1 > 0, so that outside of some large ball the contribution to the integral is very small. This limit being zero is a contradiction, so we are done.
We may now embark on the proof of Theorem 5.1. Changing scale, to prove the main theorem it would in fact suffice to show the following (which is very slightly stronger due to the bound on the scale of the density ratios) Theorem (Rescaled main theorem). There exist s 0 , δ 0 and τ such that if t ≤ δ 0 , r 2 ≤ τ and s ≤ s 0 , thenΘ s t (x 0 , r) ≤ 1 + ε 0 for all x 0 with |x 0 | ≤ (2(s + t)) −1/2 .
Thus to prove the rescaled main theorem, it suffices to show that for appropriately chosen s 0 , δ 0 and τ the following holds true: if r 2 ≤ τ , s ≤ s 0 , t ≤ δ 0 and t ≥ q 1 s thenΘ s t (y 0 , r) ≤ 1 + ε 0 whenever |y 0 | ≤ K 0 . This is what we now show.
Proof of Theorem 5.1. Define for each We now claim that we can find δ 0 > 0 and s 0 > 0 such that T s ≥ δ 0 for all s ≤ s 0 . Indeed, with τ defined as before, we choose a > 1 with a < (1 + 2τ ). Let C be the constant of Brian White's local regularity theorem, and set C := C 2(a + 3) We next let r 3 := max{r 0 , r 1 , r 2 , 1}, where r 0 , r 1 , and r 2 are as in, respectively, the construction of the approximating family, Lemma 5.4, and Lemma 5.5. Let R := √ 1 + 2q 1 K 0 + r 3 , and ε = ε(Σ, ε 0 , α) as given by Lemma 8.2. We apply the stability result, Theorem 4.2 with R = R; r = r 3 ; C = max{C 1 , C} the constants from Lemma 5.4, and the construction of the approximating family respectively; M =C; τ = τ ; Σ = Σ and ε = ε. Thus we obtainR ≥ R, η > 0 and ν ≥ 0 as in the theorem. Apply Lemma 5.7 with η = η/2 and R =R. This gives s 5 and δ 5 such that the lemma holds. Next apply Lemma 5.4 with ν to obtain s 2 and δ 2 . We now let s 0 := min{s 1 , s 2 , s 3 , s 4 , s 5 } and δ 0 := min{δ 1 , δ 2 , δ 3 , δ 4 , δ 5 }. We finally possibly decrease s 0 and δ 0 slightly to ensure that This will ensure that in the annular region A(r 3 ,R) we have all of the estimates of the intermediate lemmas of this section. We now claim that these s 0 and δ 0 are the required constants. Specifically we claim that for all s ≤ s 0 we have T s ≥ δ 0 . Indeed, suppose that this were not the case and that for some s ≤ s 0 we have T s < δ 0 . Define T := T s /a, then since T < T s we have for all t ∈ [T, T s ) Θ s t (x, r) ≤ 1 + ε 0 for all r 2 ≤ τ and x ∈ B K 0 . In fact, as has already been observed, the same is true for all |x| ≤ (s + t) −1/8 , so in particular for all |x| ≤ 2R. LetL s l denote the Lagrangian mean curvature flow with initial conditionL s T . Then it is easy to verify thatL s l = √ 1 + 2lL s T +σ 2 l , where σ 2 = 2(s + T ). This implies the density ratio control Θ s l (x, r) ≤ 1 + ε 0 , for all l such that T + σ 2 l ∈ [T, T s ), r 2 ≤ τ and x ∈ B 2R . By the local regularity theorem of Brian White this means we get curvature bounds of the form or, scaled back to the original scale this means on B σR for all t < T s with T ≤ t ≤ T + 2(s + T )τ = (1 + 2τ )T + 2τ s. Notice in particular that T s = aT ≤ aT 0 ≤ (1 + 2τ )t 0 + 2τ s, so the above estimate always holds up to time T s . Let t 0 := T (a + 1)/2. Then In other words, for each t ∈ [t 0 , T s ) we have which implies that for each t ∈ [t 0 , T s ) we have |Ã s t | ≤C on BR. Applying Lemma 5.7, we may select t 1 ∈ [t 0 , T s ) with Condition (iv) of Theorem 4.2 holds forL s t 1 by Lemma 5.4, and condition (ii) holds by definition of T s as t 1 < T s . Hence by Theorem 4.2 we know thatL s t 1 is ε-close to Σ in C 1,α (BR). RedefineL s l to be the Lagrangian mean curvature flow with initial conditionL s t 1 , then Lemma 8.2 says that Θ s l (x, r) ≤ 1 + ε 0 r 2 , l ≤ q 1 for |x| ≤R − 1. By definition ofR this means that the same is true for |x| ≤ √ 1 + 2q 1 K 0 . Rescaling this is equivalent tõ for r 2 , l ≤ q 1 and |x| ≤ √ 1 + 2q 1 K 0 . Or in other words Θ s t (x, r) ≤ 1 + ε 0 for r 2 ≤ q 1 /(1 + 2q 1 ) = τ , |x| ≤ K 0 and t 1 ≤ t ≤ (1 + 2q 1 )t 1 + 2q 1 s. However, (1 + 2q 1 )t 1 + 2q 1 s > at 1 > aT = T s , a contradiction.

Short-time Existence
In this section we prove the following short time existence result using Theorem 5.1. Theorem 6.1. Suppose that L ⊂ C n is a compact Lagrangian submanifold of C n with a finite number of singularities, each of which is asymptotic to a pair of transversally intersecting planes P 1 + P 2 where neither P 1 + P 2 nor P 1 − P 2 are area minimizing. Then there exists T > 0 and a Lagrangian mean curvature flow (L t ) 0<t<T such that as t ց 0, L t → L as varifolds and in C ∞ loc away from the singularities.
Proof. For simplicity we suppose that L has only one singularity at the origin. The case where L has more than one follows by entirely analogous arguments. By standard short time existence theory for smooth compact mean curvature flow, for all s ∈ (0, c] there exists a Lagrangian mean curvature flow (L s t ) 0≤t≤Ts with T s > 0. We claim that there exists a T 0 > 0 such that T s ≥ T 0 for all s sufficiently small, and that furthermore, we have interior estimates on |A| and its higher derivatives for all t > 0, which are independent of s. By virtue of Lemma 8.1, we can apply Corollary 8.4 on small balls everywhere outside B 1/3 to get uniform curvature bounds outside of B 1/2 up to time min{T s , δ} where δ > 0 is independent of s. Uniform estimates on the higher derivatives then immediately follow by standard parabolic PDE theory. To obtain the desired bounds on B 1/2 we use Theorem 5.1. Let ε 0 > 0 be the constant of Brian White's local regularity theorem. Then Theorem 5.1 says that there exist s 0 , δ 0 and τ such that for all s ≤ s 0 , t ≤ δ 0 and r 2 ≤ τ t we have Θ s t (x 0 , r) = Θ s (x, t + r 2 , r) ≤ 1 + ε 0 . This implies that for all s ≤ s 0 , t ≤ δ 0 and r 2 ≤ τ t we have Θ s (x, t, r) ≤ 1 + ε. We now fix s ≤ s 0 , t 0 < min{δ 0 , T s }, and ρ ≤ min{1/4, √ t 0 }. Then it follows that B 2ρ (x 0 ) ⊂ B 1 , and furthermore that Θ s (x, t, r) ≤ 1 + ε 0 for all r ≤ τ ρ 2 , and (x, t) ∈ B 2ρ (x 0 ) × (t 0 − ρ 2 , t 0 ]. Then it immediately follows from White's theorem that , where C depends only on τ and ε 0 . These estimates are then uniform in s for s ≤ s 0 . Moreover, these curvature bounds, along with those outside of the ball B 1/2 , imply that T s ≥ min{δ, δ 0 }. Because the estimates are independent of s, they pass to the limit in the varifold topology when we take a subsequential limit of the flows and so we obtain a limiting flow (L t ) 0<t<T 0 , for which L t → L as varifolds. Note that away from the singularities, we can obtain uniform curvature estimates on |A| thanks to Corollary 8.4, so it follows that (L t ) attains the initial data L in C ∞ loc away from the singular points.

Construction of Approximating Family
In this section, we consider a Lagrangian submanifold L of C n with a singularity at the origin which is asymptotic to the pair of planes P considered in Section 4. We approximate L by gluing in the self-expander Σ which is asymptotic to P at smaller and smaller scales in place of the singularity. We will show that this yields a family of compact Lagrangians, exact in B 4 , which satisfy the hypotheses (H1)-(H4) given in Section 5 which are required to implement the analysis in that section. Since L is conically singular we may write L ∩ B 4 as a graph over P ∩ B 4 (possibly rescaling L so that this is the case). We may further apply the Lagrangian neighbourhood theorem (its extension to cones was proved by Joyce, [7,Theorem 4.1]), so that we may identify L ∩ B 4 with the graph of a one-form γ on P . Recall that the manifold corresponding to the graph of such a one-form is Lagrangian if and only if the one-form is closed. Moreover, since we have assumed that L is exact inside B 4 , there exists u ∈ C ∞ (P ∩ B 4 ) such that du = γ. Since we know that γ must decay quadratically, we can choose a primitive for γ which has cubic decay, i.e., We saw in Theorem 4.1 that there exists a unique, smooth zero-Maslov self-expander asymptotic to P . We may also identify the self-expander outside a ball of radius r 0 with the graph of a one-form over P and, since a zero-Maslov class Lagrangian self-expander is globally exact, there exists a function v ∈ C ∞ (P \B r 0 ) such that the self-expander is described by the exact one-form ψ = dv on P \B r 0 . Further, Lotay and Neves proved [9, Theorem 3.1] We will glue Σ s := √ 2sΣ into the initial condition L to resolve the singularity. Our new manifold, L s , will be the rescaled self-expander Σ s inside B r 0 √ 2s , the manifold L outside B 4 and will smoothly interpolate between the two on the annulus A(r 0 √ 2s, 4). To do this, we will glue together the primitives of the one-forms corresponding to these manifolds, before taking the exterior derivative. This gives us a one-form that will describe L s on the annulus A(r 0 √ 2s, 4), which ensures L s is still Lagrangian and is exact in B 4 . We will then show that this family satisfies the properties (H1)-(H4). Let ϕ : R + → [0, 1] be a smooth function satisfying ϕ ≡ 1 on [0, 1] and ϕ ≡ 0 on [2, ∞). Consider the one-form given by, for where we have that r 0 √ 2s < s 1/4 < 2s 1/4 < 4 holds for all s ≤ c. Notice that in particular we must have c < 1. Then γ s (x) ≡ ψ s (x) := √ 2sψ(x/ √ 2s), the one-form corresponding to the rescaled self-expander Σ s for |x| < s 1/4 and γ s ≡ γ for |x| > 2s 1/4 . Notice that since γ s is exact, it is closed and therefore its graph corresponds to an exact Lagrangian. We define L s by We will now show that L s satisfies (H1)-(H4). For (H1), notice that both the self-expander and the initial condition individually satisfy (H1), and so for the rescaled self-expander, we have that Since L s interpolates between Σ s and L on a compact region, L s satisfies (H1). We see that (H2) is satisfied because the Lagrangian angle of the initial condition L and the self-expander Σ are bounded, as is that of the rescaled self-expander Σ s by Lemma 3.1 (i) and the maximum principle, since the Lagrangian angle of P is locally constant. When we interpolate between the two, we may consider the formula for the Lagrangian angle of a Lagrangian graph, as seen in [1, pg. 5]. This tells us that a Lagrangian graph in C n (over R n ) given by (x 1 , ..., x n , u 1 (x), ..., u n (x)), where u : R n → R, u i := ∂u ∂x i , has Lagrangian angle where the λ i 's are the eigenvalues of the Hessian of u. Since the eigenvalues of the Hessian of u are some non-linear function of the second derivatives of u, if the C 2 norm of u is small we have that the Lagrangian angle of the graph is close to that of the Lagrangian angle of the plane that u is a graph over. So we can uniformly bound the Lagrangian angle of the graph. Since in our case, the Lagrangian angle of γ s is given by the sum of arctangents of the eigenvalues of the Hessian of the function w s , and, as we will show when we prove (H4), the C 2 norm of w s is small, this means that we can uniformly bound the Lagrangian angle of the graph γ s , and so the Lagrangian angle of L s . On the initial condition, since λ = Jx, we have that dβ L = λ| L = (Jx) T . Therefore, β L is bounded quadratically, and so is the primitive for the Liouville form of L s \B(2s 1/4 ). On the self-expander, applying the maximum principle to Lemma 3.1 (ii), we have β s (the primitive of λ| Σs ) is bounded by β P , and so |β s (x)| ≤ |β P (x)| ≤ C|x| 2 for |x| < s 1/4 . So it remains to check this still holds where we interpolate. We perform a calculation similar to that in the proof of Lemma 3.1(ii). We have that, for L s t the manifold described by the graph of the one-form tdw s , d dt λ| L s t =: L J∇ws λ| L s t = d(J∇w s λ| L s t ) + J∇w s dλ| L s t . Since dλ = ω and J∇w s ω = dw s and possibly adding constant to β s t dependent on s and t, we have that dβ s t dt = −2w s + x, ∇w s | L s t , where dβ s t is equal to the restriction of the Liouville form λ to graph of tγ s . Integrating, we find that where β P is the primitive for λ on P . Now, w s is bounded independently of s by D(1 + |x| 2 ), using (7.1) and (7.2), as is x, ∇w s , using Cauchy-Schwarz and the estimates (7.1) and (7.2) so we find that β s is bounded independently of s on the annulus A(s 1/4 , 2s 1/4 ). Therefore, we have that |θ s (x)| + |β s (x)| ≤ D 2 (|x| 2 + 1). and so (H2) is satisfied.
To show that (H3) is satisfied, recall that we define L s as L s ∩B r 0 √ 2s = Σ s ∩B r 0 √ 2s , L s \B 4 = L\B 4 and we interpolate smoothly between the two, which exactly happens when s 1/4 ≤ |x| ≤ 2s 1/4 . Therefore when we rescale by 1/ √ 2s, we have that L s ∩ B r 0 ≡ Σ. So it remains to check convergence outside this ball.
On the annulus r 0 ≤ |x| ≤ 4/ √ 2s,L s is identified with the graph of the following one-formγ From this expression, noticing that we see that as s → 0,γ s → dv = ψ, the one-form whose graph is identified with Σ. This says that, outside B r 0 ,L s → Σ as s → 0 smoothly. Therefore we actually have stronger than the required C 1,α loc convergence. Finally, we check that the second fundamental form ofL s is uniformly bounded in s. We have that the second fundamental form of Σ must be bounded, and if A is the second fundamental form of L, rescaling L by 1/ √ 2s means that the second fundamental form scales by √ 2s. Since √ 2s < 1, we can uniformly bound both second fundamental forms so thatL s , which is a combination of both Σ and 1/ √ 2sL, has second fundamental form uniformly bounded in s. To see (H4), first notice that since we can write L s ∩ A(r 0 √ 2s, 4) as a graph over P ∩ A(r 0 √ 2s, 4), we have that L s has the same number of connected components as P in the annulus A(r 0 √ 2s, 4). We now must estimate γ s . Firstly, note that we have where we have used (7.2). We will need different estimates on 2s∇ 2 v(x/ √ 2s) and 2s∇ 3 v(x/ √ 2s), which we find as follows.

Appendix
We collect in the appendix a few technical results about Mean curvature flow in high codimension that were used throughout the paper. The first is a graphical estimate. Specifically, if the initial manifold can be written locally as a graph with small gradient in some cylinder, then the submanifold remains graphical in a smaller cylinder and we retain control on the gradient. To state this more rigorously we first introduce some notation. The notation and statement of the result are as in [5]. Given any point x ∈ R n+k we write x = (x,x), wherex is the projection onto R n andx is the projection onto R k . We define the cylinder C R (x 0 ) ⊂ R n+k by C r (x) = {x ∈ R n+k ||x −x 0 | < r, |x −x 0 | < r}.
Lemma 8.1. Let (M n t ) 0≤t<T be a smooth mean curvature flow of embedded ndimensional submanifolds in R n+k with area ratios bounded by D. Then for any η > 0, then there exists ε, δ > 0, depending only on n, k, η, D, such that if x 0 ∈ M 0 and M 0 ∩ C 1 (x 0 ) can be written as graph(u), where u : B n 1 (x 0 ) → R k with Lipschitz constant less than ε, then is a graph over B n δ (x 0 ) with Lipschitz constant less than η and height bounded by ηδ. The proof can be found in [5] Next we prove that if an initial manifold M is close to some smooth manifold Σ in C 1,α , then one gets estimates on the density ratios that are independent of M.
Lemma 8.2. Let Σ be a smooth manifold with bounded curvature and let (M t ) t∈[0,T ) be a solution of mean curvature flow. Fix ε 0 > 0, α < 1. There are ε = ε(Σ, ε 0 , α) > 0 and q 1 = q 1 (Σ, ε 0 , α) > 0 such that for every R ≥ 2, if M 0 is ε-close to Σ in C 1,α (B R ) then for every r 2 , t ≤ q 1 and y ∈ B R−1 we have Θ t (y, r) ≤ 1 + ε 0 Proof. This follows immediately from Lemma 8.1. Indeed the curvature bound on Σ means that there is a uniform radius r such that for any x ∈ Σ, Σ ∩ C r (x) is (after maybe rotating) a graph with small gradient. By requiring that ε is small enough we can therefore ensure that any M 0 which is ε-close to Σ in C 1,α (B r (x)) is also a graph with small gradient. It only remains to apply Lemma 8.1.
8.1. Local curvature estimates for high codimension graphical MCF. In [2] Ecker and Huisken proved celebrated curvature estimates for entire graphs moving by mean curvature in codimension one. Also in [3] they proved interior curvature estimates for hypersurfaces moving by mean curvature flow. Analogous results in higher codimension have been cited in various places in the literature, but to our knowledge an explicit statement of these results has not appeared. In this section we localise estimates of Mu-Tao Wang in [16] using the methods of Ecker and Huisken. The calculations are totally analogous to those appearing in [3] so we will just give the outline and not the detail. We consider a function u : R n → R k whose graph in R n × R k evolves by mean curvature. Ω is the volume form of R n which we can extend to a parallel form on R n × R k . As show by Wang, one can calculate that * Ω = 1 det(δ ij + D i u · D j u) = 1 n i=1 (1 + λ 2 i ) where λ i are the eigenvalues of (du t ) T du t . Moreover, for ε > 0 small (depending only on the dimensions n and k), we have that if det(δ ij + D i u · D j u) < 1 + ε, then * Ω satisfies the evolution inequality d dt * Ω ≥ ∆ * Ω + 1 2 * Ω|A| 2 . Theorem 8.3 (High codimension interior estimate). Let R > 0 and suppose that K R 2 := {(x, t) ∈ M t |r(x, t) ≤ R 2 } is compact and can be written as a graph over some plane for t ∈ [0, T ]. Suppose further that if the graph function is denoted by u, that det(δ ij + D i u · D j u) < 1 + ε, where ε > 0 depends only on n and k. Then for any t ∈ [0, T ] and θ ∈ (0, 1) we have If we denote by · T projection onto the plane over which M t is graphical, then it's easy to see that d dt − ∆ |x T | = 0 for x = F (p, t) some point in M t . Therefore, defining r(x, t) := |x T | 2 we have d dt − ∆ r = 2|(∇x) T | 2 ≤ c(n, k) With this choice of r we have the following corollary where B R (y 0 ) denotes a ball centred at y 0 with radius R in the plane.