Ancient solutions and translators of Lagrangian mean curvature flow

Suppose that $\mathcal{M}$ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbb{C}^n$. We show that if $\mathcal{M}$ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then $\mathcal{M}$ is a translator. In particular in $\mathbb{C}^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.


Introduction
An important problem in complex and symplectic geometry is to find special Lagrangian submanifolds in Calabi-Yau manifolds. Szmoczyk [23] showed that the mean curvature flow preserves the class of Lagrangian submanifolds in Calabi-Yau manifolds, and so one can attempt to use the flow to deform any Lagrangian into a special Lagrangian. The Thomas-Yau conjecture [25], motivated by mirror symmetry [24], predicts that this is indeed possible, assuming that the initial Lagrangian satisfies a certain stability condition. More recently Joyce [17] formulated a detailed conjectural picture, relating singularity formation along the Lagrangian mean curvature flow to Bridgeland stability conditions.
To motivate our main result, recall that along the mean curvature flow of zero-Maslov Lagrangians, all tangent flows at singularities are given by unions of minimal Lagrangian cones, according to Neves [20]. In particular all such tangent flows are singular, or have higher multiplicity. In order to understand how such singularities form, it is therefore crucial to study a general class of ancient solutions of the flow, such as Type II blow-ups.
The simplest ancient solutions are those whose blow-down at −∞ is special Lagrangian. In this case [19,Proposition 4.5] implies that the ancient solution itself is special Lagrangian, and in particular static. Our main result is the following, addressing the next simplest situation. See Section 2 for the basic definitions. Theorem 1.1. Let P 1 , P 2 ⊂ C n be Lagrangian subspaces which intersect along a line ℓ and have distinct Lagrangian angles. Let M be a smooth, zero-Maslov, exact, ancient solution to Lagrangian mean curvature flow in C n with uniformly bounded variation of the Lagrangian angle and uniformly bounded area ratios. For n ≥ 3 assume in addition that M is almost calibrated.
If M has a blow-down at −∞ given by the static flow consisting of the union P 1 ∪ P 2 , then M is a translator.
Two possibilities for the translator in Theorem 1.1 are the static flows given by unions of translates of P 1 and P 2 , and the non-trivial translators constructed Date: May 2, 2022. 1 by Joyce-Lee-Tsui [18], which play an important role in Joyce's conjectural picture [17]. It is an interesting question whether there are any more possibilities.
Combining Theorem 1.1 with the work in [19] shows the following. (Again, see Section 2 for the definitions.) Corollary 1.2. Let L be a rational Lagrangian in C 2 with entropy less than 3 and bounded Lagrangian angle. Then along the Lagrangian mean curvature flow starting at L, any Type II blow-up at a singularity is either special Lagrangian (and given in [19, Theorems 1.1 and 1.3]) or a non-trivial translator.
This results follows from Theorem 1.1 since for a zero-Maslov ancient solution of the flow in C 2 with entropy less than 3 the only possible blow-downs are a union of two planes. If the two planes have the same Lagrangian angle, then [19, Theorem 1.1 or Theorem 1.3] applies. If the two planes have different Lagrangian angle, then using [19,Proposition 4.1] the planes must meet along a line, and Theorem 1.1 applies. An analogous result holds also for Lagrangian mean curvature flow in a compact Calabi-Yau surface under a suitable rationality assumption, such as in Fukaya [14,Definition 2.2], at singularities with density less than 3.
In singularity analysis it is important to consider arbitrary blow-up limits of the flow, not just those that are smooth. Theorem 5.1 provides an extension of Theorem 1.1 in the case n = 2 to Brakke flows obtained as blow-up limits of smooth flows.
Recently there has been great progress in classifying ancient solutions of geometric flows such as Ricci flow and mean curvature flow (see e.g. [2,10,3,4,5]). A crucial new difficulty in our work is that the blow-down P 1 ∪ P 2 is singular along the line of intersection ℓ. As a result, an approach based on the analysis of the linearized operator on the blow-down faces substantial difficulties. An earlier result characterizing translators among eternal solutions to the mean curvature flow of hypersurfaces is due to Hamilton [15], relying on a differential Harnack estimate. It is not known if this approach can be extended to higher codimension flows. Our approach is completely different and relies on additional structure present in the Lagrangian setting. In particular, translators are characterized by the condition that one of the coordinate functions, w, is a linear combination of 1 and the Lagrangian angle θ, i.e. w = a + bθ, see Proposition 2.10. The function w is an ancient solution of the heat equation along M, and the basic idea of the proof is to obtain information about w through solutions of the heat equation on the possible blow-downs. This is related to work of Colding-Minicozzi [9] on mean curvature flow, and also to earlier works on harmonic functions [7,11] and holomorphic functions [12].
To illustrate the basic ideas, suppose that n = 2 and let w be a coordinate function vanishing on P 1 ∪ P 2 . Define z so that on C 2 we have ∇w = J∇z, i.e. the line ℓ is parallel to ∇z. The ancient solutions of the heat equation on P 1 ∪ P 2 with at most linear growth, allowing a different smooth solution on each plane, are spanned by 1, θ, x, y, z, zθ (see Lemma 3.6). Here θ is simply a different constant on each plane. At the same time, 1, θ and the coordinate functions x, y, z, w are ancient solutions of the heat equation on our ancient flow M.
The first main step of the proof is to show that either M is a translator, or along a suitable sequence of scales t → −∞, the normalized projection of w orthogonal to x, y, z converges to zθ on the blow-down P 1 ∪ P 2 (see Proposition 3.12). The main technical difficulty at this stage is that the singular set given by the line ℓ has codimension one in P 1 ∪ P 2 , and we need to exploit that the angle θ takes on different values on P 1 and P 2 in order to pass solutions of the heat equation along M to solutions on P 1 ∪ P 2 in the limit. This is the content of Proposition 3.7.
The proof of Theorem 1.1 is completed using Proposition 4.5, based on the idea that if along the flow at some scale w behaves like zθ at some scale, then the flow must break into two pieces, which roughly look like the two planes P 1 , P 2 rotated in such a way that their intersections with the unit spheres are linked. Here the fact that θ is a different constant on each plane P 1 , P 2 is crucial. This linking behaviour is used to show that the flow must have a point of density two, but the monotonicity formula then implies that the flow is a static union of planes.
Acknowledgements. This project grew out of discussions at the AIM workshop "Stability in mirror symmetry" in December 2020. JDL and FS were partially supported by a Leverhulme Trust Research Project Grant RPG-2016-174. GSz was supported in part by NSF grant DMS-1906216.

Preliminaries
In this section we introduce various key definitions and notation that we shall require throughout the article. In particular, we introduce the set-up for our study.
2.1. Lagrangians in C n . We first recall some basic definitions concerning Lagrangian submanifolds in C n . Definition 2.1. An oriented Lagrangian L in C n is zero-Maslov if there exists a function θ on L (called the Lagrangian angle) so that where H is the mean curvature vector of L and J is the complex structure on C n . We further say that L is almost calibrated if θ can be chosen so that for some ǫ > 0. Definition 2.2. An oriented Lagrangian L in C n is exact if there exists a function β on L so that Jx ⊥ = ∇β, where x ⊥ is the normal projection of the position vector x ∈ C n . Equivalently, where λ is the Liouville form on C n , which is a 1-form on C n so that 1 2 λ is a primitive for the Kähler form ω on C n . The Lagrangian L is rational if the set λ(H 1 (L, Z)) is discrete in R. An exact Lagrangian is clearly rational.
2.2. Spacetime track. Throughout we consider a smooth, zero-Maslov, exact, ancient solution to Lagrangian mean curvature flow (LMCF) (−∞, 0) ∋ t → L t ⊂ C n which evolves with normal speed given by H, with uniformly bounded variation of the Lagrangian angle. We assume that L t has uniformly bounded area ratios, i.e. there exists C > 0 such that sup x,t H n (L t ∩ B(x, r)) ≤ Cr n for all r > 0, where B(x, r) is the Euclidean ball of radius r about x ∈ C n . We call the spacetime track of the flow, and write M(t) = L t . Remark 2.3. For n > 2 we will additionally need to assume that the flow M is almost calibrated, so that one can apply the structure theory in [19] and [21].
Since our focus is on planes arising as blow-ups or blow-downs, it is useful to consider them as trivial static flows as follows.
Definition 2.4. For a pair of n-dimensional planes P 1 , P 2 ⊂ C n , we let M P1∪P2 denote the static flow corresponding to P 1 ∪ P 2 .

2.3.
Rescalings. It will be useful to perform parabolic rescalings of our flows, so we shall introduce the following notation.
Definition 2.5. For λ > 0 we shall denote the parabolic rescaling Note that for a (Lagrangian) mean curvature flow M, we have that D λ M is again a (Lagrangian) mean curvature flow.
It turns out to be helpful to consider a further rescaling, which turns self-similarly shrinking solutions into static points of the flow.
We recall Huisken's monotonicity formula [16]: for t < t 0 , where f is a function on L t with at most polynomial growth, and is the backwards heat kernel (centred at (x 0 , t 0 )). The density of a point (x 0 , t 0 ) along the flow L t is defined to be Recall also the entropy λ(L) defined by Colding-Minicozzi [8]: By virtue of Huisken's monotonicity formula, t → λ(L t ) is non-increasing along any n-dimensional mean curvature flow in C n .
2.4. Set-up. We now describe the main set-up that we shall have throughout the article. In particular, this will be useful to fix notation. We consider two oriented Lagrangian planes P 1 , P 2 ⊂ R 2n = C n which intersect along an oriented (real) line ℓ through 0. Suppose further that P 1 , P 2 have distinct Lagrangian angles, which we denote by θ 1 , θ 2 . (Note that this must be the case if n = 2.) Changing the Lagrangian angles by a fixed constant, we can in addition assume that θ 1 = −θ 2 .
We assume that the unit vector in the direction of ℓ is given by e z , corresponding to the (real) coordinate z. We let e w = Je z , corresponding to the real coordinate w, noting that e w is necessarily orthogonal to P 1 ∪ P 2 . We will think of w as the "height " function, since w = 0 on P 1 ∪ P 2 . We choose coordinates x 1 , . . . , x 2n−2 such that x 1 , . . . , x n−1 , w vanish along P 2 and x n , . . . , x 2n−2 , w vanish along P 1 .
Our key assumption is that our ancient solution M to LMCF has a blow-down at −∞ given by P 1 ∪ P 2 , i.e.
for λ i ց 0, where the convergence is in the sense of Brakke flows. We note that this is equivalent to the assumption that the sequence of smooth flows At this point we make the following observation about blow-downs of M. Proof. We have the set of angles {θ 1 , θ 2 } for two multiplicity one planes in one blow-down and the set of angles is the same for any blow-down by [19,Theorem 3.1]. Therefore, since θ 1 = θ 2 , and by (2.3) the (Gaussian) density at −∞ is two, any blow-down has to consist of two distinct unit multiplicity planes with the given angles. Again using θ 1 = θ 2 , the case that a blow-down is a pair of transverse planes is ruled out since in this case [19,Proposition 4.1] would force the ancient solution M to be the static flow consisting of the two transverse planes, which contradicts one blow-down being two planes meeting along a line.

2.5.
Translators. There is a smooth, connected, zero-Maslov (in fact, almost calibrated), exact, ancient (in fact, eternal) solution to Lagrangian mean curvature flow in C n whose blow-down at −∞ is P 1 ∪ P 2 , which is a translator constructed by Joyce-Lee-Tsui [18]. We recall the definition of a translator as follows. Remark 2.9. It is worth noting that any Lagrangian plane P so that e z is tangent to P will give a trivial example of a Lagrangian translator, since it will satisfy (2.4).
Using (2.5) we deduce the following result. Proof. If M is a translator in the e z direction then (2.6) is satisfied by (2.5). We now suppose that (2.6) is satisfied on M. If b = 0 we deduce that M is a translator by differentiating (2. If n ≥ 3, then M ′ is almost calibrated and by (2.3) has a blow-down at −∞ given by If n = 2, then M ′ is an ancient solution γ to curve shortening flow in R 2 , which has a blow-down at −∞ which is a pair of non-parallel lines. We now show that γ must in fact be the asymptotic lines.
Lemma 2.11. Let γ = (γ(t)) −∞<t<T be an ancient smooth curve shortening flow in R 2 . Assume that a blow-down D λi (γ) (for λ i ց 0) is either a pair of unit density static lines ℓ 1 ∪ ℓ 2 meeting at one point or a single unit density line. Then the flow γ is the static line(s).
Proof. The case where the blow-down is a single unit density line follows from the monotonicity formula, so we only consider the case of a pair of transverse lines in the blow-down.
If γ were almost calibrated, then the classification of almost calibrated ancient solutions to Lagrangian mean curvature flow in [19,Proposition 4.1] implies that γ must be the lines, since they have distinct angles.
Let γ i := D λi (γ). Using the right-hand side of the integrated monotonicity formula ((2.2) with f = 1) and Fatou's lemma we can pick a time t < 0 (and a subsequence in i) such that on γ i (t) the curvature of the curve is locally uniformly bounded in L 2 . This implies locally uniform convergence in C 1,α from which it follows that, for i and R sufficiently large, γ i (t) ∩ B R (0) is given by the union of two small C 1,α graphs over (ℓ 1 ∪ ℓ 2 )(t). Since the flow is smooth (and using pseudolocality), this description of the flow has to persist for a short time. We deduce that the flow has a point with Gaussian density two (where the two graphs intersect) and thus the flow is backwards self-similar around that point. Since we have assumed that one blow-down is ℓ 1 ∪ ℓ 2 , the result follows.
By Lemma 2.11 we deduce that each component of M is a plane which has e z tangent to it, and hence M is trivially a translator.

The drift heat equation
It is well known that the functions 1, θ and the coordinate functions x i all satisfy the heat equation along the mean curvature flow. Along the rescaled flow we instead consider rescaled coordinate functions as follows.
Definition 3.1. For any coordinate function x i on C n we have the rescaled coordinate functionx i = e −τ /2 x i along the rescaled flow M τ . In particular, we have the rescaled heightw = e −τ /2 w.
Using the above definition, the next result, which is key for our purposes, follows from a straightforward rescaling.
Lemma 3.2. The functions 1, θ and the rescaled coordinate functionsx i satisfy the drift heat equation Note that, when computing derivatives ∂f ∂τ , the rescaled flow has velocity H + 1 2 x ⊥ . We will compare solutions of the drift heat equation along the rescaled flow with solutions on the blow-downs. By Proposition 2.7 all possible blow-downs are unions P ′ 1 ∪ P ′ 2 of two n-dimensional subspaces of C n meeting along a subspace of dimension less than n. We therefore study solutions of (3.1) on Euclidean spaces.
On an n-dimensional space P = R n we define the drift heat equation and drift Laplacian by (3.1) and (3.2). For a solution f (x, τ ) of the drift heat equation on P , we define the weighted norm f τ by By [6, Theorem 0.6] the function log f τ is convex in τ , and it is linear if and In this case log f 2 τ = −2λ + log h 2 and we say that f has degree 2λ. The eigenfunctions of the Ornstein-Uhlenbeck operator L = ∆−x·∇ on Euclidean space are well-studied, see e.g. Bogachev [1,Chapter 1]. The eigenvalues of L are non-negative integers k, and the corresponding eigenfunctions are degree k homogeneous polynomials given by products of Hermite polynomials. If H k is an eigenfunction of L with eigenvalue k, then the function h k (x) = H k (x/ √ 2) is an eigenfunction of L 0 with eigenvalue k/2. This leads to the following. Lemma 3.3. Let P = R n and let x i be coordinate functions on P . The eigenvalues of L 0 on P are given by non-negative half integers, and so the homogeneous solutions of the drift heat equation on P have non-negative integer degrees. The homogeneous solutions with degree 0 are the constants, while those with degree 1 are spanned by the rescaled coordinate functions e −τ /2 x i .

We will be interested in solutions to the drift heat equation on the blow-downs
where P ′ j are two distinct n-dimensional subspaces of C n . We define these as follows.
τ . We observe that the function θ, equal to the constant θ j on P ′ j , is a solution of the (drift) heat equation on P ′ 1 ∪ P ′ 2 in this sense. Note that we can see u = (u 1 , u 2 ) as one solution to the (drift) heat equation on the (immersed) shrinker P ′ 1 ∪ P ′ 2 , so we still have by [6, Theorem 0.6] that log u τ is convex, and it is linear if and only if u is homogeneous. Lemma 3.3 implies the following.
Lemma 3.5. On any blow-down P ′ 1 ∪ P ′ 2 the homogeneous solutions of the drift heat equation have non-negative integer degrees.
Recall our basic assumption that one blow-down is given by where the coordinate along ℓ is z and w is the height function vanishing along P 1 ∪ P 2 . We then have the following, which also uses the assumption that the Lagrangian angles of P 1 and P 2 are different. Proof. The degree 0 solutions on P 1 ∪ P 2 are given by pairs (c 1 , c 2 ) of constants. These are spanned by the functions 1, θ since θ equals two distinct constants θ j on the subspaces P j .
The degree 1 solutions on P 1 ∪ P 2 are given by pairs (f 1 , f 2 ) of linear functions on C n restricted to the subspaces. According to our choice of coordinates in §2.4, f 1 is in the span of x 1 , . . . , x n−1 , z and f 2 is in the span of x n , . . . , x 2n−2 , z. Since x 1 , . . . , x n−1 vanish on P 2 , and x n , . . . , x 2n−2 vanish on P 1 , the collection of functions x 1 , . . . , x 2n−2 on P 1 ∪P 2 define the pairs (x i , 0) and (0, x j ), where 1 ≤ i ≤ n−1 and n ≤ j ≤ 2n − 2. At the same time z, zθ contain the pairs (z, 0) and (0, z) in their span (again since θ takes distinct values on P 1 , P 2 ). given by a union of distinct n-dimensional subspaces, then along a subsequence we can extract a normalized limit of u, determining a solution of the heat equation on both planes separately. Standard methods allow us to extract limits on compact sets away from the intersection E = P ′ 1 ∩ P ′ 2 , and in the limit we obtain solutions of the heat equation on P ′ j \ E for j = 1, 2, which are in L ∞ across E. The main difficulty is that E may have codimension 1 in P ′ j and codimension 1 sets are not removable for solutions of the heat equation. (Consider for instance the solution given by two different constants on the lower and upper half planes.) To overcome this issue it is crucial that the angle θ differs on the two subspaces (which we have by Proposition 2.7), while at the same time the space-time integral of |∇θ| 2 converges to zero as we approach the blow-down. This allows us to show that the solutions that we obtain in the limit on P ′ j \ E are distributional solutions across E, and hence smooth. To state the result, let L i t be a sequence of smooth solutions of LMCF in C n defined for t ∈ [−1, 0]. We assume that the L i i have Euclidean area growth and uniformly bounded Lagrangian angles. We assume that L i t ⇀ P ′ 1 ∪ P ′ 2 weakly as i → ∞, where as above P ′ j are n-dimensional subspaces meeting along a subspace E of dimension at most n − 1. Here, as usual, we view P ′ 1 ∪ P ′ 2 as a static flow. Proposition 3.7. In the setting above, for each i, let u i be a solution to the heat equation on L t i for t ∈ [−1, 0], with at most polynomial growth. Assume further that there is a uniform C > 0 so that Then, after passing to a subsequence, we have u i → u where u = (u 1 , u 2 ) is a solution of the heat equation on the union P ′ 1 ∪ P ′ 2 for t ∈ (−1, 0] in the sense of Definition 3.4. The convergence u i → u here means smooth convergence on compact subsets of (−1, 0] × C n \ E, i.e. on compact subsets away from t = −1 and away from the intersection We apply the monotonicity formula (2.2) to u 2 i with different centers (x 0 , t 0 ) in C n × (−1, 0]. Using the uniform bound (3.4) we find that for any R > 0 there is a constant C R > 0 so that Let θ i be the Lagrangian angle on L i −1 and θ 1 = θ 2 the (constant) Lagrangian angles on P ′ 1 , P ′ 2 . As in [20, Theorem A], we have that for all s ∈ (−1, 0), f ∈ C 2 (R) and compactly supported smooth functions φ, Since θ 1 = θ 2 , we can choose f ∈ C 2 (R) such that f (θ 1 ) = 1 and f (θ 2 ) = 0, and we fix such a function f for the rest of the proof.
We also fix a smooth function χ compactly supported in B R (0) × (−1, 0). Then we have where D denotes the ambient derivative on Euclidean space, using the fact that both u i and θ i solve the heat equation on L i t and H = J∇θ i . Note that, since χ has compact support, we may use (3.5) and the fact that the (spacetime) L 2 -norm of ∇θ i goes to zero as i → ∞ to deduce that Therefore, since χ has compact support in B R (0) × (−1, 0), if we integrate (3.6) with respect to t on [−1, 0], we have that where ǫ i → 0 as i → ∞. (Note that ǫ i will depend on χ.) Integrating by parts on the right-hand side of (3.7) we get Again from (3.5) and the fact ∇θ i converges to zero in L 2 (in spacetime), we may absorb the second integral on the right-hand side of (3.8) into ǫ i . We can then integrate by parts in the first term on the right-hand side of (3.8) and absorb another term involving ∇θ i by the same argument into ǫ i to get Recall that f (θ 1 ) = 1 and f (θ 2 ) = 0. Since the u i are uniformly bounded on the support of χ by (3.5), and we have good convergence away from the line ℓ = P 1 ∩P 2 , the contribution as we pass to the limit as i → ∞ in (3.9) near the singular set ℓ is negligible. Therefore, we can pass to the limit in (3.9) along a subsequence, and get that the subsequential limit u 1 of the u i on P 1 satisfies This means that the limit u 1 is a bounded distributional solution of the heat equation on P 1 so it follows that u 1 is a classical solution on P 1 .
Repeating the argument starting with the subsequence converging to u 1 on P 1 and changing the choice of function f so that it takes the value 1 on θ 2 and 0 on θ 1 yields the result.
Since the drift heat equation and usual heat equation are related by rescaling, one can apply Proposition 3.7 to sequences of solutions of the drift heat equation along rescaled mean curvature flows. In particular, suppose that we have a sequence of rescaled flows M i τ , for τ ∈ [−1, 0], converging to P ′ 1 ∪P ′ 2 weakly. Recall the weighted L 2 -norm defined in (3.3) and let u i be solutions of the drift heat equation on M i τ , with u i −1 ≤ 1, and such that the u i have polynomial growth. Proposition 3.7 implies that after passing to a subsequence we have u i → u, for a solution u of the drift heat equation on P ′ 1 ∪ P ′ 2 for τ ∈ (−1, 0]. We have the following additional information, saying that for τ > −1 the weighted L 2 -norms of the u i cannot concentrate near E and near infinity.
Lemma 3.8. Under the setup above we have Proof. The inequality u i τ ≤ u i −1 for τ > −1 follows immediately from the monotonicity formula (2.2) and the observation that and B r (E) denotes the r-neighbourhood of E. Let δ > 0. From Proposition 3.7 we know that for any r, R > 0, and τ ∈ [−1 + δ, 0] we have To prove (3.10) it is enough to show that for any ǫ, δ > 0, there are r, R > 0 such that for all i and τ ∈ [−1 + δ, 0] we have First, using the log Sobolev inequality due to Ecker [13, Theorem 3.4], we have a p > 1, depending on δ > 0, such that for a uniform C, as long as τ ∈ [−1 + δ, 0]. It follows using Hölder's inequality that, given R > 0, we have and so using the Euclidean area bounds for M i τ we can find an R (depending on δ, ǫ) such that Viewing R (and δ) as fixed, the uniform bound in (3.5) implies that if r is sufficiently small (depending on ǫ, δ, R), then Combined with (3.11) this implies M i τ \Ar,R |u i | 2 e −|x| 2 /4 < ǫ, as required.

Three annulus lemma.
A well-known method for controlling the growth of solutions of PDEs is the three annulus lemma, see for example [22]. In this subsection we prove a version of the three annulus lemma for solutions of the drift heat equation along the rescaled flow. We use an argument by contradiction, similar to Simon [22], based on the monotonicity of frequency shown by Colding-Minicozzi [6]. Related ideas are also applied in [9]. In this subsection we assume that L τ is a rescaled Lagrangian mean curvature flow such that, along a sequence τ i → −∞, we have L τi ⇀ P 1 ∪ P 2 . In addition we assume, as before, that the L τ have uniformly bounded area ratios, uniformly bounded Lagrangian angle and are almost calibrated for n ≥ 3.
For the statement of the three annulus lemma, we say that a solution u of the drift heat equation (3.1) on the rescaled flow M τ has polynomial growth if there are τ -independent constants C, k > 0 such that |u(x, τ )| ≤ C(1 + |x| k ).  By rescaling we can assume that u i −Ti−1 = 1 for all i. It follows from (3.13) that then u i −Ti−2 < e s/2 . We can apply Proposition 3.7 to time translations of the u i , and along a subsequence we can extract a limit u satisfying the drift heat equation along a blow-down P ′ 1 ∪ P ′ 2 of the flow L τ on the interval (−2, 0]. Using (3.10) we have By [6, Theorem 0.6] we know that log u 2 τ is convex. From (3.14) and (3.15) it follows that log u 2 τ is linear with slope s. By [6, Theorem 0.6] u must be homogeneous with degree s. By Lemma 3.5 the homogeneous solutions on any blow-down have integer degrees, so since s ∈ Z, we have a contradiction.
We can use the three annulus lemma to extract the leading order behaviour of ancient solutions to the heat equation as follows. Proof. Let s 0 > d for some s 0 ∈ Z. We claim that there is a τ 0 < 0 such that we then have (3.16) u τ −1 ≤ e s0/2 u τ for all τ < τ 0 . If this were not the case, then Proposition 3.9 would imply that in fact u τ −1 ≥ e s0/2 u τ for all sufficiently negative τ , but this would eventually contradict the assumption u 2 τ ≤ Ce −dτ . The growth condition (3.16) together with the normalization of u i implies that u i −3 ≤ e 3s0/2 . Using Proposition 3.7 we can extract a limit u along a subsequence on P ′ 1 ∪ P ′ 2 . The convergence is locally smooth on (−3, 0] away from P ′ 1 ∩ P ′ 2 , and using Lemma 3.8 the convergence is in L 2 for τ ∈ [−2, 0] as required.
It remains to argue that u is homogeneous. For this note that Proposition 3.9 implies that for any s ∈ Z one of the following must hold: (1) u τ −1 ≥ e s/2 u τ for all sufficiently negative τ , (2) u τ −1 ≤ s s/2 u τ for all sufficiently negative τ , since if (1) holds for some sufficiently negative τ then it must hold for all more negative τ as well. It follows that there is some s 1 ∈ R such that (1) holds for all s < s 1 , and (2) holds for all s > s 1 . We deduce that in the limit we have u −2 = e s1/2 u −1 , u −1 = e s1/2 u 0 .
The convexity of log u τ then implies that log u τ is linear, from which it follows that u is homogeneous.
We will also need the following variant of the three annulus lemma, similar to Donaldson-Sun [12,Proposition 3.11].
Given s ∈ Z, there is a T 0 > 0 with the following property. If Proof. The proof is by contradiction, similar to that of Proposition 3.9. Suppose that we have a sequence T i → ∞ and corresponding u i such that and at the same time Let By scaling we can assume that v i −Ti−1 = 1. It follows that Π −Ti−1 v i −Ti−1 ≤ 1, and so by (3.19) we have v i −Ti−2 ≤ e s/2 . We claim that for sufficiently large i we also have If We claim that we also have (3.21) v 0 ≤ e −s/2 , in which case we will reach a contradiction just like in the proof of Proposition 3.9. Note that the new difficulty is that we only have the bound Π −Ti v i −Ti ≤ e −s/2 , and the norm of v i can be larger than that of its projection Π −Ti v i . To see that (3.21) holds we show that under our assumption that v i is orthogonal to V at time −T i − 2, we have that v i is also approximately orthogonal to V at time −T i . Let g ∈ V and consider normalizations g i of g such that g i −Ti = 1. By Proposition 3.10, after taking a further subsequence we can assume that the g i converge to a homogeneous limit g on P ′ 1 ∪ P ′ 2 , on the time interval [−2, 0], satisfying the drift heat equation. We can apply the L 2 -convergence (3.10) to v i ± g i , together with our assumption v i , g i −Ti−2 = 0 to find that v, g −2 = 0. Since g is homogeneous, this implies that v, g τ = 0 for all τ ∈ [−2, 0]. It follows from the L 2 -convergence we have v i , g i −Ti → 0. Since this applies to all g ∈ V , we find that and it follows that v 0 ≤ e −s/2 . This leads to a contradiction as discussed above.
Let us use coordinates x 1 , . . . , x 2n−2 , z, w as in §2.4. Recall that a blow-down of our ancient rescaled flow M τ along a sequence of scales τ i → −∞ is given by P 1 ∪ P 2 , where P 1 ∩ P 2 = ℓ is a line, the coordinate w vanishes on P 1 ∪ P 2 and J∇z = ∇w. Without loss of generality we can assume that the τ i are all integers.
Let us write L i t for the corresponding sequence of flows for t ∈ [−2, 0] converging weakly to P 1 ∪ P 2 . We then have the following dichotomy.
Proposition 3.12. Either we have that w = a + bθ for some constants a, b along our flow L t , so L t is a translator, or up to choosing a subsequence of the τ i we can find a sequence of linear functions φ i ∈ Span{x 1 , . . . , x 2n−2 , z} with φ i → 0 and a sequence σ i → 0 such that along the sequence L i t converging to P 1 ∪ P 2 we have where the convergence is in L 2 and locally uniformly away from the line ℓ.
Proof. Recall Definition 3.1 and let V = Span{1, θ,x 1 , . . . ,x 2n−2 ,z}. Suppose first that the rescaled heightw is in V . Note that M τi ⇀ P 1 ∪ P 2 and w vanishes on P 1 ∪ P 2 , but non-trivial linear combinations of x 1 , . . . , x 2n−2 , z do not vanish on P 1 ∪ P 2 . This implies that we must havew = a + bθ for constants a, b. By Proposition 2.10 the flow L t is a translator.
Suppose now thatw is not in V . We apply Proposition 3.11 tow along the flow with V as chosen. For any integer k < 0 let us writẽ Note that by our assumption Π kw = 0 for all k. Using Proposition 3.11 together with the argument in the proof of Proposition 3.10 we find that along a subsequence k i = τ i → −∞, time translations of thew ki converge to a homogeneous solution w of the drift heat equation on P 1 ∪ P 2 , which is orthogonal to the solutions 1, θ,x 1 , . . . ,x 2n−2 ,z. At the same time the growth rate of w can be at most degree 1, so by Lemma 3.6 we must have w = ce −τ /2 zθ for a non-zero constant c.
To finish the proof we need to consider how thew k are related for different k. By definition we have where γ k , a k , b k are constants, and F k ∈ Span{x 1 , . . . ,x 2n−2 ,z}. Using Proposition 3.11, and arguing as in the proof of Proposition 3.10, we know that for any subsequence k j → −∞ there is a further subsequence along which thew kj (translated in time) converge to a homogeneous solution along some blow-down P ′ 1 ∪ P ′ 2 , with degree 1 which is orthogonal to V . Since w k k = 1 for all k, it follows that γ k → e −1/2 and a k k , b k θ k , F k k → 0 as k → −∞. Note that the norms 1 k , θ k are uniformly bounded away from 0 and ∞ for all k, using the fact that on all blow-downs P ′ 1 ∪ P ′ 2 the angle θ equals the same constants θ 1 , θ 2 on the two subspaces P ′ 1 , P ′ 2 . Therefore a k , b k → 0. Let us define the constants µ k by µ 0 = 1 and γ k = µ k+1 /µ k for all sufficiently negative integers k. From (3.22) we have Using that µ k+1 /µ k → e −1/2 and a k , b k → 0, it follows that At the same time, sincew 0 is the normalized L 2 -projection of w orthogonal to V (at time τ = 0), we havew 0 = c 0w + c 1 + c 2 θ + F , where c 0 , c 1 , c 2 are constants with c 0 = 0 and F ∈ Span{x 1 , . . . ,x 2n−2 ,z}. Using (3.23) and (3.24) we can write where E k k → 0 andφ k is in the span ofx 1 , . . . ,x 2n−2 ,z. Along our subsequence k i we havew ki → w = ce −τ /2 zθ, and so as required we obtain a sequence L i t converging to P 1 ∪ P 2 , and σ i = 0, φ i ∈ Span{x 1 , . . . , x 2n−2 , z} satisfying are bounded away from 0 and ∞. It follows that if φ i → 0, along a subsequence, then also σ i → 0 along this subsequence, and we would have σ −1 i φ i → zθ in L 2 , but this contradicts the fact that on P 1 ∪ P 2 the function zθ is L 2 -orthogonal to x 1 , . . . , x 2n−2 , z. Therefore we must have φ i → 0, which implies that w − φ i L i −1 → 0 and so σ i → 0 as well. In the next section we will show using a topological argument that the second alternative in Proposition 3.12 leads to a contradiction. This will complete the proof of our main result.

Linking argument
In this section we use a topological argument to rule out the second alternative in Proposition 3.12. Throughout this section we let (−∞, 0) ∋ t → L t ⊂ C n be a smooth, exact, ancient solution of LMCF with uniformly bounded area ratios and Lagrangian angle and which is almost calibrated for n ≥ 3. Recall that for a positive sequence λ i → 0, we consider the sequence of parabolically rescaled flows We assume that as i → ∞ the flows t → L i t converge weakly to the static flow (−∞, 0) ∋ t → P 1 ∪ P 2 , where P 1 , P 2 are n-dimensional Lagrangian subspaces meeting along a line ℓ. We write θ j for the Lagrangian angles of P j as before, where θ 1 = −θ 2 .
Since the L i t are exact, they admit primitives β i of the Liouville form as in Definition 2.2. We have the following (see Neves [20, Proposition 6.1]). Since |∇β i | = |x ⊥ | and L i −1 converges to the union P 1 ∪ P 2 locally smoothly away from ℓ, we have that β i | L i −1 → β j as i → ∞ locally smoothly on each plane P j away from ℓ, for suitable constants β j . Similarly θ i → θ j locally smoothly on P j away from ℓ, as i → ∞. Given this, we make the following definition. Definition 4.2. Since the L i t are exact, and almost calibrated for n ≥ 3, by [21,Theorem 4.2] there exists a set E ⊆ (−2, 0) of measure zero so that whenever s ′ ∈ (−2, 0)\E, we have two distinct connected components Σ i 1,s ′ , Σ i 2,s ′ of B 3 (0)∩L i s ′ (after possibly passing to a subsequence) intersecting B 2 (0) and converging (as Radon measures) to the planes P 1 , P 2 respectively in B 2 (0). Note that there might be more connected components of B 3 (0) ∩ L i s ′ , but the components Σ i 1,s ′ , Σ i 2,s ′ are uniquely determined, and the remaining components converge to zero as Radon measures. Let This is always possible since θ 1 = θ 2 and E has measure 0. We then let Σ i j = Σ i j,s1 in the notation above.   Remark 4.4. The idea of Proposition 4.3 is that in the limit as i → ∞ the functions B i approximate the constants b j on the two planes P j . Therefore B i z approximately satisfies the heat equation as i → ∞ and as such should stay close to a genuine solution h i with the same initial condition. In addition we have a good pointwise estimate for the difference between B i and the constants b j on the two components Σ i j at the specific time t = s 1 , as in Neves [21,Theorem 4.2]. Proof. Let Our goal is to show that E i is small as i becomes large. At t = −1 we have E i = 0, so we compute the evolution of E i . We have and since β i + 2(t − s 1 )θ i satisfies the heat equation we get Since |B i | ≤ 1, at t = −1 we have |h i | ≤ (1 + |x| 2 ). Using the maximum principle and the evolution equation (∂ t − ∆)(1 + |x| 2 ) = −2n we find that |h i | ≤ C(1 + |x| 2 ) for t ∈ [−1, 0) for a dimensional constant C > 0. Below the constant C may change from line to line but is independent of i, t. In particular we also have |E i | ≤ C(1 + |x| 2 ).
Since z and h i satisfy the heat equation along the flow L i t , we have the evolution equation We deduce that From this, together with the estimate |E i | ≤ C(1 + |x| 2 ), we get . Using that θ i satisfies the heat equation and |∇θ i | = |H|, as well as (∂ t −∆)|x| 4 ≤ 0, we also have Let κ > 0 be small. Combining (4.5) with the previous inequality, for t ∈ (−1, 0) we have Suppose that x 0 ∈ B 1 (0) ∩ L i s1 and denote by ρ x0,s1 the backwards heat kernel centred at (x 0 , s 1 ). For t ∈ [s 0 , s 1 ) we have from (4.6), using the monotonicity formula (2.2), that Integrating (4.7) with respect to t from −1 to s 1 yields: Note that E i = 0 at t = −1 by the definition of h i , and hence the first term on the right-hand side in (4.8) vanishes. We now estimate the second term in (4.8). Note that for t ∈ [−1, s 1 − κ] we have for some κ-dependent constant C κ > 0, since s 1 < 0 and thus ρ x0,s1 will decay more rapidly at infinity than ρ 0,0 for any t ∈ [s 0 , s 1 − κ]. Therefore, We now notice that where C > 0 is a constant and κ is chosen sufficiently small that (t + 1) ≥ 2Cκ for t ∈ [s 1 − κ, s 1 ]. Equation (4.10) shows that the integral on the left-hand side of the inequality can be compensated for using the last term in (4.8).
Our remaining concern is (4.11) The first integral can clearly be estimated as s1 s1−κ L i t ∩B κ −1/10 (0) for some constant C > 0, using the uniform area bounds for L i t . Using the area bounds again for t ∈ [−1, s 1 ], we can estimate our remaining spacetime integral by the integral over an n-plane P for κ sufficiently small: for constants C 1 , C 2 > 0. Combining (4.9)-(4.13) shows that, for κ sufficiently small, we have for some constant C > 0 and a constant C κ > 0 depending on κ.
Noting also that θ 2 i is uniformly bounded, we may therefore combine (4.8) and (4.14) to obtain if κ > 0 is sufficiently small. For fixed κ > 0, the first term on the right-hand side of (4.15) converges to zero as i → ∞, as in [20,Lemma 5.4]. It follows that for any κ > 0 we can choose i sufficiently large so that (4.4), and the fact that B i = cos(β i ) at t = s 1 , we have that As in [20,Lemma 7.3], we now use that the limiting behaviour of the functions B i in (4.2) as i → ∞ is t-independent. More precisely, for all φ with compact support in B 2 (0) and f ∈ C 2 (R) we have (4.17) lim On L i s1 we have B i = cos(β i ) and so we have the pointwise bound |∇B i | ≤ |x ⊥ |. Using the Poincaré type inequality [20, Proposition A.1], we deduce that there are constantsb 1 ,b 2 such that sup Σ i j |B i −b j | → 0 as i → ∞. At the same time from (4.17) we find thatb j = b j for the constants in (4.1), since on L i −1 we have B i = cos(β i − 2(1 + s 1 )θ i ). Note that by construction on L i −1 we have β i → β j and θ i → θ j on the plane P j , locally smoothly away from ℓ. It follows then from (4.16) that lim i→∞ sup Σ i j |b j z − h i | = 0, as required. 4.2. The linking argument. Continuing the setup from the previous subsection, we now show that indeed the second possibility in Proposition 3.12 leads to a contradiction, if our flow t → L t is smooth and embedded.
Proposition 4.5. Suppose that we have φ i ∈ Span{x 1 , . . . , x 2n−2 , z} with φ i → 0 and a sequence λ i → 0 such that along the sequence L i t we have (4.18) where the convergence is in L 2 and locally uniform away from ℓ. Then for sufficiently large i the flow L i t is not embedded.
Remark 4.6. Recall that Σ i j are the components of B 2 (0) ∩ L i s1 , as in Definition 4.2, and let us suppose for simplicity that C i j = Σ i j ∩ ∂B 1 (0) are smooth (n − 1)dimensional submanifolds of the sphere (which can always be done by changing the radius of the ball slightly if necessary). The key to the argument is to show that the submanifolds C i j in the (2n − 1)-sphere are linked for i sufficiently large, which implies that the Σ i j intersect in B 2 (0). Then L i s1 cannot be embedded.
Proof. Since θ equals the distinct constants θ 1 , θ 2 on the planes P 1 , P 2 , by modifying the λ i and adding multiples of z to the φ i , we can assume for simplicity that where b j are given in (4.1). We also assume without loss of generality that λ i > 0.
Recall the notation of Definition 4.2 and Proposition 4.3. Using the L 2 convergence of the u i at t = −1 and the monotonicity formula applied with points (x 0 , s 1 ) for different x 0 ∈ B 2 (0), we then have Applying Proposition 4.3 then yields We deduce that, given any ǫ > 0, once i is sufficiently large we will have (4.19) |w − φ i − λ i b j z| < ǫλ i on Σ i j . Suppose without loss of generality that b 1 > b 2 and choose and, recalling that φ i ∈ Span{x 1 , . . . , x 2n−2 , z}, define half-spaces . , x 2n−2 , z, w) ∈ C n : w < φ i + λ i b 0 z}. The inequality (4.19) implies that, for all i sufficiently large, In other words, the relative positions of the components Σ i j in terms of the halfspaces H i ± must switch as we pass from z > 1/2 to z < −1/2. We can choose R = 1 + δ for δ ≥ 0 small such that (4.21) C i j = Σ i j ∩ ∂B R (0) are smooth. Our aim now is to show that the submanifolds C i j in ∂B R (0) are linked for sufficiently large i, which will imply that the Σ i j intersect in B R (0). Consider the two points p − , p + whose only non-zero entries are ±R in the zcomponent in coordinates (x 1 , . . . , x 2n−2 , z, w) on C n . So p − , p + lie on ℓ ∩ ∂B R (0) where ℓ = P 1 ∩ P 2 . Since the Σ i j converge smoothly to P j away from the singular line ℓ as i → ∞, we can assume that outside of B 1/20 (p ± ) the submanifolds C i j are smooth perturbations of P j ∩ ∂B R (0).
Any connected components of the C i j contained entirely inside B 1/10 (p ± ) must lie in different half-spaces H i ± by (4.20) for i sufficiently large, and so do not contribute to the linking number of the C i j . We may therefore discard these components, if there are any, and assume from now on that the C i j are connected. For j = 1, 2, letP i j be the graph of w = φ i + λ i b j z over P j . Since φ i , λ i → 0, theP i j are small perturbations of the P j for i sufficiently large. Moreover, since b 1 = b 2 by (4.1) we have that theP i j intersect transversely at the origin. Hence, the spheres (4.22)C i j =P i j ∩ ∂B R (0) have linking number 1.
We now claim that, for i sufficiently large, the submanifolds C i j in (4.21) can be deformed to theC i j in (4.22) without any crossings. Outside of the balls B 1/20 (p ± ) this is clear since there the C i j are smooth perturbations of theC i j . At the same time, for i sufficiently large, inside the balls B 1/10 (p ± ) the pairs of submanifolds {C i j ,C i j } are contained in disjoint half-spaces for j = 1, 2 by (4.20), so the submanifolds in each pair can be deformed to coincide without intersecting the submanifolds in the other pair.
We conclude that the linking number of the submanifolds C i j in (4.21) is therefore also 1 for sufficiently large i, which implies that the flow is not embedded.

Proof of Main Theorem
We first show that combining the results from Section 3.2 and Section 4.2 yields a proof of the main theorem: Proof of Theorem 1.1. We can assume that the ancient flow M is defined for t < 0. We note that the assumption that the flow has a blow-down given by the static union of the planes P 1 ∪ P 2 implies that the entropy is bounded above by 2. This implies that if the flow has an immersed point (x 0 , t 0 ), then the monotonicity formula yields that the flow is backwards self-similar around (x 0 , t 0 ), i.e. the flow is given by the static flow (M P1∪P2 + (x 0 , t 0 )) ∩ {t < 0}.
We can thus assume that M is embedded. Combining Proposition 3.12 and Proposition 4.5 yields the statement.
Since in many geometric applications it is essential to classify not only smooth ancient solutions to mean curvature flow, but also more general Brakke flows arising as limit flows, we also record the following extension of Theorem 1.1.
Theorem 5.1. Let P 1 , P 2 ⊂ C 2 be Lagrangian subspaces which intersect along a line ℓ through 0. Let M be an ancient 2-dimensional Brakke flow which is the (weak) limit of smooth, zero-Maslov, exact Lagrangian mean curvature flows (L i t ) −R 2 i <t<0 defined on B(0, R i ) ⊂ C n , where R i → ∞, with uniformly bounded variation of the Lagrangian angle and uniformly bounded area ratios. If M has a blow-down at −∞ given by the static flow consisting of the union of the planes P 1 ∪ P 2 , then M is a smooth translator.
Proof. We again assume that M is defined and non-vanishing for t < 0. Note that the assumptions imply that M has uniformly bounded area ratios and is unit regular, meaning that every point of Gaussian density one has a space-time neighbourhood where the flow is smooth. Furthermore, as in the proof of Theorem 1.1, it follows that the entropy is bounded above by 2. Assume now that there is a point (x 0 , t 0 ) where the Gaussian density of M is 2. Then as above we see that M = (M P1∪P2 + (x 0 , t 0 )) ∩ {t < 0} (using unit regularity to conclude that neither of the two planes can vanish before t = 0). We can thus assume that all Gaussian density ratios of M are strictly less than 2. Neves structure theory [20] then implies that we obtain uniform local curvature bounds along the sequence (L i t ) −R 2 i <t<0 , and thus the convergence is smooth. This yields that M is a smooth, ancient, zero-Maslov, exact Lagrangian mean curvature flow with uniformly bounded variation of the Lagrangian angle and uniformly bounded area ratios. Theorem 1.1 then implies the statement. are defined on the Riemannian manifolds (B(p i , R i ), g i ) where B(p i , R i ) ⊂ R 2n are geodesic balls with respect to g i and g i is a sequence of Calabi-Yau metrics on B(p, R i ) converging smoothly to the standard Euclidean metric on C n .