Proof of the Caratheodory Conjecture

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PROOF OF THE CARATHEODORY CONJECTURE

BY MEAN CURVATURE FLOW IN THE SPACE

OF ORIENTED AFFINE LINES

BRENDAN GUILFOYLE AND WILHELM KLINGENBERG

Contents

1. Introduction 22. Strategy and Proof 32.1. Reformulation of the Conjecture in TS2 32.2. The manifold of J-holomorphic discs with boundary 42.3. Proof of the Main Theorem 53. Mean Curvature Flow of Spacelike Submanifolds 73.1. Setting 73.2. Background frame 73.3. Immersed spacelike submanifolds 83.4. The group O(n,m) 83.5. The height functions 113.6. Mean curvature flow 133.7. Boundary conditions 174. Mean Curvature Flow in TS2 184.1. The neutral Kahler metric 184.2. Orthonormal frames 204.3. The second fundamental form 214.4. Mean curvature flow 224.5. The multi-time function 284.6. Asymptotic holomorphicity 294.7. Boundary and initial conditions 335. Convergence of the Evolution to a J-Holomorphic Disc 355.1. Compactness of the manifold of J-holomorphic discs with boundary 355.2. Convergence of the flow to a J-holomorphic bubble-disc in TS2 366. Concluding Remarks 376.1. Background 376.2. Mean curvature flow 386.3. J-holomorphic curves 386.4. Neutral Kahler surfaces 39References 39

Date: 6th August, 2008.

1

http://arXiv.org/abs/0808.0851v1

2 BRENDAN GUILFOYLE AND WILHELM KLINGENBERG

1. Introduction

A conjecture attributed to Constantin Caratheodory1 states:

The Caratheodory Conjecture: Every closed convex C3-smooth surface inEuclidean 3-space has at least two umbilic points.

In this paper we prove the following:

Main Theorem. Let p be an isolated umbilic point on a C3-smooth surface S inEuclidean 3-space. Then the winding number of the principal foliation about p isless than or equal to 1.

Aside from establishing the Caratheodory conjecture, this result has interest inits own right, as it is well-known to be equivalent to a conjecture of Loewner onsolutions of the wave equation. For more on these conjectures and their history, seeBerger [2] and references therein.

Recall that an umbilic point is a point where the second fundamental form(a symmetric 2-tensor) of the surface has a double eigenvalue. Away from suchpoints, the eigenvectors of the second fundamental form define a pair of orthogonalfoliations of the surface - the principal foliations. The winding number of theprincipal foliations about an isolated umbilic point is called the index of the umbilicpoint - which, due to the fact that the principal foliations may not be orientable,is an element of 1

2Z.Our proof of the Main Theorem depends upon a reformulation of the problem

in terms of isolated complex points on Lagrangian surfaces in the space of orientedgeodesics of Euclidean 3-space, which we identify with TS2. Here complex andLagrangian refer to the neutral Kahler structure on TS2 introduced by the authorsin [7].

We relate the index of an isolated complex point on an otherwise totally realsurface in any almost-complex 4-manifold to the space of holomorphic discs whoseboundary is contained in the surface. Indeed, if there exists such a holomorphicdisc, and it persists for all small deformations of the boundary surface, the indexof the associated elliptic boundary value problem is greater than zero. We showthrough the Keller-Maslov index of the boundary that, in our setting, this is exactlythe condition required for the Caratheodory conjecture to hold. We thus seek toprove the existence of stable holomorphic discs attached along the boundary to theLagrangian surface in TS2.

To establish the existence of such holomorphic discs we flow to them by the meancurvature flow associated to the neutral Kahler metric on TS2. In particular, weprove long-time existence of the mean curvature flow for n-dimensional spacelikesubmanifolds in n+m-dimensional manifolds with metrics of signature (n,m) undercertain conditions. In our case these conditions are met, and so we establish long-time existence for the mean curvature flow in TS2 for positive discs with boundarylying in a totally real surface.

Moreover, we prove that for small enough initial disc, the flow is asymptoticallyholomorphic in time. By altering the complex structure we construct a sequenceof holomorphic discs which, by virtue of our estimates, remain in a compact set.Crucially, we establish an a priori area bound for positive discs with boundary

1Hans Hamburger, Berliner Mathematische Gesellschaft, Berlin, 26th March 1924

CARATHEODORY CONJECTURE 3

contained on the deformation of a Lagrangian surface. Thus, the compactness ofthe space of J-holomorphic curves with boundary leads to a convergent sequenceof holomorphic discs. While the limit disc may be bubbled - bubbling off otherholomorphic discs - continuity up to the boundary ensures that the Keller-Maslovindex is retained in the limit. In fact, this is all we require for our proof.

Throughout, we prove existence of the flow by a priori estimates. This allows usto conclude that the solution persists under perturbation of the boundary surface.Thus we have demonstrated the existence of stable holomorphic discs and the MainTheorem follows.

The next section contains the reformulation and proof of the Main Theorem. Inthe following sections we supply the details of the proof.

2. Strategy and Proof

2.1. Reformulation of the Conjecture in TS2. The space of oriented geodesicsin Euclidean 3-space E3, namely the oriented affine lines in the underlying R3, maybe identified with the total space of the tangent bundle to the 2-sphere.

This 4-manifold is endowed with a natural Kahler structure (J,Ω,G) which isinvariant under the action induced on TS2 by the Euclidean group acting on E3.Throughout this paper, we denote the Kahler surface (TS2, J,Ω,G) simply by TS2.We now briefly summarize the properties of this structure we require - furtherdetails can be found in [7] and references therein.

Given an oriented Ck-smooth surface S in E3 with k ≥ 1, the set of orientedlines normal to S gives rise to a Ck−1-smooth surface Σ in TS2. Such a surfaceΣ is Lagrangian: Ω|Σ = 0. Indeed, the well-known converse holds by Frobeniusintegrability (see for example [6]):

Proposition 1. A surface Σ in TS2 is Lagrangian iff there exists a surface S inE3 which is orthogonal to the oriented lines of Σ.

Given one such orthogonal surface S in E3, we have a one parameter family ofparallel surfaces which are also orthogonal to the oriented lines of Σ. Moreover, apoint on S is umbilic iff the corresponding points on the parallel surfaces are alsoumbilic. Indeed, this property allows us to reformulate the Conjecture entirely inTS2, as we will see below.

Let S be a C3-smooth oriented surface in E3, and Σ the corresponding surfacein TS2 formed by the normal lines to S. The canonical projection π : TS2 → S2

restricted to Σ is just the Gauss map of the surface, and so we have:

Proposition 2. The surface S is non-flat (has non-zero Gauss curvature) iff Σ isthe graph of a section of the bundle π : TS2 → S2. In particular, the surface Σin TS2 formed by the oriented normal lines of a convex surface S is the graph of asection.

A point p on a surface Σ in an almost complex 4-manifold (M, J) is said to becomplex if J : TpM → TpM leaves TpΣ invariant. In our situation:

Proposition 3. A point γ on a Lagrangian surface Σ in TS2 is complex iff thepoint on the orthogonal surface S in E3 with oriented normal line γ is umbilic.

Moreover, the index of an isolated umbilic point on S is precisely the index ofthe associated complex point on the Lagrangian surface Σ as defined in [6].


Here, an umbilic point on S ⊂ E3 is a point where the second fundamental formhas a double eigenvalue and the 1

2Z-valued index of an isolated umbilic point is thewinding number of the eigen-directions about the point.

Thus the Main Theorem is equivalent to:

Main Theorem. An isolated complex point γ on the graph of a Lagrangian sectionof TS2 → S2 has index less than or equal to 1.

Note that, given a surface S ⊂ E3 with isolated umbilic point, by a Mobiustransformation of E3, S can be locally mapped to a convex surface with isolatedumbilic point of the same index. Thus, while we prove the Main Theorem in theconvex case, we can in fact drop the assumption of convexity of the surface in E3.

2.2. The manifold of J-holomorphic discs with boundary. We now discussholomorphic discs with boundary lying on a totally real surface in an almost com-plex surface (M, J). Throughout, we denote by Ck,α the usual Holder spaces in theappropriate function spaces.

Definition 1. For an almost complex surface (M, J) define the set of Holder bound-ary conditions by

B = Σ ⊂ M | Σ is a totally real embedded surface of smoothness Ck,α ,and the set of parameterized discs with boundary condition by

F = (f,Σ) | f : D → M, f ∈ Ck,α(D) ∩ C0(D) and f(∂D) ⊂ Σ ∈ B.The spaces F and B are Banach manifolds with respect to the Ck,α norm and

the projection π : F → B : π(f,Σ) = Σ is a Banach bundle. For (f,Σ) ∈ F define∂Jf = 1

2 (df j − J df) where j is the standard complex structure on D. Then,

∂Jf ∈ Ω01(f∗TM) and we define

Ω01(F) =⋃

(f,Σ)∈F

Ω01(f∗TM).

This is a Banach vector bundle over F and the operator ∂J is a section of thisbundle.

Definition 2. The set of holomorphic discs with boundary condition is defined by

MJ = (f,Σ) ∈ F | ∂Jf = 0 .Consider the linearization of ∂J at (f,Σ) ∈ F with respect to any connection on

Ω01(F):∇(f,Σ)∂J : Γ(f∗TM, f∗TΣ) → Ω01(f∗TM),

where Γ(f∗TM, f∗TΣ) denotes sections of f∗TM → D with boundary values in thesub-bundle f∗TΣ → ∂D. The key points about this operator are:

Proposition 4. [10] ∇(f,Σ)∂J is Fredholm. If ∇(f,Σ)∂J is surjective then MJ ⊂ Fis a Banach manifold with T(f,Σ)MJ = TΣB ⊕ ker∇(f,Σ)∂J .

There exists B0 ⊂ B dense and open such that for all (f,Σ0) ∈ π−1(B0) ∩MJ

we have: ∇(f,Σ0)∂J is surjective.


Definition 3. [10] For (f,Σ) ∈ F , define the Keller-Maslov index µ(f,Σ) ∈ Z tobe the relative Chern class

µ(f,Σ) = c1(f∗TM, f∗TΣ) = c1

f∗T 10M⋃

f∗TΣ

f∗T 10M

,

where we double the base D to D ∪ D and identify the fibres by conjugation aboutTΣ ⊂ T 10M over ∂D.

Proposition 5. The index of the Fredholm operator ∇(f,Σ)∂J is ind(∇(f,Σ)∂) =

µ(f,Σ) − 1. In addition, if ∇(f,Σ)∂J is surjective, then the projection π : MJ → Bis Fredholm with index ind(∇(f,Σ)π) = µ(f,Σ) − 1.

Proof. The first claim follows from the index formula for the corresponding Riemann-Hilbert boundary value problem, see [10].

The second claim is seen as follows. By Proposition 4, MJ is a Banach manifoldand since ∇(f,Σ)π : T(f,Σ)MJ ≡ TΣB⊕ker∇(f,Σ)∂J → TΣB we have ker(∇(f,Σ)π) =

ker∇(f,Σ)∂J from which the claim follows.

Theorem 1 (Sard-Smale). If ind(∇(f,Σ)π) ≤ 0 for (f,Σ) ∈ U ⊂ MJ , then π(U) ⊂B has measure zero.

2.3. Proof of the Main Theorem. Given a Lagrangian surface Σ ⊂ TS2 withisolated complex point γ ∈ Σ, we seek to construct a holomorphic disc with bound-ary lying in Σ by mean curvature flow. In particular, we consider the followingparabolic boundary value problem.

P.B.V.P.

Consider a family of immersed positive discs in TS2 given by fs : D → TS2 :fs(ξ, ξ) = (ξ, η = F (s, ξ, ξ)) such that

dfs

ds

⊥

= H,

with initial condition

F (0, ξ, ξ) = F0(ξ, ξ), that is D0 = Σ0,

and boundary conditions

(i) fs(∂D) ⊂ Σ,(ii) −ǫ|λ| + |σ| = |σ| on fs(∂D),

where H is the mean curvature vector associated with the immersion fs in (TS2,G),

Σ0 and Σ are some given positive discs, ǫ > 0 and λ, σ and σ are the twist andshear of fs(D) and Σ (cf. Definition 9).

We claim that for small enough initial discs the above flow converges to a holo-morphic disc with boundary on Σ. We prove this as follows.

In Theorem 2 we establish long-time existence of the mean curvature flow of acompact n-dimensional spacelike submanifold Σ of a manifold M endowed with ametric of signature (n,m), subject to the conditions:


(A) the submanifold Σ remains within a compact set of M

(B) timelike curvature condition (3.1) holds

This result is extended in Theorem 3 from compact submanifolds to subman-ifolds with boundary constrained to lie in an n-dimensional submanifold, and anadditional Neumann condition.

In order to apply this result to mean curvature flow of positive discs in the spaceof oriented affine lines TS2, a number of technical issues must be addressed, namely:

(1) show that conditions (A) and (B) hold for spacelike graphs in TS2

(2) determine boundary and initial surfaces(3) ensure parabolicity of the boundary conditions

To resolve issue (1) we show that (A) holds in Proposition 25. While the timelikecurvature condition does not hold in full generality for TS2, we prove in Proposition30 that it does hold along a solution of P.B.V.P.

Proposition 18 and Proposition 36 with C0 = 0 demonstrate that we cannot usethe Lagrangian surface Σ generated by normals to the convex surface in E3 as eitherinitial or boundary surface in our flow, as it is neither positive definite nor allowspositive discs to be attached to it.

Rather, for the boundary condition, we add a linear holomorphic twist about thecomplex point (equation (4.9)), which produces a disc Σ that is positive for smallenough radius and allows other positive discs to be attached to it (Proposition32). This perturbed surface has an isolated complex point of the same index as theoriginal Lagrangian surface and so we aim to find stable holomorphic discs attachedto it. The existence of a compatible initial disc is proven in Proposition 35 and sowe have addressed issue (2).

The C3 smoothness requirement on S ⊂ E3 ensures that the mean curvaturevectors of Σ, Σ adn Σ0 are C0, the minimum requirement for P.B.V.P. to admita solution.

To resolve issue (3), we formulate a general set of higher codimensional boundaryvalue problems and prove in Proposition 15 that they are parabolic. Then, inProposition 33, we show that the boundary conditions we implement in the TS2

flow are of this type.We have therefore proven long-time existence in the setting in which we require

it. Moreover, in Theorem 4 we prove that, for small enough initial disc, the flowingdisc is asymptotically holomorphic. Thus we can deform the complex structure sothat, for late enough time, the flowing disc is holomorphic. From this we extract asequence of J-holomorphic discs for which, if there exists an a priori area bound,compactness results (Theorem 5) imply that we can extract a holomorphic disc inthe limit. In Proposition 36 such an area bound is shown to exist precisely whenthe boundary surface is Lagrangian with a linear holomorphic twist. The limitdisc f∞(D), which may be bubbled, is holomorphic with respect to our originalJ and continuity up to the boundary implies that the Keller-Maslov index of theholomorphic discs is retained in the limit. These results are established in Theorem6. Interpolation arguments show that the flow converges to this disc.

To complete the proof of the Main Theorem we prove in Proposition 34 thatthe boundary of the flowing disc does not cross the complex point. Thus we canattach a holomorphic disc f(D) to Σ about the complex point γ. Moreover, for any

small perturbation of the boundary surface Σ we can carry out the mean curvature


flow and arrive at a holomorphic disc attached to the perturbation. By virtue ofProposition 5 and Theorem 1 this implies that ind(∇(f,Σ)π) = µ(f,Σ) − 1 > 0.

Let I be the index of the isolated complex point and µ be the Keller-Maslovindex of (f∞, Σ). Then, considering the Euler number of the sphere consisting of

f∞(D) and the Σ joined along the boundary of D we have

I + 12µ = 2.

Thus,I = 2 − 1

2µ <32 ,

which establishes the Main Theorem.

3. Mean Curvature Flow of Spacelike Submanifolds

In this section we establish a long-time existence result for mean curvature flowof spacelike surfaces in indefinite manifolds. Throughout we utilize the summationconvention on repeated indices, except for the quantity ψα, defined below. In someinstances we include summation signs for clarity. Note that raising and loweringnormal indices (Greek indices) changes the sign of the component.

3.1. Setting. Let M be an n+m-dimensional manifold endowed with a metric G

of signature (n,m). We assume throughout that there exists a multi-time functiont : M → Rm of maximal rank with components tα for α = 1...m such that

G(∇tα,∇tα) < 0 ∀α = 1...m,

where all geometric quantities associated with G will be denoted with a bar.

Definition 4. The manifold (M,G) is said to satisfy the timelike curvature condi-tion if, for any spacelike n-plane P at a point in M, the Riemann curvature tensorsatisfies

G(R(X, τi)X, τi) ≥ k G(X,X), (3.1)

for some positive constant k, where τini=1 form an orthonormal basis for P and

X is any timelike vector orthogonal to P .

Note 1. This generalises the timelike convergence condition of the codimensionone case employed in [4]:

Ric(X,X) ≥ 0.

3.2. Background frame. We fix an orthonormal frame on (M,G):

ei, Tαn,mi,α=1 s.t. G(ei, ej) = δij G(Tα, Tβ) = −δαβ G(ei, Tα) = 0,

withTα = −ψαG(∇tα, ·) ψ−2

α = −G(∇tα,∇tα).

Definition 5. Given a contravariant tensor B on M we define its norm by

‖B‖2 =n∑

i1,...il=1

[B(ei1 , ei2 , ..., eil)]2 +

m∑

β1,...βl=1

[B(Tβ1, Tβ2

, ..., Tβl)]2.

Similarly, for a covariant tensor B we dualise it with the metric G and define itsnorm as above.


Higher derivative norms are also defined:

‖B‖2k =

k∑

j=0

‖∇jB‖2.

For a mixed tensor, we occasionally use the induced metric on the spacelikecomponents to define a norm on the timelike components. That is, if Bαβijk is atensor of the indicated type, then we define

|Bαβ |2 =

n∑

i=1

[Bαβ(ei, ei, ei)]2.

3.3. Immersed spacelike submanifolds. Let f : Σ → M be a spacelike immer-sion of an n-dimensional manifold Σ, and let g be the metric induced on Σ byG.

Definition 6. A second orthonormal basis for (M,G) along Σ is adapted to thesubmanifold if:

τi, ναn,mi,α=1 s.t. G(τi, τj) = δij G(να, νβ) = −δαβ G(τi, να) = 0,

where τini=1 form an orthonormal basis for (Σ, g), and ναm

α=1 span the normalspace.

The second fundamental form of the immersion is

Aijα = G(∇τiνα, τj) = −G(∇τi

τj , να),

while the mean curvature vector is

Hα = gijAijα.

We have the following two equations for the splitting of the connection

∇τiτj = ∇τi

τj −Aαijνα (3.2)

∇τiνα = Aj

iατj + Cβiανβ , (3.3)

where Cβiα are the components of the normal connection

∇⊥τiνα = Cβ

iανβ .

We also have the Gauss and Codazzi-Mainardi equations

Rijkl = Rijkl +AαkjAilα −Aα

ljAikα, (3.4)

− Rαkij = ∇iA

αjk −∇jA

αik +Aβ

jkCαiβ −Aβ

ikCαjβ . (3.5)

3.4. The group O(n,m). Introduce the notation

Xij = G(τi, ej) Wiβ = G(τi, Tβ) Uαj = −G(να, ej) Vαβ = −G(να, Tβ).

Thus

ei = Xijτj + Uαiνα Tβ = Wiβτi + Vαβνα,

and the (n+m) × (n+m) dimensional matrix

M =

(

X W−U −V

)

,

is an element of the orthogonal group O(n,m).


Proposition 6. The O(n,m) condition on M is

XTX = In + UTU V TV = Im +WTW UTV = XTW. (3.6)

Proof. This follows from the condition

MT

(

In 00 −Im

)

M =

(

In 00 −Im

)

.

The vectors τin1 span the tangent space of Σ, while ναm

1 span the normalbundle. We are free to rotate these frames within these two spaces, and this corre-sponds to left action of O(n) and O(m) within O(n,m).

Similarly, we consider rotations of ein1 that preserve the n-dimensional vec-

tor space that they span, along with rotations of Tβm1 that preserves the m-

dimensional space they span. These correspond to right actions of O(n) and O(m)within O(n,m). Note that the positive definite norm in Definition 5 is preservedby these rotations.

Proposition 7. By rotations of the frames ei, Tα and τj , νβ which preservethe tangent and normal bundles of Σ, as well as the tensor norm of Definition 5,we can simplify the matrix M ∈ O(n,m) for n ≥ m to

M =

In−m 0 00 D1 ±D4A

T

0 D3A D2

,

where A ∈ O(m) is a transposition matrix, D1, D2, D3 and D4 are diagonal ma-trices satisfying

D21 = Im +D2

3 D22 = Im +D2

4 |D1|2 = |D2|2,and ± of a diagonal matrix means a free choice of sign on the entries of the matrix.

Proof. Consider first the matrix Xij =< τi, ej >. The matrix XTX is symmetricand non-negative definite and so it has a well-defined square root, namely a sym-

metric n×n matrix which we denote by√XTX. By the first equation of (3.6), X is

invertible since det(X) ≥ 1 and so we can define the n×n matrix A =√XTXX−1.

Then

AT InA = (X−1)T√XTX

√XTXX−1 = (X−1)TXTXX−1 = In,

so that A ∈ O(n). Define a new frame by Aijτj , να and then

Xij = Aik < τk, ej >=√XTXX−1X =

√XTX,

which is symmetric. Now we can act on both the left and right of X by O(n) todiagonalise it.

A similar argument yields a diagonalisation of Vαβ .After diagonalisation of X , the first of equations (3.6) implies that the matrix

UTU is diagonal. Thus the n m-dimensional vectors Uαiναni=1 are mutually

orthogonal and, since n ≥ m, we conclude that n−m of these vectors must be zero.After a reordering of the basis elements, the matrix M then decomposes into

M =

In−m 0 W2

0 X1 W1

0 U1 V

.


The last of equations (3.6) now implies that W2 = 0 and we reduce the problem tothe square case:

XT1 X1 = Im + UT

1 U1 V TV = Im +WT1 W1 UT

1 V = XT1 W1.

In fact, to indicate that X1 and V are diagonal, let us write X1 = D1 andV = D2. Thus

D21 = Im + UT

1 U1, (3.7)

D22 = Im +WT

1 W1, (3.8)

UT1 D2 = D1W1. (3.9)

Equations (3.7) and (3.8) imply that there exists diagonal matrices D3 and D4

(with entries defined up to a sign) such that

U1 = D3A W1 = D4B for A,B ∈ O(m).

Thus equations (3.7), (3.8) and (3.9) now read

D21 = Im +D2

3 , (3.10)

D22 = Im +D2

4 , (3.11)

ATD2D3 = D1D4B. (3.12)

Taking the transpose of this last equation and multiplying back on the right wefind that

ATD22D

23A = D2

1D24. (3.13)

However, if A ∈ O(m) that sends a diagonal matrix to a diagonal matrix, then Amust be a transposition. Similarly

BTD21D

24B = D2

2D23 ,

and so A = ±BT .Denote the diagonal elements of D1, D2, D3 and D4 by λi, µi, ai and bi, re-

spectively, where i = n − m + 1, ...n. Then equations (3.10), (3.11) and (3.13)read

λ2i = 1 + a2

i µ2i = 1 + b2i µ2

i a2i = λ2

p(i)b2p(i),

where p is the permutation of (n−m+ 1, ...n) determined by the transposition A.Combining these three equations we get

a2i + a2

i b2i = a2

p(i) + a2p(i)b

2p(i),

which when summed yields∑

i

a2i =

∑

i

b2i and∑

i

λ2i =

∑

i

µ2i .

Thus |D1|2 = |D2|2 as claimed.


Definition 7. The function v is defined to be

v2 =∑

α,β

V αβVαβ .

This is a generalization of the tilt function in the codimension one case [1].

We now use the normal form to construct estimates of the norms of the adaptedframes:

Proposition 8. For an adapted frame τi, να we have

‖τi‖2 ≤ [n2 +m(m− 1)2]v2 ‖να‖2 ≤ m3v2,

for all i = 1, 2, ...n and α = 1, 2, ...m.

Proof. First consider an adapted frame τ i,

να for which the matrix M has the

form given in Proposition 7. For a general adapted frame τi, νατi = Aj

i

τ j να = Bβ

α

νβ,

where A ∈ O(n) and B ∈ O(m). Then

‖τi‖2 =∑

j

(G(τi, ej))2 +

∑

α

(G(τi, Tα))2

=∑

j

[

∑

k

Aki G(

τk, ej)

]2

+∑

α

[

∑

k

Aki G(

τ i, Tα)

]2

≤∑

j

[

∑

k

|Aki | |G(

τk, ej)|

]2

+∑

α

[

∑

k

|Aki | |G(

τ i, Tα)|

]2

≤∑

j

[

∑

k

|G(τk, ej)|

]2

+∑

α

[

∑

k

|G(τ i, Tα)|

]2

≤∑

j

[

∑

k

|Xkj |]2

+∑

α

[

∑

k

|Wkα|]2

≤[

∑

k

|Xkk|]2

+∑

α

[

n∑

k=n−m+1

|Wkα|]2

≤ n2v2 +m(m− 1)2v2

≤ [n2 +m(m− 1)2]v2.

Similarly for να.

3.5. The height functions. Let uα : Σ → R be the height function uα = tα f .Then

Proposition 9. For all α = 1...m

∇uα = ∇tα + ψ−1α

∑

β

Vβανβ ,


∇uα · ∇uβ = ψ−1α ψ−1

β

(

∑

γ

VγαVγβ − δαβ

)

.

Proof. From the definition of uα and Tα we have

∇uα = ∇tα + ψ−1α

∑

β

Vβανβ = ψ−1α (∑

β

Vβανβ − Tα),

and so

∇uα · ∇uβ =ψ−1α ψ−1

β G

(

∑

γ

Vγανγ − Tα,∑

δ

Vδβνδ − Tβ

)

= ψ−1α ψ−1

β

(

∑

γ

VγαVγβ − δαβ

)

.

as claimed.

Proposition 10.

uγ = −ψ−1γ VαγH

α + gij∇i∇jtγ .

Vαβ =Vγβ(AijγAijα − < R(τi, νγ)τi, να >) − ∇Tβ

Hα −Aijα Tβ(gij)

+ 12 (∇LTβ

G)(να, τi, τi) − (∇LTβG)(τi, να, τi) − (∇Tβ)(H, να)

− 2C γiα < νγ ,∇Tβ

τi > +(∇iCγ

iα + C δiα C γ

iδ )Vγβ ,

where is the Laplacian of the induced metric = gij∇i∇j .

Proof. The first statement follows from a straightforward generalization of the codi-mension one case [4].

For the second statement we follow Bartnik [1] and fix a point p ∈ Σ and choosean orthonormal frame τi on Σ such that (∇iτj)(p) = 0. Extend this frame in aneighbourhood of Σ by LTβ

τi = 0 for fixed β. Then

−Vαβ = < να, Tβ >

= τiτi < να, Tβ >

= τi(< ∇τiνα, Tβ > + < να,∇τi

Tβ >)

= τi(Ajiα < τj , Tβ > +Cγ

iα < νγ , Tβ > + < να,∇τiTβ >)

=< R(τi, Tβ)τi, να > + < να,∇Tβ∇τi

τi > + < ∇τiνα,∇Tβ

τi >)

+ (∇τiHα+ < R(τi, τj)να, τi > −Aγ

ijCαiγ +HγCα

jγ) < τj , Tβ >

+ Cγiα(< ∇τi

νγ , Tβ > + < νγ ,∇τiTβ >)+ < νγ , Tβ > ∇τi

Cγiα

+Ajiα(< ∇τi

τj , Tβ > + < τj ,∇τiTβ >)

=< R(τi, νγ)τi, να >< νγ , Tβ > + < να,∇Tβ∇τi

τi > +2Ajiα < τj ,∇Tβ

τi >

+ 2Cγiα < νγ ,∇τi

Tβ > + < τi, Tβ > ∇iHα +HγCiγα < τi, Tβ >

+AijαA

γij < νγ , Tβ > +Cγ

iαCδiγ < νδ, Tβ > + < νγ , Tβ > ∇τi

Cγiα

= −Vγβ(AijγAijα + < R(τi, νγ)τi, να >) + ∇Tβ

Hα

< να,∇Tβ∇iτi > +Aij

α Tβ < τi, τj >

+ 2C γiα < νγ ,∇Tβ

τi > +(∇iCγ

iα − C δiα C γ

iδ )Vγβ . (3.14)


To complete the proof we now use the following:

Lemma 1.

Tβ < τi,∇iνα > = − < ∇iτi,∇Tβνα > + 1

2 (∇LTβG)(να, τi, τi)

− (∇LTβG)(τi, να, τi)− < ∇HTβ, να > .

Proof. The proof of this follows the codimension one case (Proposition 2.1 of [1]).

To complete the proof of the proposition we note that

< να,∇Tβ∇iτi > = Tβ < να,∇iτi > − < ∇Tβ

να,∇iτi >

= −Tβ < ∇iνα, τi > − < ∇Tβνα,∇iτi >

= − 12 (∇LTβ

G)(να, τi, τi) + (∇LTβG)(τi, να, τi)

+ < ∇HTβ , να >,

where in the last equailty we have used Lemma 1. Substituting this in equation(3.14) then yields the result.

3.6. Mean curvature flow. Let fs : Σ → M be a family of n-dimensional space-like immersed surface in an n + m-dimensional manifold M with a metric G ofsignature (n,m). In addition, we assume that n ≥ m, the case n < m follows bysimilar arguments.

Then fs moves by mean curvature flow if it satisfies the equation

df

ds

⊥

=

(

gjk∇ ∂f∂xj

∂f

∂xk

)⊥

= H, (3.15)

where ∇ is the Levi-Civita connection associated with the ambient metric on M,H is the mean curvature vector H = Hανα and ⊥ is the projection perpendicularto the tangent space of Σ.

The flow of the functions uγ and v are given by

Proposition 11.(

d

ds−

)

uγ = −gij∇i∇jtγ , (3.16)

v

(

d

ds−

)

v ≤− V αβVγβ(AijγAijα − < R(τi, νγ)τi, να >) +Aij

α LTβgijV

αβ

− 12 (∇LTβ

G)(να, τi, τi)Vαβ + (∇LTβ

G)(τi, να, τi)Vαβ

+ 2C γiα < νγ ,∇Tβ

τi > V αβ − C δiα C γ

iδ VγβVαβ .

Proof. Generalizing Proposition 3.1 of [4], note the time derivatives are

duγ

ds= −ψ−1

γ VαγHα,

dVαβ

ds= −∇Tβ

Hα −Hγ < ∇νγTβ, να > .

The flow of uγ then follows immediately from Proposition 10.


To find the flow of v note that

v

(

d

ds−

)

v = V αβ

(

d

ds−

)

Vαβ +1

v2

[

(vα∇Vα) · (vβ∇Vβ) − V 2α |∇Vβ |2

]

,

where we sum over α and β and diagonalised Vαβ = diag(V1, ...Vm). By the Cauchy-Schwarz inequality we have

v

(

d

ds−

)

v ≤ V αβ

(

d

ds−

)

Vαβ .

Now contracting the second equation of Proposition 10 with V αβ yields the result.

Proposition 12. Assume that M satisfies the timelike curvature condition. Let Σs

be a smooth solution of (3.15) on the interval 0 ≤ s < s0 such that Σs is containedin a smooth compact subset of M for all 0 ≤ s < s0. Then the function v satisfiesthe a priori estimate

v(p, s) ≤ (m+ supΣ0

v) sup(q,s)∈Σ×[0,s0]

exp[K(u(q, s) − u(p, s))],

for some positive constant K(n,m, ‖t‖3, |ψ|, ‖R‖, |H |, k), where u =∑

α uα.

Proof. Let K>0 be a constant to be determined later and set

CK = (1 + supΣ0

v) supΣ×[0,s0]

exp(Ku).

Consider the function f = v exp(Ku). Suppose, for the sake of contradiction, thatthe function f reaches CK for the first time at (p1, s1) ∈M × (0, s0]. Then at thispoint v ≥ m+ 1 and

(

d

ds−

)

f·≥ 0 ∇f ·

= 0.

Here and throughout a dot over an inequality or equality will refer to evaluation atthe point (p1, s1). Working out these two equations we have

(

d

ds−

)

v +Kv

(

d

ds−

)

u− 2K∇u · ∇v −K2v|∇u|2·≥ 0,

(3.17)

∇v +Kv∇u ·= 0. (3.18)

Substituting the second of these in the first we obtain

Kv

(

d

ds−

)

u·≥ −

(

d

ds−

)

v −K2v|∇u|2. (3.19)

Now, from Proposition 11 and the estimates in Proposition 8(

d

ds−

)

u = −gij∇i∇jt ≤ ‖∇i∇jt‖.‖τi‖.‖τj‖ ≤ C1v2, (3.20)

where C1 = C1(n,m, ‖t‖2).


At p1 we can set C βiα = 0 and then, Proposition 11 and the timelike curvature

condition imply that

v

(

d

ds−

)

v·≤− V 2

α |Aα|2 + C2(‖T ‖1)|Aα|Vα + C3(n,m, ‖T ‖2)v4

≤− (1 − ǫ)V 2α |Aα|2 + C4(ǫ, n,m, ‖T ‖2)v

4, (3.21)

for any ǫ > 0. Here we have utilised the gauge choice Vαβ = Vαδαβ on some of theterms and summation is over α.

Now, from the Schwartz and arithmetic-geometric mean inequalities

V 2α |Aα|2 ≥

(

1 +1

n

)

λ2αV

2α −H2

αV2α , (3.22)

where λα is the eigenvalue of Aijα with the maximum absolute value.On the other hand we compute

∇iVαβ = −Ajiα < τj , Tβ > − < να,∇iTβ >,

and so

v∇iv = V αβ∇iVαβ = −AjiαWjβV

αβ− < να,∇iTβ > V αβ .

Thus

|∇v| ≤ |λα|Vα + C5(n,m, ‖T ‖1)v2,

and so

|∇v|2 ≤ (1 + ǫ)V 2αλ

2α + C6(ǫ, n,m, ‖T ‖1)v

4.

Rearranging this last inequality

V 2αλ

2α ≥ 1

1 + ǫ|∇v|2 − C6v

4. (3.23)

Combining inequalities (3.22) and (3.23) we get

V 2α |Aα|2 ≥

(

1 +1

n

)[

1

1 + ǫ|∇v|2 − C5v

4

]

− |Hα|2V 2α ,

which, when substituted in inequality (3.21) yields

v

(

d

ds−

)

v·≤ −

(

1 +1

n

)

1 − ǫ

1 + ǫ|∇v|2 + C7(n, |H |)v2 + C6v

4,

and, by virtue of equation (3.18),

|∇v|2 ·= K2v2|∇u|2,

yielding(

d

ds−

)

v·≤ −

(

1 +1

n

)

1 − ǫ

1 + ǫK2v|∇u|2 + C7v + C6v

3. (3.24)

Substituting inequalities (3.20) and (3.24) in (3.19) we get

KC1v2

·≥[(

1 +1

n

)

1 − ǫ

(1 + ǫ)− 1

]

K2|∇u|2 + C7 + C6v2,

for any ǫ > 0.Now for 0 < ǫ < 1/(1 + 2n)

(

1 +1

n

)

1 − ǫ

1 + ǫ− 1 > 0,


and so using Proposition 9

|∇u|2 =∑

α,β

∇uα · ∇uβ =∑

α

ψ−2α (V 2

α − 1) ≥ minαψ−2α (v2 −m),

we have

KC1v2

·≥ C8(ǫ, n, |ψ|)K2(v2 −m) + C7 + C6v

2,

which can be rearranged to

v2·≤ mC8K

2 − C7

C7K2 − C1K + C6,

which for large K violates v ≥ m+ 1.

For tensors Hα and Aijα we define a positive norm by

|H |2+ = −HαHα |A|2+ = −AijαA

ijα,

and similarly for their gradients.

Proposition 13. [3] Under the mean curvature flow(

d

ds−

)

|H |2+ = −2|∇H |2+ − 2|A|2+|H |2+ − 2HαHβRiαiβ ,

(

d

ds−

)

|A|2+ = −2|∇A|2+ − 2|A|4+ +A ∗A ∗R +A ∗ ∇ R,

where ∇ is the covariant derivative in both the tangent and normal bundles and ∗represents linear combinations of contractions of the tensors involved.

Proposition 14. Under the mean curvature flow

|H |2+ ≤ C1(1 + s−1),

|A|2+ ≤ C2(1 + s−1),

where C1 = C1(n, k) and C2 = C1(n, ‖R‖1).

Proof. From the previous proposition and the timelike curvature condition (3.1) weconclude that

(

d

ds−

)

|H |2+ ≤ −2n−1|H |4+ + 2k|H |2+,

while(

d

ds−

)

|A|2+ ≤ −2|A|4+ + C3|A|2+ + C4|A|+ ≤ −|A|4+ + C5,

The result then follows by a suitable modification of Lemma 4.5 of Ecker andHuisken [4].

Theorem 2. Let Σ0 be a smooth compact n-dimensional spacelike submanifoldof an n + m dimensional manifold M satisfying the timelike curvature condition.Then there exists a unique family fs(Σ) of smooth compact n-dimensional spacelikesubmanifolds satisfying the initial value problem (3.15) on an interval 0 ≤ s < s0.Moreover, if fs(Σ) remains in a smooth compact region of M as s→ s0, the solutioncan be extended beyond s0.


Proof. Having bounded the second fundamental form in Proposition 14, bounds onthe higher derivatives and hence long-time existence, follow from standard parabolicbootstrapping arguments as in [4].

3.7. Boundary conditions. While our considerations so far have been on flowingcompact submanifolds, the results can be extended to the case where the flowingsubmanifold has boundary, so long as we implement suitable boundary conditions.For dimensional reasons, and because mean curvature flow is a second order PDE,we must have m boundary conditions in total.

Consider the case ofm−1 Dirichlet conditions and one Neumann condition. Thatis, let Σ be a fixed n-dimensional submanifold in an n + m-dimensional manifoldM and consider the Dirichlet requirement that a given immersed n-dimensionalspacelike submanifold f : Σ → M has boundary in Σ.

Definition 8. An independent Neumann boundary condition is a Neumann con-dition

N(f) = 0 along f(∂Σ),

where N is some linear combination of the slopes of Σ which is linearly independentfrom the derived Dirichlet condition .

Consider the following flow.

Let fs : Σ → M be a family of immersed n-dimensional spacelike submanifoldsof M such that

df

ds

⊥

= H,

with initial condition

f0(Σ) = Σ0,

and boundary conditions

(i) ∂fs(Σ) ⊂ Σ,(ii) N(fs) = 0,

where H is the mean curvature vector associated with fs(Σ), Σ0 and Σ are somegiven n-dimensional submanifolds, and boundary condition (ii) consists of an inde-pendent Neumann condition.

Proposition 15. The above boundary value problem is parabolic.

Proof. To establish this we verify that the well-known Lopatinski-Shapiro condi-tions are satisfied. Let (x1, ..., xn−1, ν) be coordinates that flatten out the boundary,so that the boundary is given by ν = 0. The Lopatinski-Shapiro condition requiresthat there exists a unique solution of the ODE associated with the above problem,namely:

d2fα

d2t−∑

j

(ζj)2fα = 0,

limt→∞

fα = 0,


(i) Aαkdfα

dt(0) +Bαkjζ

jfα(0) = gk (ii) Cαdfα

dt(0) +Dαjζ

jfα(0) = h,

where the initial conditions (i) and (ii) are derived from the linearisation of theDirichlet condition (j, k = 1, ..., n − 1), along with the Neumann condition. Thefirst equation can be integrated to

fα = Eα1 exp(−|ζ|t) + Eα

2 exp(|ζ|t).The second condition implies that Eα

2 = 0 and it remains to satisfy the initialconditions

(i) −Aαk|ζ|Eα1 +Bαkjζ

jEα1 = gk (ii) − Cα|ζ|Eα

1 +DαjζjEα

1 = h.

Since the Neumann condition is assumed linearly independent from the linearizedDirichlet conditions, there exists a unique solution to this ODE.

Thus we can extend Theorem 2 to the case where the flowing submanifold hasthe above boundary constraints:

Theorem 3. Let Σ0, Σ be smooth n-dimensionalsubmanifolds of an n + m di-mensional manifold M satisfying the timelike curvature condition. Assume thatΣ0 is spacelike, ∂(Σ0) lies on Σ and an independent Neumann condition holdsalong the boundary. Then there exists a unique family fs(Σ) of smooth compactn-dimensional spacelike submanifolds satisfying the above parabolic boundary valueproblem on an interval 0 ≤ s < s0. Moreover, if fs(Σ) remains in a smooth compactregion of M as s→ s0, the solution can be extended beyond s0.

4. Mean Curvature Flow in TS2

In this section we establish the main estimates we require to prove convergenceof mean curvature flow in TS2 as stated in section 2.3. Throughout we use the termpositive surface to mean spacelike surface: the induced metric is positive definite.

4.1. The neutral Kahler metric. In order to compute geometric quantities weintroduce local coordinates on TS2. These are readily supplied by lifting the stan-dard complex coordinate ξ (obtained by stereographic projection from the southpole on S2) to complex coordinates (ξ, η) on TS2. In particular, we identify(ξ, η) ∈ C2 with the vector

η∂

∂ξ+ η

∂

∂ξ∈ TξS

2.

These coordinates are holomorphic with respect to the complex structure J:

J

(

∂

∂ξ

)

= i∂

∂ξJ

(

∂

∂η

)

= i∂

∂η,

and the symplectic 2-form and neutral metric have the following local expressions:

Ω = 4(1 + ξξ)−2Re

(

dη ∧ dξ − 2ξη

1 + ξξdξ ∧ dξ

)

,

G = 4(1 + ξξ)−2Im

(

dηdξ +2ξη

1 + ξξdξdξ

)

. (4.1)


We are interested in graphs of local sections of the bundle π : TS2 → S2. Suchlocal sections are given by ξ 7→ (ξ, η = F (ξ, ξ)), for some function F : C → C.

Definition 9. We introduce the complex slopes of F :

σ = −∂F ρ = θ + iλ = (1 + ξξ)2∂(F (1 + ξξ)−2).

Here, and throughout, ∂ represents differentiation with respect to ξ. The functionsλ and σ are commonly referred to as the twist and shear of the underlying familyΣ of oriented lines in E3.

We note the following two identities, which follow from these definitions:

− (1 + ξξ)2∂

[

σ

(1 + ξξ)2

]

= ∂ρ+2F

(1 + ξξ)2, (4.2)

Im ∂

(1 + ξξ)2∂

[

σ

(1 + ξξ)2

]

= ∂∂λ+2λ

(1 + ξξ)2. (4.3)

The geometric significance of λ and σ are as follows:

Proposition 16. [6] A surface Σ given by a local section η = F (ξ, ξ) is Lagrangianiff λ = 0 and is holomorphic iff σ = 0.

For the induced metric we have:

Proposition 17. The metric induced on the graph of a section by the Kahler metricis given in coordinates (ξ, ξ) by;

g =2

(1 + ξξ)2

[

iσ −λ−λ −iσ

]

,

with inverse

g−1 =(1 + ξξ)2

2(λ2 − σσ)

[

iσ −λ−λ −iσ

]

.

Proof. This follows from pulling back the neutral metric (4.1) along a local sectionη = F (ξ, ξ).

Proposition 18. The induced metric on a Lagrangian surface is Lorentz, exceptat complex points, where it is degenerate.

Proof. By the previous Proposition the determinant of the induced metric is 2(1 +ξξ)−2(λ2 − σσ) and the result follows.

Note 2. In Section 2.1 we have seen that umbilic points on surfaces in E3 giverise to complex points on Lagrangian surfaces in TS2 and now we see that thesecorrespond to degeneracies in the induced Lorentz metric. Moreover, the principalfoliation corresponds exactly to the null directions of the Lorentz metric. Thus theCaratheodory conjecture bounds the winding number of the light cone of certainLorentz surfaces with isolated degenerate points, and the hyperbolic nature (andhence difficulty) of the problem becomes evident.


Note 3. Roughly speaking, our strategy is to maximise the area of an immersedsurface by reducing the shear and increasing the twist. In fact, the previous Propo-sition was generalized [9]:

Let (M, J,Ω,G) be any neutral Kahler surface and let p ∈ M and v1, v2 ∈ TpM

span a plane. Then

Ω(v1, v2)2 − ς2(v1, v2) = det G(vi, vj),

where ς2(v1, v2) ≥ 0 with equality iff spanRv1, v2 is a complex line in TpM.

In order to continue, we introduce geometric tools which will prove useful later.

4.2. Orthonormal frames. Let Σ →TS2 be an immersed surface and assumethat the induced metric is not degenerate on Σ, so that for γ ∈ Σ we have theorthogonal splitting TγTS

2 = TγΣ ⊕NγΣ .

Definition 10. Given a surface Σ in TS2, a frame e(a)4a=1 is adapted if e(1), e(2)

is a basis for the tangent space TΣ and e(3), e(4) is a basis for the normal bundleNΣ, and

G(e(a), e(b)) = diag(1, ǫ,−ǫ,−1),

where ǫ = 1 for G|Σ positive definite and ǫ = −1 for G|Σ Lorentz. For meancurvature flow of positive discs we have ǫ = 1.

Proposition 19. If Σ is a non-degenerate surface given by the graph ξ → (ξ, η =F (ξ, ξ)), then the following vector fields form an adapted basis:

e(1) = 2Re

[

α1

(

∂

∂ξ+ ∂F

∂

∂η+ ∂F

∂

∂η

)]

,

e(2) = 2Re

[

α2

(

∂

∂ξ+ ∂F

∂

∂η+ ∂F

∂

∂η

)]

,

e(3) = 2Re

[

α2

(

∂

∂ξ+ (∂F − 2(F∂u− F ∂u))

∂

∂η− ∂F

∂

∂η

)]

,

e(4) = 2Re

[

α1

(

∂

∂ξ+ (∂F − 2(F∂u− F ∂u))

∂

∂η− ∂F

∂

∂η

)]

,

for

α1 =e−u− 1

2φi+ 1

4πi

√2[−λ− |σ|] 1

2

α2 =e−u− 1

2φi− 1

4πi

√2[ǫ(−λ+ |σ|)] 1

2

,

where ∂F = −|σ|e−iφ and we have introduced e2u = 4(1 + ξξ)−2. Note that when|σ| = 0, then φ is just a gauge freedom for the frame.

Using the same notation as above:

Proposition 20. The dual basis of 1-forms is:

θ(1) = Im[

(α1∂F + α1(∂F − 2(F∂u− F ∂u)))dξ − α1dη]

e2u,

θ(2) = ǫ Im[

(α2∂F + α2(∂F − 2(F∂u− F ∂u)))dξ − α2dη]

e2u,

θ(3) = ǫ Im[

(α2∂F − α2∂F )dξ + α2dη]

e2u,

θ(4) = Im[

(α1∂F − α1∂F )dξ + α1dη]

e2u.


4.3. The second fundamental form. Now consider the Levi-Civita connection∇ associated with G and for X,Y ∈ TΣ we have the orthogonal splitting

∇XY = ∇XY +H(X,Y ),

where H : TΣ×TΣ → NΣ is the second fundamental form of the immersed surfaceΣ.

Proposition 21. The second fundamental form is:

H(e(a), e(b)) = 2Re

[

βab

(

∂

∂ξ+ (∂F − 2(F∂u− F ∂u))

∂

∂η− ∂F

∂

∂η

)]

,

where

β11 =(

iλ∂|σ| − σ∂|σ| + iλ∂λ− σ∂λ+ |σ|(|σ| + λ)(∂φ− ieiφ∂φ+ 2i∂u− 2eiφ∂u))

/(

2e2u+iφ(|σ| + λ)2(−|σ| + λ))

,

β22 =(

−iλ∂|σ| + σ∂|σ| + iλ∂λ− σ∂λ+ |σ|(|σ| − λ)(∂φ+ ieiφ∂φ+ 2i∂u+ 2eiφ∂u))

/(

2ǫe2u+iφ(|σ| − λ)2(−|σ| − λ))

,

β12 =(

−|σ|∂|σ| + iλeiφ∂|σ| + λ∂λ− iσ∂λ)

/(

2e2u+iφ(|σ|2 − λ2)√

ǫ(λ2 − |σ|2))

.

Proof. Consider the parallel and perpendicular projection operators ‖P : TTS2 →TΣ and ⊥P : TTS2 → NΣ. These are given in terms of an adapted frame by

‖P kj = δk

j − ek(3)θ

(3)j − ek

(4)θ(4)j

⊥P kj = δk

j − ek(1)θ

(1)j − ek

(2)θ(2)j .

The parallel projection operator has the following coordinate description:

‖P ξη = − 1

2∆ σ‖P ξ

ξ= − 1

2∆ (∂F + λi)σ,

‖P ξη = − 1

2∆λi‖P ξ

ξ = 12∆ [(∂F − 2λi)λi− |σ|2],

‖P ηη = 1

2∆ σ(∂F − λi) ‖P η

ξ= 1

2∆ [−σ[∂F ∂F − |σ|2 − λi(∂F − ∂F )] + 2λ2],

‖P ηη = − 1

2∆ [λi∂F + |σ|2] ‖P ηξ = 1

2∆λi[(∂F − 2λi)∂F − |σ|2].

where ∆ = λ2 − |σ|2, while the perpendicular projection operator is

⊥P ξξ = ‖P η

η⊥P ξ

ξ= − ‖P ξ

ξ⊥P ξ

η = − ‖P ξη

⊥P ξη = − ‖P ξ

η ,

⊥P ηξ = − ‖P η

ξ⊥P η

ξ= − ‖P η

ξ⊥P η

η = ‖P ξξ

⊥P ηη = − ‖P η

η .

In terms of an adapted frame the second fundamental form

H j(ab) = ⊥P j

k el(a)∇l e

k(b).

The result follows by computation.


Proposition 22. The mean curvature vector of the surface Σ is:

H = 2Re

[

γ

(

∂

∂ξ+ (∂F − 2(F∂u− F ∂u))

∂

∂η− ∂F

∂

∂η

)]

,

where

γ =(

−λ(−iλ∂|σ| + σ∂|σ|) − |σ|(iλ∂λ − σ∂λ) − |σ|(|σ|2 − λ2)(∂φ+ 2i∂u))

/(

e2u+iφ(|σ|2 − λ2)2)

.

Proof. The mean curvature vector of the surface Σ is the trace of the second fun-damental form, which is

Hj = H j(11) + ǫH j

(22) .

The result follows from computing this with the aid of the previous Proposition.

Note 4. We can also write the mean curvature vector component (see [8] for avariational derivation of this formula)

Hξ = ǫ2e−2u

√

|λ2 − |σ|2|

[

ie−2u∂

(

σe2u

√

|λ2 − |σ|2|

)

− ∂

(

λ√

|λ2 − |σ|2|

)]

.(4.4)

Corollary 1. A holomorphic graph has vanishing mean curvature.

Proof. This follows from inserting σ = 0 in equation (4.4).

4.4. Mean curvature flow. Consider P.B.V.P. of section 2.3 for discs in TS2

attached to a real surface Σ.

Proposition 23. For a graph in TS2, the mean curvature flow is

F =gjk∂j∂kF +iσ

∆

(

(σξ − ρξ)(1 + ξξ) + F − ξ2F)

=(1 + ξξ)2

2(λ2 − σσ)

(

−2σ∂λ− iσ∂σ + 2λ∂σ + iσ∂σ +4iσ(σξ + λiξ)

1 + ξξ

)

.(4.5)

Proof. Consider a surface f : Σ×[0, s0) → M such that fs(ξ, ξ) = (ξ, ξ, Fs(ξ, ξ), Fs(ξ, ξ)).Then

∂f

∂s=∂F

∂s

∂

∂η+∂F

∂s

∂

∂η.

Projecting normal to Σ

∂f

∂s

⊥

=(⊥P ξη F + ⊥P ξ

η˙F )∂

∂ξ+ (⊥P η

η F + ⊥P ηη

˙F )∂

∂η

+ (⊥P ξη

˙F + ⊥P ξη F )

∂

∂ξ+ (⊥P η

η˙F + ⊥P η

η F )∂

∂η,

and so the mean curvature flow is⊥P ξ

η F + ⊥P ξη

˙F = Hξ,

or from the expressions of the projection operators given in the proof of Proposition21

λi

2∆F − σ

2∆˙F = Hξ.


Combining this with its complex conjugate we get

F = −2λiHξ + 2σH ξ. (4.6)

Using the expression (4.4) for the mean curvature we get that

Hξ =(1 + ξξ)2

4∆2

[

2

(

i∂σ − ∂λ− 2iξσ

1 + ξξ

)

∆

− 2iλσ∂λ+ iσσ∂σ + σ2∂σ + 2λ2∂λ− λσ∂σ − λσ∂σ]

,

and the second equality stated in the Proposition follows from inserting this inequation (4.6).

To see that the first equality in the Proposition holds, we compute

gjk∂j∂kF =(1 + ξξ)2

2∆

(

iσ∂2F − 2λ∂∂F − iσ∂2F)

=(1 + ξξ)2

2∆

[

iσ∂

(

θ + iλ+2ξF

1 + ξξ

)

+ 2λ∂σ + iσ∂σ

]

=(1 + ξξ)2

2∆

[

−2σ∂λ− iσ∂σ + iσ∂σ + 2λ∂σ

+iσ

(

2(σξ + ρξ)

1 + ξξ− 2(F − ξ2F )

(1 + ξξ)2

)]

,

where we have used identity (4.3) in the more convenient form

∂θ = i∂λ− (1 + ξξ)2∂

(

σ

(1 + ξξ)2

)

− 2F

(1 + ξξ)2.

Thus

gjk∂j∂kF +iσ

∆

(

(σξ − ρξ)(1 + ξξ) + F − ξ2F)

=(1 + ξξ)2

2(λ2 − σσ)

(

−2σ∂λ− iσ∂σ + iσ∂σ + 2λ∂σ +4iσ(σξ + λiξ)

1 + ξξ

)

.

as claimed.

Proposition 24. Under the mean curvature flow the shear evolves by:

∂σ

∂t= gjk∂j∂kσ +

H1(1 + ξξ)2 + 2H2(1 + ξξ) + 2H3

2∆2,

where

H1 = − 4λσ∂λ∂λ− 2iλσ∂λ∂σ + 2(λ2 + y)∂λ∂σ + 2iλσ∂λ∂σ + 2λ2∂λ∂σ

+ 2σ2∂λ∂σ + iσ2 (∂σ)2 − 2λσ∂σ∂σ − 2λσ∂σ∂σ − iσ2 (∂σ)2 ,

H2 = −2σ∂λ(

2iλσξ + (λ2 + σσ)ξ)

+ 2σ∂λ(

λ2 − σσ)

ξ + ∂σ(

iσξ(3λ2 − σσ) + 2λ3ξ)

+ ∂σ(

iσξ − 2λξ)

(λ2 − σσ) + 2∂σ(

iσ2σξ + λσ2ξ)

,

and

H3 = −σ(

iσξ2 − 3iσξ2 − 4λξξ)

(λ2 − σσ).

In addition,

∂ρ

∂t= gjk∂j∂kρ+

H4(1 + ξξ)3 +H5(1 + ξξ)2 +H6(1 + ξξ) +H7

2(1 + ξξ)∆2,


where

H4 = 4λσ (∂λ)2 − 2σ2∂λ∂σ + 2iλσ∂λ∂σ − 4λ2∂λ∂σ − 2σσ∂λ∂σ − 2iλσ∂λ∂σ

− iσ2∂σ∂σ + 2λσ∂σ∂σ + iλ2∂σ∂σ − iλ2∂σ∂σ + 2λσ (∂σ)2

+ iσ2∂σ∂σ,

H5 = −4∂λ(

2iλσσξ − λ2σξ − σσ2ξ)

+ 2∂σ(

iλ2σξ + iσσ2ξ − 2λσ2ξ)

+ 2(

2∂σiλ2σξ − 2∂σλσσξ + ∂σiλ2σξ − ∂σiσ2σξ)

,

H6 = −4(i(λ2 − σσ) + λθ)(λ2 − σσ),

and

H7 = 4i(Fσξ − F σξ)(λ2 − σσ).

Proof. The proofs of these statements follow from differentiation of the flow equa-tion (4.5). We illustrate this for the flow of σ, leaving the flow of ρ to the reader.

We start by splitting the expression into convenient terms:

− ˙σ = ∂F = E1 + E2 + E3 + E4,

where E1 is second order in the derivatives of λ and σ, E2 and E3 are the quadraticand linear first order terms, and E4 is the zeroth order terms. We now computeeach of these terms in turn.

So, differentiating equation (4.5) we have

E1 =(1 + ξξ)2

2∆

(

−2σ∂∂λ− iσ∂∂σ + 2λ∂∂σ + iσ∂∂σ)

.

At this point we exploit the 3-jet identity (4.3) which we write in the morefavorable form

∂∂λ = Im

[

∂∂σ − 2ξ

1 + ξξ∂σ +

2ξ2

(1 + ξξ)2σ

]

− 2λ

(1 + ξξ)2.

Inserting this in the expression for E1 yields

E1 =(1 + ξξ)2

2∆

(

−iσ∂∂σ + 2λ∂∂σ + iσ∂∂σ − 2iσ(ξ∂σ − ξ∂σ)

1 + ξξ+

2iσ(ξ2σ − ξ2σ − 2iλ)

(1 + ξξ)2

)

.

The first three terms of this, noting the expression for g−1 in Proposition 17, areeasily seen to be the rough Laplacian of −σ:

E1 = −gjk∂j∂kσ +iσ

∆

(

(ξ∂σ − ξ∂σ)(1 + ξξ) + ξ2σ − ξ2σ − 2iλ)

.

We note that in the final sets of expressions, the lower order terms introduced intoE1 by the 3-jet identity will have to be added to E3 and E4.


Moving to the quadratic first order term we compute from differentiating equa-tion (4.5) that

E2 =(1 + ξξ)2

2∆

(

− 2∂σ∂λ− i∂σ∂σ + 2∂λ∂σ + i∂σ∂σ

− 1

∆∂∆(−2σ∂λ− iσ∂σ + 2λ∂σ + iσ∂σ)

)

=(1 + ξξ)2

2∆2

(

4λσ∂λ∂λ+ 2iλσ∂λ∂σ − 2(λ2 + σσ)∂λ∂σ − 2iλσ∂λ∂σ

− 2λ2∂σ∂λ+ 2λσ∂σ∂σ + iσ2(∂σ)2 − iσ2(∂σ)2

− 2σ2∂σ∂λ+ 2λσ∂σ∂σ)

= − (1 + ξξ)2

2∆2H1.

This establishes the quadratic first order term, once we recall that ∂F = − ˙σ.Moving to the linear first order term

E3 =(1 + ξξ)ξ

∆

(

−2σ∂λ− iσ∂σ + 2λ∂σ + iσ∂σ)

+1 + ξξ

∆

(

2i(σξ + iλξ)∂σ + 2iσ(ξ∂σ + iξ∂λ))

− 1 + ξξ

∆2

[

2iσ(σξ + iλξ)(2λ∂ − σ∂σ − σ∂σ)]

=1 + ξξ

∆2

− 2ξσ(λ2 − σσ)∂λ+ [2σ2(iσξ − λξ) + iξσ(λ2 − σσ)]∂σ

+ 2ξλ(λ2 − σσ)∂σ + [iξσ(3λ2 − σσ) − 2ξλ3]∂σ

+ 2[ξσ(λ2 + σσ) − 2iξλσσ]∂λ

.

Adding the linear term from E1 we compute that

−H3 =∆2

1 + ξξE3 + iξσ(λ2 − σσ)∂σ − iξσ(λ2 − σσ)∂σ

=1 + ξξ

∆2

− 2ξσ(λ2 − σσ)∂λ+ 2σ2(iσξ − λξ)∂σ

+ (λ2 − σσ)(2ξλ + iξσ)∂σ + [iξσ(3λ2 − σσ) − 2ξλ3]∂σ

+ 2[ξσ(λ2 + σσ) − 2iξλσσ]∂λ

,

as claimed in the Proposition.Finally, we work out the zero order term by looking again at the derivative of

equation (4.5) :

E4 =2iσ

∆

(

σξ2 + iλ(1 + 2ξξ))

,

and taking into account the zero order term of E1, we have

−H4 = ∆2 E4 + iσ(ξ2σ − ξ2σ − 2iλ)∆ = σ(3iξ2σ − iξ2σ − 4ξξλ)∆,

as claimed.


For later use, introduce the flat metric

< dξ, dξ >= 1 < dξ, dξ >=< dξ, dξ >= 0,

on Σ via its coordinate ξ. If we denote the flat norm and inner product by |.| and< ·, · >, and the norm and inner product of g by ‖.‖ and << ·, · >>, we have thefollowing estimates which will prove useful:

Lemma 2.

(1 + ξξ)2(−λ− |σ|)λ2 − |σ|2 |X |2 ≤ ‖X‖2 ≤ (1 + ξξ)2(−λ+ |σ|)

λ2 − |σ|2 |X |2.

We now prove that the flow remains in a compact set.

Proposition 25. For the mean curvature flow in TS2 there exists positive constantsC1 and C2 depending only on the size of the initial disc and F0 such that

FF + C1ξξ < C2,

for all time.

Proof. A straightforward computation shows that(

d

ds−0

)

ξξ =λ(1 + ξξ)2

λ2 − σσ,

and(

d

ds−0

)

FF = −2gjk∂jF∂kF

+[F (iσσξ − (iσθ + λσ)ξ) − F (iσσξ − (iσθ − λσ)ξ)](1 + ξξ)

λ2 − σσ

− F 2iσ − FF iσξ2 + FF iσξ2 − F 2iσ

λ2 − σσ,

where we introduce 0 for the rough Laplacian:

0 = gjk∂j∂k.

Let

I1 =[F (iσσξ − (iσθ + λσ)ξ) − F (iσσξ − (iσθ − λσ)ξ)](1 + ξξ)

λ2 − σσ.

Define the function D : C2 → R by D(η, ξ) = ηη + C1ξξ for some positiveconstant C1 to be determined later. We consider the pullback of this function tothe flowing disc. At a turning point p of D we have

∂D = F∂F + F ∂F + C1ξ = F ∂F + F ∂F + C1ξ = 0,

which can be written

F θ = Fσ − iFλ− C1ξ −2ξF F

1 + ξξ,

Fθ = F σ + iFλ− C1ξ −2ξF F

1 + ξξ.


Substituting these in the definition of I1 we find that, at a turning point,

I1 =2[F σ(iσξ − λξ) − Fσ(iσξ + λξ)](1 + ξξ)

λ2 − σσ

+i(C1(1 + ξξ) + 2FF )(σξ2 − σξ2)

λ2 − σσ.

Moreover, at p

F dF = F ∂Fdξ + F ∂F dξ = (−F∂F −C1ξ)dξ + F ∂F dξ = (Fσ − C1ξ)dξ − F σdξ,

and so

<< FdF, ξdξ − ξdξ >> = − [F σ(iσξ − λξ) − Fσ(iσξ + λξ)](1 + ξξ)2

2(λ2 − σσ)

− C1(1 + ξξ)2(iσξ2 + λξξ)

2(λ2 − σσ).

We conclude that, at p,

I1 = − 2

1 + ξξ<< FdF, ξdξ − ξdξ >> +

2

1 + ξξ<< FdF , ξdξ − ξdξ >>

+2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ+

2iF F (σξ2 − σξ2)

λ2 − σσ.

Now combining these, we find that at a turning point(

d

ds−0

)

FF = −2 << dF, dF >> −2 << FdF,ξdξ − ξdξ

1 + ξξ>>

+ 2 << FdF ,ξdξ − ξdξ

1 + ξξ>> +

2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ

− F 2iσ + FF iσξ2 − FF iσξ2 − F 2iσ

λ2 − σσ

= −2 << dF − Fξdξ − ξdξ

1 + ξξ, dF + F

ξdξ − ξdξ

1 + ξξ>>

− 2FF

(1 + ξξ)2<< ξdξ − ξdξ, ξdξ − ξdξ >>

+2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ

− F 2iσ + FF iσξ2 − FF iσξ2 − F 2iσ

λ2 − σσ.

However

<< ξdξ − ξdξ, ξdξ − ξdξ >>=(1 + ξξ)2

2(λ2 − σσ)(iσξ2 + 2λξξ − iσξ2),

and so(

d

ds−0

)

FF = −2 << dF − Fξdξ − ξdξ

1 + ξξ, dF + F

ξdξ − ξdξ

1 + ξξ>>

+2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ− F 2iσ − F 2iσ

λ2 − σσ

− 2λξξF F

λ2 − σσ.


From the negativity of the first term on the right hand side of the last equation wehave that(

d

ds−0

)

(FF + C1ξξ) ≤2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ− F 2iσ − F 2iσ

λ2 − σσ

− 2λξξF F

λ2 − σσ+C1λ(1 + ξξ)2

λ2 − σσ,

and we now estimate each of these terms.First, comparing with the flat metric

2C1(1 + ξξ)(iσξ2 + λξξ − iσξ2)

λ2 − σσ=

4C1

1 + ξξ‖ξdξ + ξdξ‖2 +

6λC1(1 + ξξ)ξξ

λ2 − σσ

≤ 4C1(1 + ξξ)(−λ− |σ|)λ2 − σσ

|ξdξ + ξdξ|2

=8C1ξξ(1 + ξξ)(−λ− |σ|)

λ2 − σσ

≤ 8C1ξξ(1 + ξξ)(−λ+ |σ|)λ2 − σσ

,

and

−F2iσ − F 2iσ

λ2 − σσ− 2λξξF F

λ2 − σσ=

2

(1 + ξξ)2‖F dξ + F dξ‖2 +

2λ(1 − ξξ)FF

λ2 − σσ

≤ 4FF (−λ+ |σ|)(1 + ξξ)2

λ2 − σσ+

2λ(1 − ξξ)FF

λ2 − σσ

≤ 8FF (−λ+ |σ|)(1 + ξξ)2

λ2 − σσ.

Finally, we conclude that(

d

ds−0

)

(FF + C1ξξ) ≤8(FF + C1ξξ)(−λ+ |σ|)(1 + ξξ)2

λ2 − σσ+C1λ(1 + ξξ)2

λ2 − σσ,

which is negative for

C1 >16|Fmax|2

1 − 16R2max

,

where |Fmax|2 is the maximum modulus of F and Rmax the maximum radius ofthe initial disc (assumed to be less than 1/16).

Thus, at a maximum value the function D is decreasing and the bound follows.

4.5. The multi-time function. We now construct the multi-time function on thecompact set in which the flow stays (cf. section 3.1).

Proposition 26. Define the open ball

B = (ξ, η) ∈ C2 | C1ξξ + ηη < C2 and ξξ < 1 ,

for positive constants C1 and C2. Then the function t : B → R2 = C defined by

t = η + iC3ξ,

for any positive constant C3 satisfying C23 > C2 is a multi-time function on B.


Proof. Let ξ = Reiϑ, t1 = Re(t) and t2 = Im(t). For any α = A1 + iA2 ∈ C,consider the surface t−1(α), or equivalently, η = α− iC3ξ. On this surface

G(∇t1,∇t1) = G(∇t2,∇t2) = 12 (1 +R2)(C3R

2 − 2R(A2 cosϑ−A1 sinϑ) − C3).(4.7)

On t−1(α) ∩B we have

C1R2+(α−iC3ξ)(α+iC3ξ) = (C1+C

23 )R2−2RC3(A2 cosϑ−A1 sinϑ)+A2

1+A22 < C2.

Thus

0 >C1 + C2

3

C3R2 − 2R(A2 cosϑ−A1 sinϑ) +

A21 +A2

2 − C2

C3

> C3R2 − 2R(A2 cosϑ−A1 sinϑ) − C2

C3,

and so if C23 > C2 we conclude that on t−1(α) ∩B

C3R2 − 2R(A2 cosϑ−A1 sinϑ) − C3 < 0.

From equation (4.7) we have then that for t restricted to B

G(∇t1,∇t1) = G(∇t2,∇t2) < 0.

Thus t is a multi-time function on B.

4.6. Asymptotic holomorphicity. Consider again the mean curvature flow onTS2.

Proposition 27. For the flow there exists a positive constant C1 such that

|σ|2λ2

< C1 = supΣ0

|σ|2λ2

.

Proof. We compute the flow of this quantity to be(

d

ds−0

) |σ|2λ2

= −2λ2 − 3|σ|2λ2 − |σ|2

∥

∥

∥d(

|σ|λ

)∥

∥

∥

2

+4|σ|

λ(1 + ξξ)<< d

(

|σ|λ

)

, d(1 + ξξ) >>

− 2|σ|2λ2

∥

∥

∥dφ− 2jd(1 + ξξ)

1 + ξξ

∥

∥

∥

2

+2|σ|(1 + ξξ)

λ2 − |σ|2 < d(

|σ|λ

)

, d(1 + ξξ) >

+4|σ|2λ3

.

At a maximum of this quantity d(|σ|/λ) = 0 and thus(

d

ds−0

) |σ|2λ2

= −2|σ|2λ2

∥

∥

∥dφ− 2jd(1 + ξξ)

1 + ξξ

∥

∥

∥

2

+4|σ|2λ3

≤ 0.

Thus it is decreasing at a maximum and the result follows.


Proposition 28. The flow satisfies(

d

ds−0

) |σ|2λ2 − |σ|2 = −2

(λ2 + |σ|2)(λ2 − |σ|2)3

∥

∥

∥λd|σ| − |σ|dλ∥

∥

∥

2

− 2λ2|σ|2

(λ2 − |σ|2)2∥

∥

∥dφ∥

∥

∥

2

+4λ|σ|

(λ2 − |σ|2)3 Re [H8], (4.8)

where

H8 =(1 + ξξ)[

iλσξ∂|σ| − i|σ|σξ∂λ+ 2λ|σ|(iλξ − σξ)∂φ]

− 2iλσ|σ|ξ2 − 2|σ|3 + 2(1 + 2ξξ)λ2|σ|,and φ is the argument of σ. Here the norm ‖.‖ is taken with respect to the inducedmetric given in Proposition 17.

Proof. This follows from the fact that(

d

ds−0

) |σ|2λ2 − |σ|2 =

λ

(λ2 − |σ|2)2[

λσ

(

d

ds−0

)

σ + λσ

(

d

ds−0

)

σ − 2|σ|2(

d

ds−0

)

λ

]

− 2|σ|2(3λ2 + |σ|2)(λ− |σ|2)3 ‖dλ‖2 − 2λ2(λ2 + |σ|2)

(λ− |σ|2)3 << dσ, dσ >>

+4λ(λ2 + |σ|2)(λ− |σ|2)3 << σdσ + σdσ, dλ >> − 2λ2σ2

(λ− |σ|2)2 ‖dσ‖2

− 2λ2σ2

(λ− |σ|2)2 ‖dσ‖2,

and the flow equations given in Proposition 24.

We now prove:

Proposition 29. By choosing a small enough initial disc, under the mean curva-ture flow we have the following estimate:

(

d

ds−0

) |σ|2λ2 − |σ|2 ≤ 4λ

(λ2 + |σ|2)|σ|4

(λ2 − |σ|2)2 ,

Proof. Our starting point is equation (4.8), which we rewrite in the form(

d

ds−0

) |σ|2λ2 − |σ|2 = I1 + I2 + I3 + I4,

where

I1 = −2(λ2 + |σ|2)(λ2 − |σ|2)3

∥

∥

∥λd|σ| − |σ|dλ − λ2 − |σ|2λ2 + |σ|2

λ|σ|1 + ξξ

d(1 + ξξ)∥

∥

∥

2

,

I2 = 2(1 + ξξ)λ2|σ|(λ2 − |σ|2)3 < d(1 + ξξ), λd|σ| − |σ|dλ >,

I3 = −2λ2|σ|2

(λ2 − |σ|2)3∥

∥

∥dφ− 2(1 + ξξ)−1j[d(1 + ξξ)]∥

∥

∥

2

,

I4 =|σ|2

2(λ2 − |σ|2)2(λ2 + |σ|2)

− i|σ|λ2(ξ2eiφ − ξ2e−iφ) + 4λ[2 − ξξ]λ2 + 2|σ|2

.

Here σ = |σ|eiφ and we have introduced the flat complex structure j(dξ) = idξ.


First we estimate I1 using the flat metric and Lemma 2:

I1 ≤ −2(1 + ξξ)2(λ2 + |σ|2)

(λ2 − |σ|2)3(−λ+ |σ|)∣

∣

∣λd|σ| − |σ|dλ− λ2 − |σ|2λ2 + |σ|2

λ|σ|1 + ξξ

d(1 + ξξ)∣

∣

∣

2

,

and so after completing the squares

I1 + I2 ≤ −2 (1+ξξ)2(λ2+|σ|2)(λ2−|σ|2)3(−λ+|σ|)

∣

∣

∣λd|σ| − |σ|dλ + (−λ+|σ|)(−λ+2|σ|)λ|σ|

2(λ2+|σ|2)(1+ξξ)d(1 + ξξ)

∣

∣

∣

2

− λ(−λ+|σ|)2(−3λ−4|σ|)4(λ2+|σ|2)2

λ2|σ|2

(1+ξξ)2

∣

∣

∣d(1 + ξξ)∣

∣

∣

2

.

Clearly I3 is negative, so we discard it. To estimate I4 we use

−2ξξ ≤ i(ξ2eiφ − ξ2e−iφ) ≤ 2ξξ.

Thus

I4 ≤ |σ|2

2(λ2−|σ|2)2(λ2+|σ|2)

2|σ|λ2ξξ + 2λ[4 − 2ξξ]λ2 + 4|σ|2

.

Combining the estimates(

d

ds−0

) |σ|2λ2 − |σ|2 ≤ 4λ|σ|2

(λ2 − |σ|2)2 +λ2|σ|2(λ3 + 2λ2|σ| − 2λ|σ|2 − |σ|3)ξξ

(λ2 − |σ|2)3(λ2 + |σ|2) .

By choosing a small enough disc we can achieve −λ ≥ 3|σ| throughout the flow (cf.Proposition 27), so that

(

d

ds−0

) |σ|2λ2 − |σ|2 ≤ 4λ|σ|2

(λ2 − |σ|2)2 +λ5|σ|2ξξ

9(λ2 − |σ|2)3(λ2 + |σ|2)

≤ 4λ

λ2 + |σ|2|σ|4

(λ2 − |σ|2)2 .

This completes the proof.

Proposition 30. The timelike curvature condition holds during the flow.

Proof. Given a P , chose an adapted orthonormal frame (e(i) = τi, e(3), e(4)) and for

a timelike vector X = X(3)e(3) +X(4)e(4) compute

< R(X, τi)X, τi >=2|σ|

λ2 − |σ|2 (−(X(3))2 + (X(3))2) ≥ 2|σ|λ2 − |σ|2 |X |2.

From the previous Proposition we have that(

d

ds−0

) |σ|2λ2 − |σ|2 ≤ 0,

and so we have the a priori bound

|σ|2λ2 − |σ|2 ≤ C1.

In addition, since the area form increases point wise under the flow

λ2 − |σ|2 ≥ C2,

and therefore( |σ|λ2 − |σ|2

)2

=|σ|2

λ2 − |σ|21

λ2 − |σ|2 ≤ C1

C2.


ThusG(R(X, τi)X, τi) ≥ k|X |2,

as claimed.

Proposition 31. The following estimates hold

|λ| < C10 |σ| < C11 |θ| < C12,

for positive constants C10, C11, C12, depending on n,m, ‖t‖3, |ψ|, ‖R‖, f0.Proof. This follows from the main gradient estimate for v, established in Proposi-tion 12, which we can now apply since we know that the timelike curvature con-dition holds. For completeness, we prove this by obtaining the explicit expressionfor v with respect to the background frame e(1), e(2), T(1), T(2) associated to themulti-time function t defined in Proposition 26.

We find that (cf. Section 3.2):

ψ−2 ≡ ψ−21 = ψ−2

2 = 12 (1 +R2)(C3 + 2R(A2 cos θ −A1 sin θ) − C3R

2),

and the background orthonormal frame is

e(1) = 12 (1 +R2)2ψ Re

[

i∂

∂ξ+ C3

∂

∂η

]

,

e(2) = 12 (1 +R2)2ψ Re

[

∂

∂ξ− iC3

∂

∂η

]

,

T(1) = 12ψ Re

[

−i(1 +R2)2∂

∂ξ+ (4ψ−2 − C3(1 +R2)2)

∂

∂η

]

,

T(2) = 12ψ Re

[

(1 +R2)2∂

∂ξ+ i(4ψ−2 − C3(1 +R2)2)

∂

∂η

]

.

Now, consider a frame τ1, τ2, ν1, ν2 adapted to the flowing surface (cf. Defini-tion 6). As the surface is a graph, we use the expressions for the frame supplied byProposition 19 with τ1 = e(1), τ2 = e(2), ν1 = e(3), ν2 = e(4). The transformationmatrix between the timelike vectors of the background and adapted frame is (cf.section 3.3):

Vαβ = −G(να, Tβ),

andv2 = V 2

33 + V 234 + V 2

43 + V 244.

Thus an a priori bound on v implies an a priori bound on each of the componentsof the matrix Vαβ .

Suppose for the sake of contradiction that at some point and time λ goes toinfinity. By a translation we can assume that this occurs at (ξ, η) = (0, 0). At theorigin we have the following expressions for the transition matrix:

V33 + iV34 =(1 + i)(|σ| + λ− C3) − (1 − i)θ

[2C3(−|σ| − λ)]1

2

e−iφ/2,

V44 + iV43 =(1 − i)(−|σ| + λ− C3) − (1 + i)θ

[2C3(|σ| − λ)]1

2

eiφ/2.

Thus, if λ goes to infinity, then so does v, which contradicts Proposition 12. Simi-larly for the other slopes |σ| and θ.


Theorem 4. By choosing a small enough initial disc, under the mean curvatureflow the disc becomes asymptotically holomorphic:

lims→∞

|σ| = 0.

Proof. By the previous propositions, our general results on mean curvature flowcan be applied, in particular we can apply Proposition 31.

Then by Propositions 29 and 31 we have(

d

ds−0

) |σ|2λ2 − |σ|2 ≤ 4λ

(λ2 + |σ|2)|σ|4

(λ2 − |σ|2)2 ≤ −C |σ|4(λ2 − |σ|2)2 ,

and we can again apply Lemma 4.5 of [4] to get the result.

4.7. Boundary and initial conditions. In our case we would like the boundarydisc to be the Lagrangian disc Σ with the isolated complex point, but, as themetric will be Lorentz or degenerate on such a disc (see Proposition 18), it will notbe spacelike and cannot be used as a boundary condition. Instead we perturb thedisc around the complex point to make it positive, and attach the flow disc to thissurface.

More specifically, suppose Σ is given by η = F (ξ, ξ) and the point γ lies at the

origin ξ = 0. Define the perturbed surface Σ by “adding a linear holomorphictwist”:

η = F = F − iC0ξ, (4.9)

where C0 is a real positive constant.

Proposition 32. The induced metric on the surface Σ is positive definite in aneighbourhood of the origin. The origin is an isolated complex point on Σ with thesame index as that of the complex point on Σ.

Proof. The fact that Σ has an isolated complex point of the same index at theorigin follows from the fact that the perturbation is holomorphic, and hence σ = σ.

In addition, computing the twist

λ = Im (1 + ξξ)2∂

(

F

(1 + ξξ)2

)

= Im (1 + ξξ)2∂

(

F − iC0ξ

(1 + ξξ)2

)

= −C01 − ξξ

1 + ξξ,

where we have used the fact that Σ is Lagrangian. Thus, at the origin λ2(0) −|σ(0)|2 = C2

0 > 0 and so in a small enough neighbourhood, the metric is positivedefinite.

Note 5. In order for the induced metric to be positive (rather than negative)

definite we have arranged that λ < 0.

We now prove that:

Proposition 33. The boundary condition −ǫλ + |σ| = |σ| in P.B.V.P. is anindependent Neumann condition.


Proof. By Proposition 15, this follows if the Neumann condition is linearly inde-pendent of the derived Dirichlet condition. To see that this is the case, considerthe condition that two graphs f, f : D → TS2 intersect on a curve. If the graphfunctions are F and F , then the tangent spaces are spanned by

TγΣ = Spanα∈C Re α

(

∂

∂ξ+ ∂F

∂

∂η+ ∂F

∂

∂η

)

,

TγΣ = Spanα∈C Re α

(

∂

∂ξ+ ∂F

∂

∂η+ ∂ ¯F

∂

∂η

)

.

If these tangent spaces intersect in a one dimensional set, then there exists α andα such that

Re α

(

∂

∂ξ+ ∂F

∂

∂η+ ∂F

∂

∂η

)

= Re α

(

∂

∂ξ+ ∂F

∂

∂η+ ∂ ¯F

∂

∂η

)

.

Comparing components we find that α = α and

α∂F + α∂F = α∂F + α∂F ,

or

α∂(F − F ) + α∂(F − F ) = 0.

Thus, a necessary and sufficient condition for the tangent planes to intersect on aline is that |∂(F − F )| = |∂(F − F )|, or, in our notation, |ρ− ρ| = |σ − σ|. This isclearly linearly independent of the Neumann condition −ǫλ+ |σ| = |σ|.

Proposition 34. The boundary of the flowing disc never crosses the complex pointon the boundary surface Σ.

Proof. This follows from the boundary condition −ǫλ + |σ| = |σ|, since if at somes = s1 the boundary of the flowing disc reaches ξ = 0, then −ǫλ(s1, 0)+ |σ(s1, 0)| =|σ(0)| = 0. Thus λ(s1, 0) = 0 and the disc is no longer positive definite whichcontradicts long-time existence for the flow established earlier.

We turn now to initial conditions.

Proposition 35. For ǫ small enough, there exists an initial positive disc such thatf0(∂D) ⊂ Σ is a simple closed curve containing the complex point on Σ and which iscompatible with the given boundary condition −ǫλ+ |σ| = |σ|. Moreover, the initial

disc can be chosen arbitrarily C1-close to the boundary disc Σ, and π(f0(∂D)) iscontained in the a disc of radius less than 1/4.

Proof. Given a surface Σ with boundary ∂Σ, consider the set Uǫ1(Σ) of C1 pertur-

bations Σ of Σ that fix ∂Σ and

‖Σ − Σ‖C1< ǫ1.

For p ∈ ∂Σ, define the set of graphical planes on the boundary

Pp = P a linear plane in TpΣ | π|P has rank 2 ,and

P =⋃

p∈∂Σ

Pp.


This has a bundle structure P → ∂Σ and T∂ΣΣ is a section of this bundle whichwe denote by s.

Now, there exists δ > 0 such that if s is a section of P with

‖s− s‖C1< δ,

then there exists Σ ∈ Uδ(Σ) with s = T∂ΣΣ.Our boundary condition −ǫλ + |σ| = |σ| is satisfied at ǫ = 0 if we take the

boundary and initial disc to be the same. Thus, by the preceding result, thereexists an ǫ > 0 such that we can perturb Σ to a nearby disc satisfying the boundarycondition. Moreover, since positivity is an open condition, we can choose ǫ smallenough so that the perturbed disc is also positive.

5. Convergence of the Evolution to a J-Holomorphic Disc

5.1. Compactness of the manifold of J-holomorphic discs with boundary.

Let (M,G) be a 4-manifold with neutral metric and π2(M) = 0. Fix a Ck,α surface

Σ → M and let G be any fixed Riemannian metric on M.Define

J = J | J is a Ck,α almost complex structure on M s.t. Σ is totally real ,and

MΣ = (f, J) | (f, Σ) ∈ F , J ∈ J , and ∂Jf = 0,where F is defined as in section 2.2.

Definition 11. A sequence of immersed discs fn(D) ⊂ M is said to be uniformlypositive if ∃C2 > 0 such that

G(Xn, Xn) > C2 G(Xn, Xn) ∀Xn ∈ Tfn(D).

Note that this property is independent of the Riemannian metric G (with differentC2).

Definition 12. C ⊂ D is called a regular set of curves in D if C consists ofck : [0, 1] → D, a finite number of disjoint embedded smooth curves with boundaryin ∂D.

We now state a compactness theorem for the space of J-holomorphic discs withboundary in a totally real surface.

Theorem 5. Let (fn, Jn) ∈ MΣ be a sequence such that

(1) fn(D) lies in a fixed compact set of M for all n,(2) Jn → J uniformly in Ck,α,(3) fn(D) is uniformly positive,(4) there exists a constant C1 such that, for all n,

Area G(fn(D)) < C1.

Then there exists a subsequence n′, again denoted by n, and

(i) diffeomorphisms φn ∈ C∞(D,D),(ii) complex structures jn on D,


(iii) a regular set of curves C ⊂ D,(iv) a complex structure j∞ on D − C,

(v) a limit map f∞ ∈ C0((D, ∂D), (M, Σ)) ∩ Ck+1,α(D − C,M),

such that

(a) φn(jn) → j∞ uniformly in C∞ on compact subsets of D − C,(b) f∞|D−C = j∞-J−holomorphic,(c) the restriction of f∞ to any component of C is constant,

(d) fnφn → f∞ uniformly in C0 on the closure of D and uniformly in Ck+1,α′

for each α′ ∈ (0, α) on compact subsets of the closure of D − C,

(e) The homotopy classes of fn(∂D) and f∞(∂D) coincide in π1(Σ).

This result follows from the main theorem of [5] once the following is noted. Bythe uniform positivity of fn(D) and condition (4) we have that

C2 Area G(fn(D)) < Area G(fn(D)) < C1,

and so we have an a priori area bound for the area of the discs fn(D) with respectto the Riemannian metric G. This is all that is required to apply the referencedtheorem.

The map f∞ is called a J-holomorphic bubble-disc, as the interior of f∞(D)consists of k + 1 discs whose boundaries are joined by the k points f∞(C). Thecondition that π2(M) = 0 ensures that no holomorphic spheres bubble off.

5.2. Convergence of the flow to a J-holomorphic bubble-disc in TS2. Themain ingredient in convergence of the evolution is an a priori area bound withrespect to the indefinite metric:

Proposition 36. Let f : D → TS2 be a positive disc with boundary lying in Σ, alinear perturbation of a Lagrangian surface, as given in section 4.7. Then the areaof f(D) has the a priori bound

AreaG(f(D)) ≤ πC0.

Proof. Recall the expression for the area form of the induced metric

AreaG(f(D)) =

∫∫

D

√

λ2 − σσdξdξ

(1 + ξξ)2,

where

λ = Im (1 + ξξ)2∂

(

F

(1 + ξξ)2

)

σ = −∂F .

Thus

AreaG(f(D)) ≤∫∫

D

|λ| dξdξ

(1 + ξξ)2= −

∫∫

Ds

Ω,

where Ω is the symplectic form on TS2. Since this 2-form is closed and f(∂D) ⊂ Σwe have

∫∫

D

Ω −∫∫

Σ′

0

Ω = 0,

where Σ′0 is the disc in Σ which is contained within the boundary of f(D). In fact,

the symplectic form is exact

Ω = dΘ where Θ =2(ηdξ + ηdξ)

(1 + ξξ)2,


and so

AreaG(f(D)) ≤ −∫

∂Σ′

0

Θ.

But a straight-forward computation shows that, when Σ is a linear holomorphicdeformation of a Lagrangian disc, in terms of polar coordinates (R, ϑ), ξ = Reiϑ:

∫

∂Σ′

0

Θ = −∫

∂Σ′

0

2C0R2

(1 +R2)2dϑ.

Thus the area can be estimated by the winding number of the boundary about theorigin

AreaG(f(D)) ≤∫

∂Σ′

0

4C0R2

(1 +R2)2dϑ ≤

(

Max2C0R

2

(1 +R2)2− Min

2C0R2

(1 +R2)2

)∫

∂Σ′

0

dϑ ≤ πC0.

We now apply the above and Theorem 5 to the case of TS2:

Theorem 6. Let Σ be a linear holomorphic perturbation of a Lagrangian surfaceΣ, as in equation (4.9). Let Σ0 be an initial disc of radius R0, as established inProposition 35.

Then for C0 big enough and R0 small enough, the mean curvature flow P.B.V.P.

converges to f∞(D), a J-holomorphic bubble-disc, with continuous boundary enclos-

ing the isolated complex point on Σ. In particular, the Keller-Maslov index µ(f∞, Σ)is well defined.

Proof. In section 4 we established long-time existence of a solution ft of the P.B.V.P.

under the above assumptions. Moreover, we established that the flow is asymptot-ically holomorphic in Theorem 4.

Let fn be a sequence of immersions extracted from the evolving immersion ft.Then by Proposition 25, fn(D) lies in a fixed compact set of M for all n. For n largeenough, we can chose a sequence of almost complex structures Jn, converging inCk,α to J, such that fn is Jn-holomorphic. By our gradient estimates in Proposition27 the sequence fn(D) is uniformly positive and in the previous Proposition we haveestablished an a priori area bound for fn(D).

Since we are working in an open subset U of TS2, and π2(U) = 0. We havethus met all of the conditions of Theorem 5, which we can therefore apply to thesequence. The result then follows from properties (a)-(e) of this Theorem.

6. Concluding Remarks

6.1. Background. The affirmation of such a venerable conjecture is deserving ofsome general remarks on the methods employed. Previous efforts at proving theCaratheodory conjecture have also been focused on the Loewner conjecture, al-though these efforts have almost exclusively assumed that the surface is real ana-lytic. This has involved the use of ODE analysis, dynamics of the principal cur-vature foliation and the combinatorics of its singularities. Rather than review thehistory of these attempts, which in any case are well-documented, we will discussour proof of the C3 case in the context of PDE methods.


Our proof brings together two currents in contemporary PDE: parabolic flowsand the rigidity of J-holomorphic curves, both in a new geometric setting. Thatthese techniques are just sufficient to prove the Caratheodory conjecture gives an in-dication of the depth at which it lies. The geometric setting (that of neutral Kahlersurfaces) is sufficiently new that results from both of these fields require modifica-tion. In what follows we sketch the salient features of the proof, highlighting thekey points and offering an enlarged perspective on the work.

The basic outline of the proof is as follows. The Loewner conjecture is essentiallya hyperbolic problem. By transferring to the enlarged geometric setting of the spaceof oriented lines, the neutral Kahler metric allows us to deform it to an ellipticboundary value problem. The mean curvature flow then allows us not only to solvethis problem, but to get to the most rigid of elliptic objects: holomorphic curves.The proof of the Conjecture then follows from a generalization of the argumentprincipal for holomorphic functions.

6.2. Mean curvature flow. In the case of the mean curvature flow, we mustestablish a priori gradient estimates for long-time existence in the indefinite settingwith higher codimension. In this instance the indefinite signature of the metricassists the analysis: as long as the flowing submanifold remains in a compact set, amild curvature assumption ensures that singularity formation does not occur. Thusmany of the difficulties associated with flowing in the definite case are avoided anda general result on mean curvature flow of spacelike surfaces in indefinite manifoldsis established.

However, the higher codimension significantly complicates the gradient estimatesrequired for long-time existence. In addition, rather than working with the moreusual case of compact submanifolds, we must consider the flow with mixed Dirichletand Neumann boundary conditions. These conditions must be chosen so that theboundary of the disc does not cross the complex point on the boundary surface,while still remaining parabolic.

6.3. J-holomorphic curves. For TS2 with neutral metric a holomorphic curveis maximal [8], and perhaps the most surprising aspect of this flow is that, undersuitable circumstances, we show that it converges to a holomorphic disc. To estab-lish this we first prove that the flow is asymptotically holomorphic and then utilizecompactness results for J-holomorphic discs with boundary lying on a totally realsurface. Once again, existing results must be modified, as our symplectic structureis not tamed by the complex structure, and our area bound is with respect to anon-Hermitian metric.

In this regard, the a priori area bound enjoyed by positive discs with boundarylying in the linear holomorphic perturbation of a Lagrangian surface must be seenas a key element of our proof. Without this, not only would it be hard to proveconvergence, but it is easy to construct examples where convergence fails. Indeed,this is the only point at which the Lagrangian condition is exploited.

The Keller-Maslov index estimates of the type used originate in symplectic ge-ometry and topology. It is worth noting that the Loewner conjecture is a strongerlocal statement than would be required by the Caratheodory conjecture: an um-bilic point of index 3/2 would disprove the Loewner but not the Caratheodoryconjecture. Here we have an example of the well-known phenomenon in symplecticrigidity of gaining a half-integer more than is strictly necessary globally.


6.4. Neutral Kahler surfaces. Let us now turn to the geometric setting. Themetric employed in the proof was first noted, as far as the authors are aware, byStudy [12]. As shown in [7], it admits a neutral Kahler structure and is canonicalin the sense that it is the unique metric on the space of oriented lines (up toaddition of a spherical element) that is invariant under the action of the Euclideangroup [11]. Such an invariant metric exists on the space of oriented geodesicsof any 3-dimensional space form and so our method may well extend to a proofthe Caratheodory conjecture in the 3-sphere and hyperbolic 3-space. These wouldrequire ambient curvature conditions, an a priori area bound, confinement of theflow to a compact region and asymptotic holomorphicity - issues that we postponeto a future paper.

In fact, minor modifications of our argument establish: the index of an isolatedumbilic point on a spacelike surface in Lorentz 3-space must be less than or equalto one [9]. Given our reformulation, it might also be conjectured that the index ofan isolated complex point on a Lagrangian surface in any neutral Kahler surfacemust have index less than or equal to 1.

Perhaps some insight into the difficulty of the Conjecture in Euclidean 3-space isafforded by the following observation: our Kahler metric is not Kahler-Einstein andso mean curvature flow does not preserve the Lagrangian condition. Thus, viewedin E3, our flow twists the normal lines and we lose the orthogonal surface. It mightbe that any proof of the conjecture must follow from the argument principle in somesetting, and that, in order to approach these holomorphic objects the orthogonalsurface must be destroyed in the process. Hence, any proof that remains exclusivelyin the space of surfaces in E3 will run into serious difficulties.

It is worth noting that our proof does not require arbitrarily small initial discsfor convergence. A careful study of the estimates reveals that the proof works onany disc of radius less than one quarter. This yields:

The winding number of the prinicpal foliation of a convex surface in E3 alonga closed curve contained in a disc whose Gauss image has radius 1/4 must be lessthan or equal to one.

Since these estimates are not necessarily sharp, it may be possible to improveon them and to establish convergence on discs of radius less than one. This wouldthen imply strong global rigidity results for isolated umbilic points on closed convexsurfaces in E3.

Acknowledgements: The authors would like to thank Robert Bartnik, HelgaBaum, Viktor Fromm, Karl Luttinger and Jean-Claude Sikorav for helpful discus-sions during the evolution of this work. This research was supported by the Instituteof Technology Tralee Research Investment Plan, the Research in Pairs program ofthe Mathematisches Forschungsinstitut Oberwolfach and the Special Program onGeometric Evolution Equations at the Mathematical Sciences Research Institute,Berkeley.

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[3] J. Chen and J. Li, Mean curvature flow of surface in 4-manifolds, Adv. Math. 163 (2001)287–309.

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[5] V. Fromm, Gromov compactness in Holder spaces and minimal connections on jet bundles,Preprint (2008) [math.SG/0808.0415].

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Brendan Guilfoyle, Department of Computing and Mathematics, Institute of Tech-

nology, Tralee, Clash, Tralee, Co. Kerry, Ireland.

E-mail address: [email protected]

Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham,

Durham DH1 3LE, United Kingdom

E-mail address: [email protected]

Proof of the Caratheodory Conjecture

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