On short time existence of Lagrangian mean curvature flow
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 nonareaminimising, 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 longstanding open problem in the study of CalabiYau 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 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 [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 zeroMaslov class, and the Maslov class is preserved by Lagrangian mean curvature flow, of particular interest is the mean curvature flow of zeroMaslov class Lagrangians. In this case, the structure of singularities is relatively well understood. Indeed Neves [11] has shown that a singularity of zeroMaslov 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 nonareaminimising, transversally intersecting Lagrangian planes. Specifically we prove the following theorem which serves as a partial answer to Problem 3.14 in [8].
Theorem
(Shorttime 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 CalabiYau manifold M, and to L noncompact provided, in the latter case, that we impose suitable conditions at infinity.
In the onedimensional case all curves are Lagrangian. Ilmanen–Neves–Schulze considered the flow of planar networks, that is finite unions of embedded line segments of nonzero 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 nonregular 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 shorttime existence, we construct a smooth approximating family \(L^s\) of initial conditions via a surgery procedure. Specifically we take a singularity asymptotic to some nonareaminimising pair of planes \(P_1 + P_2\), cut it out and glue in a piece of the Lagrangian selfexpander 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 selfexpanding 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 zeroMaslov smooth selfexpanders 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 selfexpander equation, which allows us to show that the approximating family of initial conditions that we construct in the proof, which are selfexpanders in a ball, remain weakly close to the selfexpander 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 selfexpander near the singularity. Since selfexpanders 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 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 highcodimension mean curvature flow.
2 Preliminaries
2.1 Mean curvature flow
Theorem 2.1
Theorem 2.2
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 ndimensional submanifolds \(\Sigma \) and L defined in U, we say that \(\Sigma \) and L are 1close 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 ndimensional 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 1close 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
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 zeroMaslov.
 (ii)
In an open set where the flow is zeroMaslov 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
Remark
Let \(\phi \) be a cutoff 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 selfexpanders
In this section we prove a dynamic stability result for Lagrangian selfexpanders. More specifically we show that if a Lagrangian submanifold is asymptotic to some pair of planes and is almost a selfexpander in a weak sense, then the submanifold is actually close in a stronger topology to some selfexpander. 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, zeroMaslov class Lagrangian selfexpander asymptotic to P.
Theorem 4.2
 (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 \(x \in L\cap A(r, \tilde{R})\);$$\begin{aligned} \mathrm {dist}(x, P) \le \nu + C\mathrm {exp}\left( \frac{x^2}{C}\right) , \end{aligned}$$
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 \(C^{1,\alpha }(B_{R_i})\).
5 Main theorem
 (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 P and satisfies the estimate$$\begin{aligned} L^s \cap A(r_0\sqrt{s},3) \subset \{x + u_s(x)  x \in P\cap A(r_0\sqrt{s}, 3)\} \subset L^s\cap A(r_0\sqrt{s}, 4), \end{aligned}$$where \(\overline{\nabla }\) denotes the covariant derivative on P, and \(b > 0\).$$\begin{aligned} u_s(x) + x\left \overline{\nabla }u_s(x)\right + x^2\overline{\nabla }^2 u_s(x) \le D_3\left( x^2 + \sqrt{2s}e^{bx^2/2s}\right) , \end{aligned}$$
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
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
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
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
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).
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 selfexpander 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
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
6 Shorttime 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.
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 selfexpander \(\Sigma \) 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 oneform \(\gamma \) on P. Recall that the manifold corresponding to the graph of such a oneform is Lagrangian if and only if the oneform is closed.
To do this, we will glue together the primitives of the oneforms corresponding to these manifolds, before taking the exterior derivative. This gives us a oneform 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 \(\gamma _s\),

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