# On short time existence of Lagrangian mean curvature flow

- First Online:

- Received:
- Revised:

- 702 Downloads

## Abstract

We consider a short time existence problem motivated by a conjecture of Joyce (Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. arXiv:1401.4949, 2014). 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.

## 1 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 and Wolfson [14] 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 [20] found a *K*3 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 [15] 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 [13]. 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 [12].

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 [11] 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 \(\mathbb {C}^2\) every such union is simply a union of Lagrangian planes, and so the case we consider in the theorem below 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 non-area-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 \(L\subset \mathbb {C}^n\) is a compact Lagrangian submanifold of \(\mathbb {C}^n\) 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 \(t \searrow 0\), \(L_t \rightarrow L\) as varifolds and in \(C^\infty _\mathrm{loc}\) away from the singularities.

We remark that the assumptions \(L \subset \mathbb {C}^n\) 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 \(L \subset M\) 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 \(2\pi /3\), 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 Sect. 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 \(s\rightarrow 0\) 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 [19] 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 \(C^{1,\alpha }\) 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 and Neves [10] and Imagi et al. [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 self-expander, and hence estimates on the Gaussian densities of the approximating flow.

*Organisation* The paper is organised as follows. In Sect. 2 we recall key definitions and results. In Sect. 3 we derive evolution equations and monotonicity formulas for geometric quantities under the flow. In Sect. 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, Sect. A, contains miscellaneous technical results, including Ecker–Huisken style curvature estimates for high-codimension mean curvature flow.

## 2 Preliminaries

### 2.1 Mean curvature flow

*n*-dimensional embedded submanifold of \(\mathbb {R}^{n+k}\). A mean curvature flow is a one parameter family of immersions \(F:M\times [0,T)\rightarrow \mathbb {R}^{n+k}\) such that the normal velocity at any point is given by the mean curvature vector, that is

*self-expanders*. These are submanifolds \(M \subset \mathbb {R}^{n+k}\) satisfying the elliptic equation

*Gaussian density ratio*centred at \((x_0,t_0)\) and at scale

*r*by

**Theorem 2.1**

*r*. Consequently we can define the

*Gaussian density*as

**Theorem 2.2**

*A*(

*x*,

*t*) is the second fundamental form of \(M_t\) at the point

*x*.

Finally, we introduce what it means for two manifolds to be \(\varepsilon \)-close in \(C^{1,\alpha }\). Given an open set \(U\subset \mathbb {R}^{n+k}\) and two *n*-dimensional submanifolds \(\Sigma \) and *L* defined in *U*, we say that \(\Sigma \) and *L* are 1-close in \(C^{1,\alpha }(W)\) for any \(W \subset U\) with \(\mathrm {dist}(W, \partial U) \ge 1\) if for all \(x \in W\), \(B_1(x) \cap \Sigma \) and \(B_1(x) \cap L\) are both graphical over some common *n*-dimensional plane, and if *u* and *v* denote the respective graph functions then \(\Vert u - v\Vert _{1,\alpha } \le 1\). We then say that \(\Sigma \) and *L* are \(\varepsilon \)-close in *W* if after rescaling by a factor \(1/\varepsilon \), \(\Sigma \) and *L* are 1-close in \(\varepsilon ^{-1}W\) for any *W* with \(\mathrm {dist}(\varepsilon ^{-1}W, \varepsilon ^{-1}\partial U) \ge 1\).

### 2.2 Lagrangian submanifolds and Lagrangian mean curvature flow

*J*denote the standard complex structure on \(\mathbb {C}^n\) and \(\omega \) the standard symplectic form on \(\mathbb {C}^n\), defined by

*n*-dimensional submanifold of \(\mathbb {C}^n\) is

*Lagrangian*if \(\omega |_L = 0\). We also consider the closed

*n*-form \(\Omega \), called the

*holomorphic volume form*, defined by

*L*. We call \(e^{i\theta _L}:L\rightarrow S^1\) the

*Lagrangian phase*, and \(\theta _L\) the

*Lagrangian angle*, which may be a multi-valued function. We henceforth suppress the subscript

*L*. In the case that \(\theta \) is a single valued function, we say that the Lagrangian

*L*is

*zero-Maslov*. An equivalent condition is \([d\theta ] = 0\), that is, \(d\theta \) is cohomologous to 0. If \(\theta \equiv \theta _0\) is constant, then we say that

*L*is

*special Lagrangian*. In this case

*L*is calibrated by \(\mathrm {Re}(e^{-i\theta _0}\mathrm {vol}_L)\), and hence is area-minimising in its homology class.

*L*is

*exact*. In this paper we will be more interested in local exactness, that is when the Liouville form \(\lambda \) only has a primitive in some open set. It can be shown that any smooth Lagrangian is locally exact.

A *Lagrangian mean curvature flow* in \(\mathbb {C}^n\) is a mean curvature flow \((L_t)_{0\le t<T}\) with \(L_0\) Lagrangian. As proved by Smoczyk [15], it turns out that the Lagrangian condition is preserved by the mean curvature flow.

## 3 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.

**Lemma 3.1**

- (i)
\(\frac{d\theta _t}{d t} = \Delta \theta _t\), where \(\theta _t\) is the Lagrangian angle for \(L_t\). Note that since only derivatives of \(\theta _t\) appear here, this does not require the assumption that the flow is zero-Maslov.

- (ii)
In an open set where the flow is zero-Maslov and exact with \(\beta _t\) a primitive for the Liouville form, \(\frac{d\beta _t}{d t} = \Delta \beta _t - 2\theta _t\).

- (iii)
\(\left( \frac{d \rho _{(x_0,t_0)}}{d t} + \Delta \rho _{(x_0,t_0)}\right) - |\vec {H}|^2\rho _{(x_0,t_0)}= - \left| \vec {H} - \frac{(x_0 - x)^\perp }{2(t_0 -t)}\right| ^2\rho _{(x_0,t_0)}\).

*Remark*

*Proof*

*x*, and denote the coordinate tangent vectors by \(\{\partial _1,\dots ,\partial _n\}\). Then we calculate

*Remark*

Let \(\phi \) be a cut-off function supported on \(B_3\) with \(0\le \phi \le 1\), \(\phi \equiv 1\) on \(B_2\) and the estimates \(|D\phi | \le 2\) and \(|D^2\phi | \le C\). We then have the following lemma.

**Lemma 3.2**

*Remark*

*Proof*

## 4 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 \(P_1\), \(P_2\subset \mathbb {C}^n\) be Lagrangian planes intersecting transversally such that neither \(P_1 + P_2\) nor \(P_1 - P_2\) are area minimising. We denote by \(P := P_1 + P_2\). We will need the following uniqueness result, proved by Lotay and Neves [10] in dimension 2 and Imagi et al. [6] in dimensions 3 and higher.

**Theorem 4.1**

There exists a unique smooth, zero-Maslov class Lagrangian self-expander asymptotic to *P*.

**Theorem 4.2**

*R*,

*r*, \(\tau > 0\), \(\alpha \), \(\varepsilon _0 < 1\), and \(C, M < \infty \). Let \(\Sigma \) be the unique smooth zero-Maslov Lagrangian self-expander asymptotic to

*P*. Then for all \(\varepsilon > 0\) there exists \(\tilde{R} \ge R\), \(\eta \), \(\nu > 0\) each dependent on \(\varepsilon _0\), \(\varepsilon \),

*r*,

*R*, \(\tau \), \(\alpha \),

*C*,

*M*and

*P*such that if

*L*is a smooth Lagrangian submanifold which is zero-Maslov in \(B_{\tilde{R}}\) and

- (i)
\(|A|\le M\) on \(L\cap B_{\tilde{R}}\),

- (ii)
\(\int _L \rho _{(x,0)}(y,-r^2)d\mathcal {H}^n \le 1 + \varepsilon _0\) for all

*x*and \(0 < r \le \tau \), - (iii)
\(\int _{L\cap B_{\tilde{R}}} |\vec {H} - x^\perp |^2d\mathcal {H}^n \le \eta \),

- (iv)The connected components of \(L\cap A(r,\tilde{R})\) (where \(A(r, \tilde{R}) := \overline{B_{\tilde{R}}}{\setminus } B_r\)) are in one to one correspondence with the connected components of \(P \cap A(r, \tilde{R})\) andfor all \(x \in L\cap A(r, \tilde{R})\);$$\begin{aligned} \mathrm {dist}(x, P) \le \nu + C\mathrm {exp}\left( \frac{-|x|^2}{C}\right) , \end{aligned}$$

*L*is \(\varepsilon \)-close to \(\Sigma \) in \(C^{1,\alpha }(\overline{B}_{\tilde{R}})\).

*Proof*

- (1)
\(|A^{L_i}| \le M\) on \(L_i\cap B_{R_i}\),

- (2)
\(\int _{L_i} \rho _{(x,0)}(y,-r^2)d\mathcal {H}^n \le 1 + \varepsilon _0\) for all

*x*and \(0 < r \le \tau \), - (3)
\(\int _{L_i\cap B_{R_i}} |\vec {H} - x^\perp |^2 d\mathcal {H}^n \le \eta _i\)

- (4)The connected components of \(L_i\cap A(r,R_i)\) are in one to one correspondence with the connected components of \(P\cap A(r,R_i)\) andfor all \(x \in L_i\cap A(r,R_i)\),$$\begin{aligned} \mathrm {dist}(x,P) \le \nu _i + C\exp \left( \frac{-|x|^2}{C}\right) \end{aligned}$$
- (5)
\(L_i\) is not \(\varepsilon \)-close to \(\Sigma \) in \(C^{1,\alpha }(B_{R_i})\).

*P*in \(C^{1,\alpha }\). Hence, in order that (5) is satisfied, we conclude that for large

*i*, \(L_i\) is not \(\varepsilon \)-close to \(\Sigma \) in \(C^{1,\alpha }(B_\rho )\).

*P*. Theorem 4.1 then implies that \(L_\infty = \Sigma \), which contradicts (5). \(\square \)

## 5 Main theorem

*P*. We assume the existence of a family \((L^s)_{0<s\le c}\) 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 Sect. 7.

- (H1)The area ratios are uniformly bounded, i.e. there exists a constant \(D_1\) such that$$\begin{aligned} \mathcal {H}^n(L^s\cap B_r(x)) \le D_1 r^n \quad \forall r > 0, \quad \forall s \in (0,c], \quad \forall x. \end{aligned}$$
- (H2)There is a constant \(D_2\) such that for every
*s*and \(x \in L^s\cap B_4\)where \(\theta ^s\) and \(\beta ^s\) are, respectively, the Lagrangian angle of \(L^s\) and a primitive for the Liouville form on \(L^s\).$$\begin{aligned} |\theta ^s(x)| + |\beta ^s(x)| \le D_2(|x|^2 + 1), \end{aligned}$$ - (H3)For any \(\alpha \in (0, 1)\), the rescaled manifolds \(\tilde{L}^s := (2s)^{-1/2}L^s\) converge in \(C^{1,\alpha }_\mathrm{loc}\) to \(\Sigma \). Moreover the second fundamental form of \(\tilde{L}^s\) is bounded uniformly in
*s*and without loss of generality we can assume thatlocally on \(\tilde{L}^s\). (Note that \(\tilde{L}^s\) is exact in the ball \(B_{4(2s)^{-1/2}}\) so we can make sense of \(\tilde{\beta }^s\) in the limit.)$$\begin{aligned} \lim _{s\rightarrow 0} (\tilde{\theta }^s + \tilde{\beta }^s) = 0 \end{aligned}$$ - (H4)The connected components of \(P \cap A(r_0\sqrt{s}, 4)\) are in one to one correspondence with the connected components of \(L^s\cap A(r_0\sqrt{s}, 4)\), and each component can be parametrised as a graph over the corresponding plane \(P_i\)where the function \(u_s:P\cap A(r_0\sqrt{s}, 3) \rightarrow P^\perp \) is normal to$$\begin{aligned} L^s \cap A(r_0\sqrt{s},3) \subset \{x + u_s(x) | x \in P\cap A(r_0\sqrt{s}, 3)\} \subset L^s\cap A(r_0\sqrt{s}, 4), \end{aligned}$$
*P*and satisfies the estimatewhere \(\overline{\nabla }\) denotes the covariant derivative on$$\begin{aligned} |u_s(x)| + |x|\left| \overline{\nabla }u_s(x)\right| + |x|^2|\overline{\nabla }^2 u_s(x)| \le D_3\left( |x|^2 + \sqrt{2s}e^{-b|x|^2/2s}\right) , \end{aligned}$$*P*, and \(b > 0\).

**Theorem 5.1**

We start by proving estimates like the one in the above theorem hold for a short time or far from the origin.

**Lemma 5.2**

*Proof*

*s*we have

*C*we have

*i*. Next from the estimate (H4) we have that

*C*is a curvature bound for \(\Sigma \). We rescale and define

*Remark*

We observe that increasing \(K_0\) will only weaken the hypotheses, and so we may do so freely if necessary without changing the conclusions. This will be important in the next lemma, and also in the proof of the main theorem where we will assume that \(K_0\) is at least 1.

**Lemma 5.3**

*Proof*

*s*is small enough we can ensure that \(\tilde{L}^s\) is \(\varepsilon \)-close to \(\Sigma \) in \(C^{1,\alpha }(B_R(0))\). The conclusion of Lemma 8.2 then says that for \(r^2, t \le q_1\) and \(x \in B_{K_0\sqrt{q_1}}\) we have

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**

*Remark*

Note in particular that \(r_1\) does not depend on \(\nu \), which will be important later.

*Proof*

*I*. If \(|x| \ge 3(s+t)^{-1/8} \ge 3|y_0|\) then

*l*is bounded independent of

*s*and

*t*, and the estimate (H1) is scale invariant, so in particular is satisfied by \((3l)^{-1/2}\Sigma ^{(s,t)}\).

*J*. Similarly as before we find that for \(|x| \le r_1 \le |y_0|/3\) we have

*K*. 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 \(P_i^\perp \). We suppose without loss of generality that

*I*and

*J*we have that

*l*,

*s*and

*t*, thus we can estimate

*C*is independent of

*s*and

*t*. Moreover because the matrix \((D_iv^1_{(s,t)}\cdot D_jv^1_{(s,t)})\) has non-negative eigenvalues we have that \(\sqrt{\det (g_{ij})} \ge 1\), so we can estimate

*l*is bounded independently of

*s*and

*t*, and the outer radius in the definition of \(\Pi _1\) is bounded below by \(3(s_2 + \delta _2)^{-1/8}\) which, by choice of \(s_2\) and \(\delta _2\), we can assume to be greater than \(2r_1\) say. Since \(b_1\) is also constant we rearrange to obtain the above identity. We want to now control the integral terms on the right hand side. First we observe that \(|a_1| \ge c|y_0|\) for some constant depending only on

*P*. This follows from the fact that we assumed \(y_0\) was closer to \(P_1\) than \(P_2\), and hence lies in some fixed conical neighbourhood of \(P_1\). Moreover for any \(0 \le l \le 1\) we have for any

*x*, \(a_1 \in \mathbb {R}^{2n}\)

*l*, which we observed earlier is bounded by 1 / 2. Similarly, using (5.4) again, we can estimate

*b*we have that for all \(s \le s_2\) and \(t \le \delta _2\) we have

*l*and

*r*are both bounded independently of

*s*and

*t*. Therefore putting this together we have

The following two Lemmas show that we have additional control in annular regions, specifically on normal deviation, curvature, Lagrangian angle and the primitive for the Liouville form.

**Lemma 5.5**

*Proof*

*C*such that

**Lemma 5.6**

*Proof*

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 \(|\theta ^s_t| \le C\) for \(\delta _4\) 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 us to apply Corollary 8.4 to get uniform curvature bounds for some short time in *A*(1 / 3, 3).

*A*(1 / 3, 3). \(\square \)

The last of the technical lemmas in this section uses the monotonicity formula of Sect. 3 to show that after waiting for a short time dependent on *s*, we can find times at which the scaled flow \(\tilde{L}^s_t\) 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**

*Proof*

*C*a constant which depends on

*a*,

*R*and

*q*, but not on

*T*or

*s*. We estimate

*C*depending only on

*R*and

*a*, so estimating the terms in front of the integrals we have

*B*. Notice that by Lemma 5.6 we have

*A*,

*l*to denote the rescaling factor of the \(\beta ^s\). We define

*T*and

*s*such that \(l^{-2}T_0 \in [C^{-1}, C]\). We want to show that by possibly again decreasing \(s_5\) and \(\delta _5\), we can ensure

*Case 1*Suppose that (after possibly extracting a further subsequence) we have that \(\sigma _i \rightarrow \sigma > 0\). Then by (H3) we have

*i*. Moreover by (H3) we have \(\lim _{i\rightarrow \infty }|\tilde{\theta }^{s_i} + \tilde{\beta }^{s_i}|^2 = 0\) locally, so inside this large ball the integral can be made as small as desired.

*Case 2*Suppose now that after possibly passing to subsequence, which we do not relabel, we have \(\sigma _i \rightarrow 0\). Then, with \(r_0\) defined as in property (H4) of the family \(L^s\), we find

*i*is large enough, \(l_i^{-1}L^{s_i}\cap A(r_0\sqrt{\sigma _i/2}, 3l_i^{-1})\) is graphical over

*P*, and if \(v_i\) is the function arising from this decomposition we have the estimate

*j*th entry is 1, and all other entries are 0, and \(v^j_i\) is the

*j*th coordinate of \(v_i\). Then given an orthonormal basis for the normal space \(\nu _1,\dots ,\nu _n\) we have \(\nu _j = \sum _{k=1}^n\alpha _{jk}n_k\), where \(\alpha _{jk}\) are fixed real numbers denoting the coefficients in the basis expansion of \(\nu _j\) in terms of the \(n_k\). It then follows that

*C*depends only on the \(\alpha _{jk}\). Now

*i*on the annular region \(A(r_0\sqrt{\sigma _i/2}, 3l_i^{-1}) \cap l_i^{-1}L^{s_i}\). Indeed suppose that \(x \in A(r_0\sqrt{\sigma _i/2}, 3l_i^{-1}) \cap l_i^{-1} L^{s_i}\), then there is a corresponding \(x' \in A(r_0\sqrt{\sigma _i/2}, 3l_i^{-1}) \cap P\) such that \(x = x' + v_i(x')\). Define

*x*as well as

*i*. We may now define a curve in \(l^{-1}_iL^{s_i}\) by setting

*i*or the original choice of |

*x*| we have from property (H3) of \(L^s\) that

*x*. Thus

*x*as \(\tilde{\theta }^{s_i}\) is bounded and \(\sigma _i \rightarrow 0\). Therefore we may bound the term \(\beta ^{s_i}_{l_i}(x_i)\) by some sequence \(b_i\) with \(b_i \rightarrow 0\). Consequently we have the estimate

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**

Let \(q_1\) be defined as in Lemma 5.3, and recall that \(q_1 < 1\). If we set \(\tau := q_1/(2(q_1 + 1))\), then the rescaled version of Lemma 5.3 implies

**Lemma**

Similarly the rescaled Lemma 5.2 tells us that

**Lemma**

*Proof of Theorem 5.1*

*s*we define

*t*close to \(T_s\), so that we can conclude \(\tilde{L}^s_t\) is \(C^{1,\alpha }\) close to \(\Sigma \). Lemma 8.2 will then give density ratio bounds for times past \(T_s\), resulting in a contradiction. To this end we define \(T := T_s/a\), then since \(T < T_s\) we have for all \(t \in [T, T_s)\)

*l*such that \(T + \sigma ^2l \in [T, T_s)\), \(r^2 \le \tau \) and \(x \in B_{2\tilde{R}}\). By White’s local regularity theorem (Theorem 2.2) we get curvature bounds of the form

## 6 Short-time existence

In this section we prove the following short time existence result using Theorem 5.1.

**Theorem 6.1**

Suppose that \(L\subset \mathbb {C}^n\) is a compact Lagrangian submanifold of \(\mathbb {C}^n\) 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 \(t \searrow 0\), \(L_t \rightarrow L\) as varifolds and in \(C^\infty _\mathrm{loc}\) 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 \(s \in (0,c]\) there exists a Lagrangian mean curvature flow \((L^s_t)_{0 \le t \le T_s}\) with \(T_s > 0\). We claim that there exists a \(T_0 > 0\) such that \(T_s \ge T_0\) 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 \(\min \{T_s,\delta \}\) where \(\delta > 0\) is independent of *s*. Uniform estimates on the higher derivatives then immediately follow by standard parabolic PDE theory.

*C*depends only on \(\tau \) and \(\varepsilon _0\). These estimates are then uniform in

*s*for \(s \le s_0\). Moreover, these curvature bounds, along with those outside of the ball \(B_{1/2}\), imply that \(T_s \ge \min \{\delta , \delta _0\}\).

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 \(L_t \rightarrow L\) 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 \(C^\infty _\mathrm{loc}\) away from the singular points. \(\square \)

## 7 Construction of approximating family

In this section, we consider a Lagrangian submanifold *L* of \(\mathbb {C}^n\) with a singularity at the origin which is asymptotic to the pair of planes *P* considered in Sect. 4. We approximate *L* by gluing in the self-expander \(\Sigma \) 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 Sect. 5 which are required to implement the analysis in that section.

Since *L* is conically singular we may write \(L\cap B_4\) as a graph over \(P\cap B_4\) (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, Theorem4.1]), so that we may identify \(L\cap B_4\) with the graph of a one-form \(\gamma \) 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.

*L*is exact inside \(B_4\), there exists \(u\in C^\infty (P\cap B_4)\) such that \(\mathrm {d}u=\gamma \). Since we know that \(\gamma \) must decay quadratically, we can choose a primitive for \(\gamma \) which has cubic decay, i.e.,

*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 \(v\in C^\infty (P\backslash B_{r_0})\) such that the self-expander is described by the exact one-form \(\psi =\mathrm {d}v\) on \(P\backslash B_{r_0}\). Further, Lotay and Neves proved [10, Theorem 3.1]

*L*to resolve the singularity. Our new manifold, \(L^s\), will be the rescaled self-expander \(\Sigma ^s\) inside \(B_{r_0\sqrt{2s}}\), the manifold

*L*outside \(B_{4}\) and will smoothly interpolate between the two on the annulus \(A(r_0\sqrt{2s},4)\).

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 \(A(r_0\sqrt{2s},4)\), 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).

\(L^s\cap B_{r_0\sqrt{2s}}=\Sigma _s\cap B_{r_0\sqrt{2s}}\),

\(L^s\cap A(r_0\sqrt{2s},4)=\)graph \(\gamma _s\),

\(L^s\backslash B_4=L\backslash B_4\).

*L*on a compact region, \(L^s\) satisfies (H1).

*L*and the self-expander \(\Sigma \) are bounded, as is that of the rescaled self-expander \(\Sigma _s\) 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, p. 5]. This tells us that a Lagrangian graph in \(\mathbb {C}^n\) (over \(\mathbb {R}^n\)) given by \((x_1,...,x_n,u_1(x),...,u_n(x))\), where \(u:\mathbb {R}^n\rightarrow \mathbb {R}, u_i:=\frac{\partial u}{\partial x_i},\) has Lagrangian angle

*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 \(\gamma _s\) 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 \(\gamma _s\), and so the Lagrangian angle of \(L^s\).

*s*and

*t*, we have that

*P*. Now, \(w_s\) is bounded independently of

*s*by \(D(1+|x|^2),\) using (7.1) and (7.2), as is \(\langle x,\nabla w_s\rangle \), using Cauchy–Schwarz and the estimates (7.1) and (7.2) so we find that \(\beta ^s\) is bounded independently of

*s*on the annulus \(A(s^{1/4},2s^{1/4})\). Therefore, we have that

To show that (H3) is satisfied, recall that we define \(L^s\) as \(L^s\cap B_{r_0\sqrt{2s}}=\Sigma _s\cap B_{r_0\sqrt{2s}}\), \(L^s\backslash B_4=L\backslash B_4\) and we interpolate smoothly between the two, which exactly happens when \(s^{1/4}\le |x|\le 2s^{1/4}\). Therefore when we rescale by \(1/\sqrt{2s}\), we have that \(\tilde{L}^s\cap B_{r_0}\equiv \Sigma \). So it remains to check convergence outside this ball.

Finally, we check that the second fundamental form of \(\tilde{L}^s\) is uniformly bounded in *s*. We have that the second fundamental form of \(\Sigma \) must be bounded, and if *A* is the second fundamental form of *L*, rescaling *L* by \(1/\sqrt{2s}\) means that the second fundamental form scales by \(\sqrt{2s}\). Since \(\sqrt{2s}<1\), we can uniformly bound both second fundamental forms so that \(\tilde{L}^s\), which is a combination of both \(\Sigma \) and \(1/\sqrt{2s}L\), has second fundamental form uniformly bounded in *s*.

*P*in the annulus \(A(r_0\sqrt{2s},4)\). We now must estimate \(\gamma _s\). Firstly, note that we have

*y*(by \(e^{-1/2}/\sqrt{b}\)) on \(\mathbb {R}\), and so \(\tilde{C}\) is independent of

*s*.

*C*(which remains independent of

*s*) larger if necessary and

*b*smaller (which does not affect the previous estimates).

*C*larger.

*s*. Therefore (H4) is satisfied.

## Notes

### Acknowledgments

Both authors would like to thank Jason Lotay and Felix Schulze for their help, guidance and feedback. We would also like to thank our reviewer for a careful reading of this paper, and many helpful comments and corrections.

### References

- 1.Chau, A., Chen, J., He, W.: Lagrangian mean curvature flow for entire Lipschitz graphs. Calc. Var. Part. Differ. Equ.
**44**(1–2), 199–220 (2012)MathSciNetCrossRefMATHGoogle Scholar - 2.Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. (2)
**130**(3), 453–471 (1989)Google Scholar - 3.Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math.
**105**(3), 547–569 (1991)MathSciNetCrossRefMATHGoogle Scholar - 4.Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom.
**31**(1), 285–299 (1990)MathSciNetMATHGoogle Scholar - 5.Ilmanen, T., Neves, A., Schulze, F.: On short time existence for the planar network flow (2014). arXiv:1407.4756
- 6.Imagi, Y., Joyce, D., Oliveira dos Santos, J.: Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in \({C^{m}}\) (2014). arXiv:1404.0271
- 7.Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications. J. Differ. Geom.
**63**(2), 279–347 (2003)MathSciNetMATHGoogle Scholar - 8.Joyce, D.: Conjectures on bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014). arXiv:1401.4949
- 9.Lawson Jr., H.B., Osserman, R.: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math.
**139**(1–2), 1–17 (1977)MathSciNetCrossRefMATHGoogle Scholar - 10.Lotay, J.D., Neves, A.: Uniqueness of Langrangian self-expanders. Geom. Topol.
**17**(5), 2689–2729 (2013)MathSciNetCrossRefMATHGoogle Scholar - 11.Neves, A.: Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. Math.
**168**(3), 449–484 (2007)MathSciNetCrossRefMATHGoogle Scholar - 12.A. Neves. Recent progress on singularities of Lagrangian mean curvature flow. In: Surveys in Geometric Analysis and Relativity. Adv. Lect. Math. (ALM), vol. 20, pp. 413–438. Int. Press, Somerville (2011)Google Scholar
- 13.Neves, A.: Finite time singularities for Lagrangian mean curvature flow. Ann. Math. (2)
**177**(3), 1029–1076 (2013)Google Scholar - 14.Schoen, R., Wolfson, J.: Mean Curvature Flow and Lagrangian Embeddings (2002)
**(Preprint)**Google Scholar - 15.Smoczyk, K.: A canonical way to deform a Lagrangian submanifold (1996). arXiv:dg-ga/9605005
- 16.Thomas, R.P., Yau, S.-T.: Special Lagrangians, stable bundles and mean curvature flow. Commun. Anal. Geom.
**10**(5), 1075–1113 (2002)MathSciNetCrossRefMATHGoogle Scholar - 17.Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math.
**148**(3), 525–543 (2002)MathSciNetCrossRefMATHGoogle Scholar - 18.Wang, M.-T.: The mean curvature flow smoothes Lipschitz submanifolds. Commun. Anal. Geom.
**12**(3), 581–599 (2004)MathSciNetCrossRefMATHGoogle Scholar - 19.White, B.: A local regularity theorem for mean curvature flow. Ann. Math. (2)
**161**(3), 1487–1519 (2005)Google Scholar - 20.Wolfson, J.: Lagrangian homology classes without regular minimizers. J. Differ. Geom.
**71**(2), 307–313 (2005)MathSciNetMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.