LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN

CURVATURE FLOW

KNUT SMOCZYK

Abstract. Given a compact Lagrangian submanifold in flat space evolv-ing by its mean curvature, we prove uniform C2,�-bounds in space andC2-estimates in time for the underlying Monge-Ampere equation underweak and natural assumptions on the initial Lagrangian submanifold.This implies longtime existence and convergence of the Lagrangian meancurvature flow. In the 2-dimensional case we can relax our assumptionsand obtain two independent proofs for the same result.

1. Introduction

In symplectic geometry there is a destinguished class of immersions, knownas Lagrangian submanifolds. They are important in physics and of course inpure and applied mathematics as well. E.g. minimal Lagrangian immersionsin a given Calabi-Yau manifold are relevant in physics because they are re-lated to T-duality and Mirror symmetry [9]. However, to construct minimalLagrangian submanifolds is a great geometric and analytic challenge. Never-theless there is a growing community working on such problems and alreadysome very nice results have been obtained, so e.g. a version of the Bernsteinproblem [5]. Due to the high codimension many techniques which are usefulfor hypersurfaces of prescribed curvature cannot be used unchanged in thetheory of Lagrangian submanifolds. In particular the mean curvature flowfor Lagrangian submanifolds is much more complicated than for hypersur-faces and it is the aim of this article to close a gap in the understanding ofLagrangian graphs moving by their mean curvature.

Let (M,!) be a 2n-dimensional symplectic manifold with symplectic 2-form!. An n-dimensional submanifold L ⊂M is called Lagrangian if

!(V,W ) = 0 ∀ V,W ∈ TL.The most prominent examples of symplectic manifolds are Kahler manifolds(M,J, g), where

!(V,W ) = g(JV,W )

Date: August 2002.1991 Mathematics Subject Classification. Primary 53C44;

1

2 KNUT SMOCZYK

is the symplectic 2-form (Kahler form) induced by the Kahler metric g andthe complex structure J .

If L is a compact n-dimensional manifold, [0, T ) ⊂ ℝ ∪ {∞} a time intervaland

F : L× [0, T )→M

a smooth family of immersions into a Kahler-Einstein manifold (M,J, g),such that

L0 := F (L, 0)

is Lagrangian and such that F satisfies the mean curvature flow equation

(1.1)dF

dt=→H

(→H = mean curvature vector along Lt := F (L, t)), then it is well-known that

(1.1) preserves the Lagrangian condition, so that Lt is Lagrangian ∀ t ∈[0, T ) (see e.g. [7].)

We write Ft(x) instead of F (x, t). The mean curvature formH of Lagrangian

submanifolds in Kahler manifolds (M,J, g) is related to→H through

H(V ) = !(→H,V ) .

If (M,J, g) is Kahler-Einstein, then H is closed and any locally definedfunction � with

(1.2) d� = H

is called a Lagrangian angle.

In [8] we proved the following longtime existence and convergence resultfor the Lagrangian mean curvature flow in Kahler-Einstein manifolds ofnonpositive scalar curvature:

Proposition 1.1. Let L be a compact manifold and let F0 : L → L0 ⊂ Mbe a smooth immersion of L as a Lagrangian submanifold into a Kahler-Einstein manifold (M,J, g) that is either compact or complete with boundedcurvature quantities. Further assume that [0, T ), 0 < T ≤ ∞ is the maximaltime interval on which the Lagrangian mean curvature flow (1.1) admits asmooth solution. Then the following is true:

(a) Assume there exists a constant C0 <∞ such that

maxLt∣A∣2 ≤ C0, ∀ t ∈ [0, T ),

where ∣A∣2 is the squared norm of the second fundamental tensor A.Then for any m ≥ 0 there exists a constant Cm < ∞ depending onm,L0,M such that

maxLt∣∇mA∣2 ≤ Cm, ∀ t ∈ [0, T ).

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 3

(b) If T <∞, then

lim supt→T

{maxLt∣A∣2

}=∞.

(c) If in addition to (a) the initial mean curvature form of L0 is exact,the ambient Kahler-Einstein manifold has nonpositive Ricci curva-ture and the induced Riemannian metrics F ∗t g on L are all uni-formly equivalent, then T =∞ and the Lagrangian submanifolds Ltconverge smoothly and exponentially to a smooth compact minimalLagrangian immersion L∞ ⊂M .

In general one cannot expect longtime existence results without extra as-sumptions on the initial Lagrangian submanifold. In [8] we consideredLagrangian submanifolds generated by symplectic maps and proved conver-gence to minimal Lagrangian maps under very natural and sharp conditionsfor the Lagrangian angle �. Here, we will give another sufficient conditionfor longtime existence in flat ambient manifolds that entirely differs fromthose conditions. In particular we will not need the oscillation condition

osc (�) ≤ �

2that was important there.

The crucial condition in this paper can be stated in terms of certain sym-metric bilinear forms S. To explain these forms suppose (M, g, J) is a 2n-dimensional Kahler-Einstein manifold with compatible complex structure J ,i.e.

!(V,W ) = g(JV,W )

is a symplectic 2-form (Kahler form.) Let us denote the Levi-Civita connec-tion on M by D. We will consider tensors S ∈ Γ(T ∗M ⊗ T ∗M) that satisfythe following conditions

S(V,W ) = S(W,V ) (Symmetry),(1.3)

S(JV,W ) = S(V, JW ) (Anti-compatibility),(1.4)

DS = 0 (Parallelity)(1.5)

and denote the set of tensors satisfying equations (1.3), (1.4) and (1.5) byΣ(M). Note that if S ∈ Σ(M), then S defined by

S(V,W ) := −S(JV,W )

also belongs to Σ(M). An example in ℝ2n is given by S(V,W ) := ⟨V ,W ⟩,where ⟨⋅, ⋅⟩ denotes the euclidean inner product and the bar is complexconjugation. Since these conditions are very similar to the structures onhyperKahler manifolds it is clear that one must expect a special holonomyfor the underlying manifold, indeed the existence of such a bilinear form Sis so strong that it implies:

4 KNUT SMOCZYK

Proposition 1.2. Let S ∈ Σ(M) be non-degenerate. Then g is flat.

A proof of the proposition will be given in the appendix where we will alsodiscuss Σ(ℝ2n) in more detail. Here we only mention that this fact mainlyhinges on condition (1.5) and that this is the only reason why we restrictour considerations to flat ambient manifolds.

The main theorem states:

Theorem 1.3. Assume F0 : L → (M, g, J) is a compact Lagrangian im-mersion into a flat manifold such that there exists a tensor S ∈ Σ(M) with

(1.6) F ∗0 S(V, V ) > 0 ∀ V ∈ TL, V ∕= 0 .

Then there exist constants c1, c2 > 0 such that

lim supt→T

{maxLt∣→H∣}≤ c1,

lim supt→T

{maxLt∣d†H∣

}≤ c2,

on the maximal time interval [0, T ), where a smooth solution of (1.1) exists(d†H is shorthand for ∇iHi). Moreover, all induced metrics gt := F ∗t g areuniformly equivalent to the initial metric g0. If in addition n = 2 or

(1.7) F ∗0 S(V, V ) > 0 ∀ V ∈ TL, V ∕= 0 ,

then T = ∞ and the Lagrangian submanifolds converge smoothly to a flatLagrangian submanifold.

Condition (1.6) implies that the Lagrangian submanifold can be written asa graph over a flat Lagrangian subspace so that the n height functions aregiven by the n components of the gradient of a smooth function u and suchthat the eigenvalues of the Hessian of u are uniformly bounded by a constantdepending on S. We discuss this in section 2. Condition (1.7) will implyconvexity of u. This is possible even for compact Lagrangian submanifolds.An example can be given as follows:

Example 1.4. Let u : ℝn → ℝ be defined by

u(x) :=a

2∣x∣2 + b

n∑i=1

cos(xi)

with two constants a, b that satisfy a > b > 0, a+ b < 1 (see also Figure 1).Then the gradient graph

F : ℝn → ℝn ⊕ iℝn = ℂn

F (x) :=(x, y := Du(x)

)

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 5

describes the universal cover of a compact Lagrangian submanifold in flatspace (since F (ℝn) is invariant under translations in ℝ2n) and the Hessianof u is

uij = diag(a− b cos(x1), . . . , a− b cos(xn)

)so that all eigenvalues � of uij satisfy

0 < � < 1.

In particular

F ∗S(V, V ) = ⟨DF (V ), DF (V )⟩ > 0

andF ∗S(V, V ) = ⟨JDF (V ), DF (V )⟩ > 0

for all V ∈ Tℝn (the bar denotes complex conjugation) so that conditions(1.6) and (1.7) are satisfied with S(X,Y ) := ⟨X, Y ⟩.

Figure 1. The gradient graph of u(x) = x2

2 − cos(x)

In particular we get the following corollary

Corollary 1.5. Assume u : ℝn → ℝ is a smooth strictly convex functionsuch that all eigenvalues of Hess(u) are bounded by 1 and such that F (x) :=(x,Du(x)

)is the universal cover of a compact Lagrangian submanifold in

ℂn = ℝn ⊕ iℝn. Then the Lagrangian mean curvature flow deforms L intoa flat plane.

(1.6) and (1.7) roughly mean that the Lagrangian submanifold lies betweentwo different Lagrangian planes. Equation (1.1) and the Lagrangian con-dition then lead to a parabolic Monge-Ampere type equation (2.10) for uwhich describes the flow (see section 2 for details) and the theorem impliesthat under condition (1.6) one gets uniform C2-estimates for u both in spaceand time. Assumption (1.7) on the other hand guarantees that the parabolicoperator is concave so that we can use the C2,�-estimates in space due toKrylov (for example section 5.5 in [6]). This eventually gives longtime exis-tence and convergence. In case n = 2 we can drop condition (1.7) and obtainthe C2,�-estimates in two different ways, from the better regularity theoryfor nonlinear equations in 2 variables and also from a direct estimate of thefull second fundamental form. We will discuss both proofs. It is unknownwhether (1.7) is redundant in any dimension.

In this paper we will use the maximum principle for tensors due to Hamilton[3], [4]. Since there is no monotonicity formula for tensors so far, we cannotdrop the compactness condition here, although we believe that this can bedone under appropriate growth conditions as e.g. in the well known paper by

6 KNUT SMOCZYK

Ecker and Huisken [1] where hypersurfaces in ℝn+1 represented as graphsare deformed by their mean curvature. However, for the mean curvatureflow in higher codimension one cannot expect the same results as in [1] anda condition like in our theorem is natural in this context. The special natureof the Lagrangian mean curvature flow allows to obtain longtime existenceand convergence merely from uniform C1,�-estimates for the evolving mapsF : L → ℝ2n because the quasilinear parabolic system given by equation(1.1) can be integrated to the fully nonlinear parabolic equation of Monge-Ampere type (2.10) and C1,�-estimates for F correspond to C2,�-estimatesof that equation.

This work has begun while I visited the CAS in Beijing, China in 2001 andwas completed at the Max-Planck-Institute for Mathematics in the Sciencesin Leipzig, Germany. I would like to thank Prof. Li Jiayu for his invitationto China and his great hospitality and Prof. Jurgen Jost at the MPI for theopportunity to finish this work.

2. Preliminaries

2.1. The Lagrangian mean curvature flow in ℝ2n.

In this subsection we explain the notation and recall elementary equationsfor the Lagrangian mean curvature flow in ℝ2n. To begin, assume that(ℝ2n, g, J, !) is the euclidean space with compatible complex structure J = iand the standard symplectic form !. Local coordinates on ℝ2n will bedenoted by (y�)�=1,...,2n whereas local coordinates for a Lagrangian sub-manifold L will be denoted by (xi)i=1,...,n. Moreover, we use the Einsteinconvention to sum over repeated indices, the sum is taken from 1 to 2n forgreek indices and from 1 to n for latin minuscles. If

F : L→M

is an immersion, then we write

F�i := ∂F�

∂xi,with F� = y�(F ),

F�ij := ∂2F�

∂xi∂xj.

The metric g = ⟨⋅, ⋅⟩ on ℝ2n can locally be written as

g = g�� dy� ⊗ dy�

and then

F ∗g = gij dxi ⊗ dxj

with

gij := g�� F�i F

�j

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 7

is the induced Riemannian metric on L. By the Lagrangian condition wehave

!ij := !��F�i F

�j = 0.

We also set

J = J��∂

∂y�⊗ dy�,

��i := J�� F�i

and

�i := ��i∂

∂y�= J

(∂F

∂xi

),

so that �i is normal along L. The induced connection on tensor bundlesover L will be denoted by ∇. Then the second fundamental tensor A is thecovariant derivative of the differential

dF = F�i dxi ⊗ ∂

∂y�,

i.e.A = ∇dF

and we setA�ij := ∇iF�j .

The second fundamental form ℎ ∈ Γ(T ∗L⊗ T ∗L⊗ T ∗L) is the tensor givenby the components

ℎijk := −!��F�i A�jk

and by definition the mean curvature form H = Hidxi is

Hi := gklℎikl

so that H is related to−→H by

H(V ) = !(→H,V ) .

We summarize the relevant equations for the second fundamental form inthe following Lemma (compare with [7] and [8].)

Lemma 2.1. The second fundamental form ℎ = ℎijkdxi ⊗ dxj ⊗ dxk of a

Lagrangian immersion into ℝ2n satisfies

a) ℎijk = ℎjik = ℎjki,

b) Rijkl = ℎ nik ℎnjl − ℎ n

il ℎnjk,c) ∇iℎjkl −∇jℎikl = 0,d) ∇iHj = ∇jHi.

Here, Rijkl is the curvature symbol of F ∗g w.r.t. coordinates (xi)i=1,...,n andwe have set

ℎ nil := ℎilmg

mn.

8 KNUT SMOCZYK

Remark 2.2. In the sequel we will often rise and lower indices w.r.t. themetric tensor gij . From Lemma 2.1 d) (the traced Codazzi equation) itfollows that the mean curvature form H is closed.

Let us introduce the following symmetric tensors

a := aij dxi ⊗ dxj

b := bij dxi ⊗ dxj

with

aij := gklHkℎlij = H lℎlij

bij := ℎ mni ℎmnj .

It follows that

(2.1) ∣A∣2 = g��gikgjlA�ijA

�kl = ℎijkℎijk = gijbij

and

(2.2) gijaij = ∣H∣2.In addition, the Ricci curvature is given by

(2.3) Rij = aij − bij .We recall the evolution equations under the mean curvature flow for crucialgeometric objects on L (compare with Lemma 5 in [8].)

Lemma 2.3. If F : L × [0, T ) → ℝ2n is a smooth family of Lagrangianimmersions that evolves according to (1.1), then

a) ddtgij = −2aij

b) ddtd� = −∣H∣2d�

c) ddtHi = ∇id†H = ΔHi −R l

i Hl

d) ddtℎijk = Δℎijk −R l

i ℎljk −Rlj ℎlki −R l

k ℎlij − 2ℎ min ℎ l

jm ℎ nkl

e) ddt ∣H∣

2 = Δ∣H∣2 − 2∣∇H∣2 + 2∣aij ∣2f) d

dt ∣A∣2 = Δ∣A∣2 − 2∣∇A∣2 + 2∣bij ∣2 + 2∣Rijkl∣2

Here, d� denotes the volume element on L w.r.t. F ∗g.

For a proof of these equations we refer to [7] and [8].

In addition we will use the evolution equation of d†H

Lemma 2.4.d

dtd†H = Δd†H + 4aij∇iHj .

Proof. This is a direct consequence of lemma 2.3 a) and c). □

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 9

Now let S ∈ Σ(M) be one of the tensors described above. We set

Sij := S��F�i F

�j

and

F� :=d

dtF�.

We need an evolution equation for F ∗S = Sij dxi ⊗ dxj and compute

ddtSij = d

dt(S��F�i F

�j )

= D S��F F�i F

�j + S��∇iF�F �j + S��F

�i ∇jF �

= −S��(∇i(H l��l )F �j + F�i ∇j(H l��l )

)= −∇iH lS���

�l F

�j −∇jH lS��F

�i �

�l

−H lS��(∇i��l F�j +∇j��l F

�i ) .

From ∇iF�l = A�il = −ℎkil��k , DJ = 0 and J2 = −Id we obtain

(2.4) ∇i��l = ℎkilF�k ,

so that

d

dtSij = −∇iH lS���

�l F

�j −∇jH

lS��F�i �

�l(2.5)

−H lS��(ℎkilF�k F

�j + ℎkjlF

�k F

�i )

= −∇iH lS����l F

�j −∇jH

lS��F�i �

�l

−aki Skj − akjSki.

Next we compute ΔSij :

∇kSij = ∇k(S��F�i F�j )

= D S��F k F

�i F

�j + S��(∇kF�i F

�j + F�i ∇kF

�j )

= −ℎlkiS����l F�j − ℎlkjS��F�i �

�l

and then

ΔSij = −∇kℎlkiS����l F�j −∇

kℎlkjS��F�i �

�l(2.6)

−ℎlki S��(ℎmklF�mF

�j − ℎ

mkj�

�l �

�m)

−ℎlkj S��(−ℎmki��m��l + F�i ℎ

mklF

�m).

From the Codazzi equation (Lemma 2.1 c)) we deduce

∇kℎlki = ∇lHi = ∇iH l

and (1.4) implies

S����m�

�l = −Sml ,

10 KNUT SMOCZYK

so thatΔSij = −∇iH lS���

�l F

�j −∇jH lS��F

�i �

�l

−bki Skj − bkjSki−2ℎlki ℎ

mkjSlm.

Inserting this into the above expression for ddtSij we finally get

Lemma 2.5. F ∗S = Sijdxi ⊗ dxj satisfies the evolution equation

d

dtSij = ΔSij −RliSlj −RljSli + 2ℎkmi ℎnjkSmn.

We define the functions := gijSij .

Then the evolution equations for gij and Sij immediately imply

Lemma 2.6. The quantity s satisfies

d

dts = Δs+ 4bijSij .

2.2. The underlying parabolic equation of Monge-Ampere type.

We will now see that in particular any Lagrangian L ⊂ ℝ2n satisfying con-dition (1.6) must be a graph over an n-plane sitting in ℝ2n.

To see this let �S : ℝ2n → ℝ2n be the endomorphism associated to S, i.e.⟨�S(V ),W ⟩ = S(V,W ). �S must be independent of y ∈ ℝ2n because weassumed that S is parallel. If � is an eigenvalue of �S , then −� must also bean eigenvalue because �SJ = −J�S and J maps the eigenspace belongingto � to that belonging to −�. Since F ∗S > 0 and L is n-dimensional, weconclude that there exist exactly n positive eigenvalues �1, . . . , �n and nnegative eigenvalues −�1, . . . ,−�n. Consequently we can split ℝ2n into thedirect sum

ℝ2n = X ⊕ Y,where X is the linear hull of all eigenvectors e belonging to positive eigenval-ues and Y := JX is the linear hull of eigenvectors f belonging to negativeeigenvalues. In addition F ∗S > 0 implies that L can be written as a graphover X, i.e. there exist n functions u1, . . . , un : ℝn → ℝ and an orthonormalframe e1, . . . , en spanning X such that

F : ℝn → ℝ2n

F (x) := xiei + �ijuifj ,

gives an immersion of L (with fj := Jej .)

The tangent vectors Fi := ∂F∂xi

are

(2.7) Fi = ei + �klukifl

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 11

Here, L is a compact Lagrangian in a flat manifold M and we see that theuniversal cover L of L must be ℝn immersed as a Lagrangian into ℝ2n, theuniversal cover of M .

From the Lagrangian condition we further deduce that there exists a functionu : ℝn → ℝ such that

uk =∂u

∂xk

holds for any k = 1, . . . , n and on all of ℝn.

As in [7] we may then transform the mean curvature flow into a parabolicequation for u, because

d

dtF =

−→H = −Hm�m = −Hm(fm − �lpulmep)

implies the equations

(2.8)dxi

dt= Hm�liulm

and

(2.9) �ipdupdt

= −H i

so that in view ofdupdt

=∂up∂t

+ upldxl

dtwe obtain the equation

(2.10) P [u] := −�− ∂u

∂t= 0

with d� = H, where the Lagrangian angle � of L can locally be written aseither

� = −arctan(ab

),

or as

� = arctan

(b

a

)depending on whether

a := Im(detℂ(�kl + iukl)

)or

b := Re(detℂ(�kl + iukl)

)is nonzero. Either a or b must be nonzero as long as L is a graph becausefor the induced metric

gij = �ij + �kluikujl

we compute

det (gij) = a2 + b2.

12 KNUT SMOCZYK

If all induced metrics F ∗t g of the flow would be uniformly equivalent onewould obtain uniform C2-estimates w.r.t. the space variables x of (2.10)and it would also imply that the solution of (2.10) remains a graph.

A simple calculation shows that

∂P [u]

∂uij= gij

which means that P [u] is always parabolic. We will also need the secondderivatives of P [u] and compute

∂2P [u]

∂uij∂ukl=

∂gij

∂ukl

= −gisgjt gst∂ukl

= −gisgjt(�pqups�

kq �

lt + �pquqt�

kp �

ls

)= −gisgjl�pkups − gilgjt�qkuqt= −

(gisgjl + gilgjs

)�pkups.

If at a fixed point p we consider a basis b1, . . . , bn of eigenvectors for uijsuch that uij becomes diagonal at p with uij = diag(�1, . . . , �n), then forany symmetric tensor vkl we obtain

∂2P [u]

∂uij∂uklvijvkl = −2

∑i,j

�i(1 + �2i )(1 + �2j )

(vij)2.

Thus we have shown

Lemma 2.7. The operator P [u] = −�− ∂u∂t is concave, if u is strictly convex.

This lemma will become important later when we show C2,�-regularity of(2.10).

We take a closer look at conditions (1.6) and (1.7). One has

Sij = S(Fi, Fj) = ⟨�(ei), ej⟩+ ⟨�(fl), fq⟩�kluki�pquqjso that

(2.11) Sij = �i�ij −n∑k=1

�kukiukj ,

where we do not sum over i in �i�ij .

In addition

Sij = ⟨J�(Fi), Fj⟩ = −⟨�(JFi), Fj⟩= −⟨�(fi − �klukiel), ej + �pqupjfq⟩)= �iuij + �juji,(2.12)

where again there is no sum over i and j on the RHS.

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 13

Lemma 2.8. u is strictly convex if F ∗S is positive definite.

Proof. Assume there exists a nonpositive eigenvalue � of uij and let V bean eigenvector so that

n∑j=1

uijVj = �

n∑j=1

�ijVj .

Then from (2.12)

F ∗S(V, V ) =n∑

i,j=1

(�iuij + �juji)ViV j

= 2�n∑i=1

(�iV

in∑j=1

�ijVj)

= 2�n∑i=1

�i(Vi)2 ≤ 0 because �i > 0

which is a contradiction. □

3. Proof of the main theorem

In this section we will prove our main theorem. For this purpose we needthe next lemma.

Lemma 3.1. Assume there exists an " > 0 and S ∈ Σ(M) such that

(3.1) Mij := Sij − "gij > 0

holds on a compact L at t = 0. Then this is also true for t ∈ [0, T ).

Proof. First we need an evolution equation for Mij . From Lemma 2.5 andLemma 2.3 a) we obtain

ddtMij = ΔMij −RliSlj −RljSli + 2ℎkmi ℎnjkSmn + 2"aij

= ΔMij −RliMlj −RljMli + 2"bij + 2ℎkmi ℎnjkSmn.

We use the maximum principle for tensors proven in [3] (see also [4] for abetter proof.) To prove that (3.1) is preserved we must only show that

NijViV j ≥ 0

for any null eigenvector V of Mij that occurs for the first time, where

Nij := −RliMlj −RljMli + 2"bij + 2ℎkmi ℎnjkSmn .

14 KNUT SMOCZYK

If for the first time there exists a null eigenvector V of Mij , then

MijVi = 0

and

MijWiW j ≥ 0 ∀ W ∈ TL

But then

NijViV j = 2"bijV

iV j + 2ℎkmi ℎnjkSmnViV j .

The quadratic tensors bij = ℎkli ℎjkl and ℎkmi V iℎnjkVj are positive semidefi-

nite and since Mij ≥ 0 implies Smn ≥ "gmn we deduce

NijViV j ≥ 2"bijV

iV j + 2"ℎkmi ℎnjkgmnViV j

= 4"bijViV j ≥ 0.

□

Lemma 3.2. If there exists a positive constant c, such that

(3.2) Sij − cHiHj > 0

at t = 0, then this remains true ∀ t ∈ [0, T ).

Proof. Again we use the maximum principle for tensors. Here we set Mij :=Sij − cHiHj . Then the evolution equations for Sij and Hi imply

ddtMij = ΔSij −RliSlj −RljSli + 2ℎkmi ℎnjkSmn

−cHj(ΔHi −RliHl)− cHi(ΔHj −RljHl)

= ΔMij + 2c∇kHi∇kHj −RliMlj −RljMli

+2ℎkmi ℎnjkSmn

and here we set

Nij = 2c∇kHi∇kHj −RliMlj −RljMli + 2ℎkmi ℎnjkSmn.

As in Lemma 3.1 we choose the first time where a null eigenvector V of Mij

occurs so that

MijVi = 0

MijWiW j ≥ 0 ∀ W ∈ TL .

Then

NijViV j = 2c∣∇kHiV

i∣2 + 2ℎkmi V iℎnjkVjSmn

≥ 2cℎkmi V iℎnjkVjHmHn

= 2caki ViajkV

j ≥ 0

because aki ViajkV

j = ∣V ∣2 with V k = aki Vi. □

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 15

Lemma 3.3. With the assumptions made in Theorem 1.3 there exists aconstant c2 such that

(3.3) ∣d†H∣ ≤ c2holds for t ∈ [0, T ).

Proof. We define the function

f := ∣H∣2 + d†H − cswith a positive constant c to be determined. Then Lemma 2.3 e), Lemma2.4 and Lemma 2.6 imply

d

dtf = Δf − 2∣∇H∣2 + 2∣aij ∣2 + 4aij∇iHj − 4cbijSij

and with Cauchy-Schwarz

d

dtf ≤ Δf + 4∣aij ∣2 − 4cbijSij .

Since ∣aij ∣2 = bijHiHj , we obtain

d

dtf ≤ Δf + 4bij(HiHj − cSij)

and from bij ≥ 0, Sij > 0 at t = 0 and Lemma 3.2 it follows

f ≤ Bfor some constant B > 0 and all t ∈ [0, T ). So we found a uniform upperbound for d†H, if we can prove that s is bounded. But since S = S��dy

� ⊗dy� is parallel, we see that S�� must be constant in cartesian coordinatesfor the flat manifold (M, g, J, !). Thus there exists a positive constant �such that

S�� − �g�� < 0

as a tensor. This implies

Sij − �gij = (S�� − �g��)F�i F�j < 0

and then also

(3.4) gijSij < �gijgij = �n.

In the same way we can proceed with the function f := ∣H∣2−d†H− cs andobtain the lower bound for d†H. □

We can now prove Theorem 1.3.

Proof of Theorem 1.3: Since there exists a positive constant " withSij − "gij > 0 at t = 0, we can find a small positive constant c so that thesymmetric tensor

Sij − cHiHj

16 KNUT SMOCZYK

is positive definite at t = 0. Then Lemma 3.2 implies that

∣H∣2 ≤ 1

cgijSij =

s

c∀ t ∈ [0, T ).

Since s is bounded (compare with the proof of Lemma 3.3) this proves theuniform bound of ∣H∣2 and by Lemma 3.1 that all induced metrics stayuniformly equivalent. In addition, Lemma 3.3 is just the uniform boundof ∣d†H∣. It remains to prove longtime existence and convergence in casen = 2 or under the extra condition (1.7). For this we recall the underlyingMonge-Ampere equation

P [u] = −�− ∂u

∂t= 0

from section 2 which is the nonparametric version of the Lagrangian meancurvature flow. Since the Lagrangian angle � depends only on second orderderivatives of u and all induced metrics gij = �ij + �kluikujl are uniformlyequivalent, we deduce uniform C2-estimates in space directions for (2.10).In addition we have

∂2u

∂t2= −∂�

∂t

= −d�dt

+d�

dxidxi

dt

= −d�dt

+HiHm�liulm with (1.2) and (2.8)

= −d†H +HiHm�liulm with (1.2) and Lemma 2.3 c).

Since we already proved uniform bounds for ∣H∣2, ∣d†H∣ and ∣D2u∣, we getuniform C2-estimates in time as well. To obtain longtime existence weneed uniform C1,�-estimates in time and uniform C2,�-estimates in spacefor some � > 0. Hence it remains to prove C2,�-bounds in space. Sofar we did not exploit condition (1.7). From Lemma 2.8, Lemma 2.7 andLemma 3.1 applied to S we conclude that the operator P [u] is concavefor all t and the results in [6] imply uniform C2,�-estimates in x for some� > 0. Standard Schauder estimates then give C∞-estimates both in spaceand time. In particular the full norm of the second fundamental form isuniformly bounded and we may apply Proposition (1.1) to get convergence.The compactness of L implies that the limit manifold must be flat. In casen = 2 we can drop condition (1.7) because if we freeze time, we may regardF as a solution of the elliptic system

ΔF =d

dtF =

−→H

with bounded RHS and the uniform C1-estimates for F and the regularitytheory for equations in two variables (e.g. see [2] section 12) give uniformC1,�-estimates for F which amounts to uniform C2,�-bounds for u sinceF (x) =

(x,Du(x)

).

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 17

□

We want to give a more direct proof of Theorem 1.3 in case n = 2. For thiswe establish a uniform bound for ∣A∣2.

Let p = p(s) be a function depending on s only and that has to be determinedlater. We set

f := p∣A∣2

and compute the evolution equation for f

ddtf = p(Δ∣A∣2 − 2∣∇A∣2 + 2∣bij ∣2 + 2∣Rijkl∣2)

+p′∣A∣2(Δs+ 4bijSij) ,

where we used Lemma 2.3 e), Lemma 2.6 and we have set p′ = ∂p∂s . Then

Δf = (p′Δs+ p′′∣∇s∣2)∣A∣2 + 2p′⟨∇s,∇∣A∣2⟩+ pΔ∣A∣2

gives

ddtf = Δf − 2p∣∇A∣2 − p′′∣∇s∣2∣A∣2 − 2p′⟨∇s,∇∣A∣2⟩

+2p(∣bij ∣2 + ∣Rijkl∣2 + 2p′

p ∣A∣2bijSij).

To proceed we observe that for n = dim(L) = 2 we have

Rijkl = R2 (gikgjl − gilgjk),

Rij = R2 gij = aij − bij ,

R = ∣H∣2 − ∣A∣2,∣Rijkl∣2 = R2,

so that

∣bij ∣2 = −Rij(bij + aij) + ∣aij ∣2

= −R2 (∣A∣2 + ∣H∣2) + ∣aij ∣2

≤ ∣A∣42 + c1(∣A∣2∣H∣2 + ∣H∣4)

for some positive constant c1.

In the same way we have

bijSij = ∣A∣2−∣H∣22 s+ aijSij

≥ ∣A∣22 s− c2(∣H∣2 + ∣H∣∣A∣)s ,

if we assume that there are positive constants c3, c4 such that

(3.5) c4sgij ≥ Sij ≥ c3sgij(we note here, that by (3.4) and Lemma 3.1 (3.5) is valid ∀ t ∈ [0, T ), if weassume that Sij > "gij for some " > 0 at t = 0.)

18 KNUT SMOCZYK

Lemma 3.4. Assume Sij > "gij for some " > 0 holds on L at t = 0 andthat dim(L) = 2. Then there exists a smooth vector field V and a positive

constant c such that f := ∣A∣2s2

satisfies

d

dtf ≤ Δf + ⟨V,∇f⟩+

c

s2− 1

2f ∣A∣2.

Proof. By Lemma 3.1 and (3.4) we know that (3.5) is valid ∀ t ∈ [0, T ). Letus choose p := 1

s2in the above expression for f = p∣A∣2.

Since∣ℎijk∇lℎmns − ℎmns∇lℎijk∣ ≥ 0

we get

2∣A∣2∣∇A∣2 ≥ 1

2∣∇∣A∣2∣2

and since in addition

∇f = p′∣A∣2∇s+ p∇∣A∣2,there exists some vector field V so that for n = 2

d

dtf ≤ Δf + ⟨V,∇f⟩+

3

2

(p′)2

p∣A∣2∣∇s∣2 − p′′∣A∣2∣∇s∣2

+2p(∣bij ∣2 + ∣Rijkl∣2 +2p′

p∣A∣2bijSij)

= Δf + ⟨V,∇f⟩+ 2p(∣bij ∣2 + ∣Rijkl∣2 +2p′

p∣A∣2bijSij)

≤ Δf + ⟨V,∇f⟩+ 2p(3

2∣A∣4 + c5(∣A∣2∣H∣2 + ∣H∣4) + 2

p′

p∣A∣2bijSij

)so that

d

dtf ≤ Δf + ⟨V,∇f⟩+ 2p

{3

2∣A∣4 + c5(∣A∣2∣H∣2 + ∣H∣4)(3.6)

−2∣A∣4 + 4c2(∣H∣2 + ∣H∣∣A∣)∣A∣2},

where we assumed that (3.5) is valid. Now by Lemma 3.1 and Lemma 3.2we know that (3.5) is valid and ∣H∣ uniformly bounded, if we assume thatSij > "gij for some " > 0 at t = 0. We apply Schwarz inequality to (3.6)and are done. □

Lemma 3.5. Assume n = 2 and that there exists an " > 0 such thatSij > "gij at t = 0. Then the quantity

∣A∣2

s2

is uniformly bounded on [0, T ).

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 19

Proof. From Lemma 3.4 we know that

d

dtf ≤ Δf + ⟨V,∇f⟩+

c

s2− f

2∣A∣2

But since s = gijSij > "gijgij = "n we obtain also

d

dtf ≤ Δf + ⟨V,∇f⟩+

c

n2"2− n2"2

2f2

and by the maximum principle f must be uniformly bounded from above.□

We can now give a uniform upper bound for ∣A∣2: From Lemma 3.5 we get

∣A∣2 = fs2 ≤ c6s2

for some constant c6 and because s is also bounded from above by (3.4) weget a uniform bound for ∣A∣2. In view of Proposition 1.1 this gives the proofof Theorem 1.3 in case n = 2.

□

4. Appendix

Here we will give a proof of Proposition 1.2. Therefore let S ∈ Σ(M) be atensor satisfying (1.3)-(1.5). To any such bilinear form we can associate anendomorphism

�S : TM → TM

by setting �S V := g��S� V ∂∂y�

. Therefore

S(V,W ) = g(�SV,W )

and (1.3) - (1.5) imply

g(�SV,W ) = g(�SW,V ) (Symmetry),(4.1)

�S ∘ J = −J ∘ �S (Anti-compatibility),(4.2)

D�S = 0 (Parallelity).(4.3)

Conversely, if � is an endomorphism satisfying (4.1) - (4.3), then S := g(�⋅, ⋅)defines an element in Σ(M).

Remark 4.1. Moreover, if � satisfies (4.1) - (4.3), then J� satisfies theserelations too. Conditions (4.1) - (4.3) are very similar to the conditionsfor the existence of a hyper-Kahler structure on M , i.e. another complexstructure K that satisfies

g(KV,W ) = −g(KW,V ),(4.4)

K ∘ J = −J ∘K,(4.5)

DK = 0,(4.6)

K2 = −Id.(4.7)

20 KNUT SMOCZYK

It is well known that hyper-Kahler metrics are Ricci-flat and that the exis-tence of a hyper-Kahler manifold gives restrictions on the holonomy of M .Here, we do not require anything for �2 and the signs in (4.1) resp. (4.4)differ.

Lemma 4.2. Let (M, g) be a Riemannian manifold and assume that � sat-isfies (4.1) and (4.3). If RM denotes the Riemannian curvature tensor onM , then one has

(4.8) RM (�V,W,X, Y ) = −RM (�W, V,X, Y ), ∀V,W,X, Y.

(4.9) RM (�X,W, Y,W ) = RM (�Y,W,X,W ), ∀X,Y,W.

Proof. The first equation follows from (4.3) and (4.1) because

0 = DVDW� −DWDV � −D[V,W ]� = RM (V,W )�.

For the second equation we use (4.8) and the first Bianchi identity

RM (�X,W, Y,W ) = −RM (�W,X, Y,W )

= RM (�W, Y,W,X) +RM (�W,W,X, Y )

= RM (�W, Y,W,X)

= −RM (�Y,W,W,X)

= RM (�Y,W,X,W )

□

Lemma 4.3. Let (M,J, g) be a Kahler manifold and assume that � satisfies(4.1) - (4.3). Then

(4.10) RM (�X,W, Y,W ) =1

2RM (�JX, Y,W, JW ), ∀X,Y,W.

Proof. In a first step we compute

RM (�X,W, Y,W ) = RM (J�JX,W, Y,W ) from J2 = −Id and (4.2)

= −RM (�JX, JW, Y,W )

= RM (�JX, Y,W, JW ) +RM (�JX,W, JW, Y )

= RM (�JX, Y,W, JW ) +RM (�JX,W, JY,W ).

Now, by remark 4.1 we can apply (4.9) to � := �J and obtain

RM (�JX,W, JY,W ) = RM (�J2Y,W,X,W )

= −RM (�Y,W,X,W )

= −RM (�X,W, Y,W ),

so that

RM (�X,W, Y,W ) = RM (�JX, Y,W, JW ) +RM (�JX,W, JY,W )

= RM (�JX, Y,W, JW )−RM (�X,W, Y,W ).

LONGTIME EXISTENCE OF THE LAGRANGIAN MEAN CURVATURE FLOW 21

□

Corollary 4.4. Under the assumptions made in Lemma 4.3 we also have

(4.11) RM (�X, JW, Y, JW ) = RM (�X,W, Y,W ), ∀X,Y,W.

Proof. This follows from (4.10) if we replace W by JW . □

Lemma 4.5. Let (M,J, g) be a Kahler manifold and assume that � satisfies(4.1) - (4.3). Then

RM (X,Y, V,W ) = 0, ∀X,Y, V,W ∕∈ ker(�).

Proof. Let �, � be two different eigenvalues of � and assume that V ∈Eig(�), W ∈ Eig(�). We apply (4.8) and obtain

�RM (V,W,X, Y ) = �RM (V,W,X, Y )

so that

(4.12) RM (V,W,X, Y ) = 0, ∀V ∈ Eig(�),W ∈ Eig(�), ∀X,Y ∈ TM.

With this and the first Bianchi identity we need to show only

RM (V1, V2, V3, V4) = 0

whenever V1, . . . , V4 belong to the same eigenspace Eig(�) of a nonzero eigen-value �. We let X := V1,W := V2, Y := V3 and use (4.11) to obtain

RM (V1, JV2, V3, JV2) = RM (V1, V2, V3, V2).

The LHS vanishes because of (4.12) and JV2 ∈ Eig(−�). So

RM (V1, V2, V3, V2) = 0, ∀V1, V2, V3 ∈ Eig(�).

But then

0 = RM (V1, V2 + V4, V3, V2 + V4)

= RM (V1, V2, V3, V4) +RM (V1, V4, V3, V2) + 0 + 0

gives

RM (V1, V2, V3, V4) = −RM (V1, V4, V3, V2)

= RM (V1, V3, V2, V4) +RM (V1, V2, V4, V3),

where we used Bianchi’s identity in the last step. Hence

RM (V1, V2, V3, V4) =1

2RM (V1, V3, V2, V4), ∀V1, V2, V3, V4 ∈ Eig(�).

Applying the last identity once again we find

RM (V1, V2, V3, V4) =1

4RM (V1, V2, V3, V4), ∀V1, V2, V3, V4 ∈ Eig(�)

and consequently RM (V1, V2, V3, V4) = 0. □

22 KNUT SMOCZYK

Proof of Proposition 1.2: This is now a direct consequence of Lemma4.5.

□

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Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103Leipzig, Germany

E-mail address: Knut.Smoczyk@mis.mpg.de