On Short Time Existence of Lagrangian Mean CurvatureFlow

Tom Begley

Joint work with Kim Moore

March 15, 2016

Mean Curvature Flow

Let Mn be an n-dimensional closed smooth manifold. A mean curvatureflow in Rn+k is a one-parameter family of immersions

F : M × [0, T )→ Rn+k

satisfying (dF

dt(p, t)

)⊥= ~H(p, t).

Where ~H(p, t) is the mean curvature vector of Mt := F (M, t) at F (p, t).

This can be interpreted as a geometric heat equation

dF

dt(p, t) = ∆MtF (p, t)

Given an initial immersion F0 : M → Rn+k, there exists a mean curvatureflow with F (·, 0) = F0(·).

Mean Curvature Flow

Let Mn be an n-dimensional closed smooth manifold. A mean curvatureflow in Rn+k is a one-parameter family of immersions

F : M × [0, T )→ Rn+k

satisfying (dF

dt(p, t)

)⊥= ~H(p, t).

Where ~H(p, t) is the mean curvature vector of Mt := F (M, t) at F (p, t).This can be interpreted as a geometric heat equation

dF

dt(p, t) = ∆MtF (p, t)

Given an initial immersion F0 : M → Rn+k, there exists a mean curvatureflow with F (·, 0) = F0(·).

Mean Curvature Flow

Let Mn be an n-dimensional closed smooth manifold. A mean curvatureflow in Rn+k is a one-parameter family of immersions

F : M × [0, T )→ Rn+k

satisfying (dF

dt(p, t)

)⊥= ~H(p, t).

Where ~H(p, t) is the mean curvature vector of Mt := F (M, t) at F (p, t).This can be interpreted as a geometric heat equation

dF

dt(p, t) = ∆MtF (p, t)

Given an initial immersion F0 : M → Rn+k, there exists a mean curvatureflow with F (·, 0) = F0(·).

Mean Curvature Flow

For an evolving surface Mt, whose velocity at each point is described by avectorfield X, the first variation of area is given by

d

dtHn(Mt) = −

∫Mt

X · ~HdHn

So the mean curvature flow is like the gradient descent for area.

Of particular interest to us are self-expanders. These are submanifoldsM ⊂ Rn+k satisfying the elliptic equation

~H − x⊥ = 0.

In this case Mt =√

2tM is a solution of mean curvature flow.

Mean Curvature Flow

For an evolving surface Mt, whose velocity at each point is described by avectorfield X, the first variation of area is given by

d

dtHn(Mt) = −

∫Mt

X · ~HdHn

So the mean curvature flow is like the gradient descent for area.

Of particular interest to us are self-expanders. These are submanifoldsM ⊂ Rn+k satisfying the elliptic equation

~H − x⊥ = 0.

In this case Mt =√

2tM is a solution of mean curvature flow.

Monotonicity Formula

Define the backwards heat kernel

ρ(x0,t0)(x, t) :=1

(4π(t0 − t))n/2exp

(−|x− x0|

2

4(t0 − t)

)Then Huisken’s monotonicity formula says that for a mean curvature flowMt (with 0 ≤ t < t0)

d

dt

∫Mt

ρ(x0,t0)(x, t)dHn = −

∫Mt

∣∣∣∣ ~H − (x0 − x)⊥

2(t0 − t)

∣∣∣∣2 ρ(x0,t0)dHn ≤ 0

We define

Θ(x0, t0, r) :=

∫Mt0−r2

ρ(x0,t0)(x, t0 − r2)dHn 0 < r ≤

√t0

Θ(x0, t0) := limr0

Θ(x0, t0, r)

the Gaussian density ratios and Gaussian density respectively.

Monotonicity Formula

Define the backwards heat kernel

ρ(x0,t0)(x, t) :=1

(4π(t0 − t))n/2exp

(−|x− x0|

2

4(t0 − t)

)Then Huisken’s monotonicity formula says that for a mean curvature flowMt (with 0 ≤ t < t0)

d

dt

∫Mt

ρ(x0,t0)(x, t)dHn = −

∫Mt

∣∣∣∣ ~H − (x0 − x)⊥

2(t0 − t)

∣∣∣∣2 ρ(x0,t0)dHn ≤ 0

We define

Θ(x0, t0, r) :=

∫Mt0−r2

ρ(x0,t0)(x, t0 − r2)dHn 0 < r ≤

√t0

Θ(x0, t0) := limr0

Θ(x0, t0, r)

the Gaussian density ratios and Gaussian density respectively.

Local Regularity Theorem

Theorem (White ’05)

There are constants ε0(n, k) > 0 and C0(n, k) <∞ such that if∂Mt ∩B2r = ∅ for t ∈ [0, r2) and

Θ(x, t, ρ) ≤ 1 + ε0 x ∈ B2r(x) ρ ≤√t t ∈ (0, r2]

then

|A|(x, t) ≤ C0√t

x ∈ Br(x) t ∈ (0, r2]

White’s theorem implies a quantitative short-time existence result.

Local Regularity Theorem

Theorem (White ’05)

There are constants ε0(n, k) > 0 and C0(n, k) <∞ such that if∂Mt ∩B2r = ∅ for t ∈ [0, r2) and

Θ(x, t, ρ) ≤ 1 + ε0 x ∈ B2r(x) ρ ≤√t t ∈ (0, r2]

then

|A|(x, t) ≤ C0√t

x ∈ Br(x) t ∈ (0, r2]

White’s theorem implies a quantitative short-time existence result.

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.

We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.

For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.

A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.

Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Geometric Preliminaries

Consider Cn with standard complex coordinates zj = xj + iyj . Letω :=

∑nj=1 dxj ∧ dyj and J the standard complex structure defined

J∂

∂xj=

∂

∂yjJ∂

∂yj= − ∂

∂xj

A Lagrangian submanifold is an n-dimensional submanifold L ⊂ Cn suchthat ωL ≡ 0.We define the holomorphic volume form Ω := dz1 ∧ · · · ∧ dzn.For any Lagrangian L, one can show Ω|L = eiθLvolL, we call θL theLagrange angle. If θL is a single valued function, we say L is zero-Maslovclass.A special Lagrangian is a Lagrangian with θL constant.Special Lagrangians are minimal, due to the remarkable formula

~HL = J∇θL

Question: Given a Lagrangian, is there a special Lagrangian in itshomology class?

Problem Statement

Neves showed that if a zero-Maslov flow develops a singularity, thatsingularity will be asymptotic to a finite collection of Lagrangian planeswith multiplicities.

Can we push through these singularities in some way? We can reformulatethis as a short time existence problem.

Problem

Given a compact Lagrangian L with a finite number of singularities, eachasymptotic to a pair of transversely intersecting planes, does there exist asmooth Lagrangian mean curvature flow attaining the singular Lagrangianas its initial condition in some suitable sense?

Problem Statement

Neves showed that if a zero-Maslov flow develops a singularity, thatsingularity will be asymptotic to a finite collection of Lagrangian planeswith multiplicities.

Can we push through these singularities in some way? We can reformulatethis as a short time existence problem.

Problem

Given a compact Lagrangian L with a finite number of singularities, eachasymptotic to a pair of transversely intersecting planes, does there exist asmooth Lagrangian mean curvature flow attaining the singular Lagrangianas its initial condition in some suitable sense?

Construction of approximating initial conditions

Strategy motivated by work of Ilmanen-Neves-Schulze on short timeexistence of planar network flows.

Theorem (Lotay-Neves, Imagi-Joyce-Oliveira dos Santos)

If P1 and P2 are planes such that neither P1 + P2 or P1 − P2 arearea-minimising, then there is a unique smooth, zero-Maslov classLagrangian self-expander asymptotic to P := P1 + P2

Given L with a singularity at the origin, we glue in√

2sΣ, where Σ is theunique zero-Maslov self-expander asymptotic to P , to get Ls.Standard short-time existence implies existence of smooth flows Lstexisting up to some time Ts.Goal: show that there is δ > 0 such that Ts ≥ δ for every s.

Construction of approximating initial conditions

Strategy motivated by work of Ilmanen-Neves-Schulze on short timeexistence of planar network flows.

Theorem (Lotay-Neves, Imagi-Joyce-Oliveira dos Santos)

If P1 and P2 are planes such that neither P1 + P2 or P1 − P2 arearea-minimising, then there is a unique smooth, zero-Maslov classLagrangian self-expander asymptotic to P := P1 + P2

Given L with a singularity at the origin, we glue in√

2sΣ, where Σ is theunique zero-Maslov self-expander asymptotic to P , to get Ls.Standard short-time existence implies existence of smooth flows Lstexisting up to some time Ts.Goal: show that there is δ > 0 such that Ts ≥ δ for every s.

Stability

There are two key results that we make use of:

Theorem

For any ε, if R is large enough and η, ν are small enough then for anysmooth zero-Maslov Lagrangian L in BR satisfying

(i) |A| ≤M on L ∩BR,

(ii)∫L ρ(x,0)(y,−r

2)dHn ≤ 1 + ε0 for all x and 0 < r ≤ 1,

(iii)∫L∩BR

| ~H − x⊥|2dHn ≤ η,

(iv) the connected components of L ∩ (BR \B1) correspond to those ofP ∩ (BR \B1) and on L ∩ (BR \B1)

dist(x, P ) ≤ ν + C exp

(−|x|

2

C

);

L is ε-close in C1,α to Σ in BR.

Monotonicity

If θt represents the Lagrange angle then

d

dtθt = ∆θt

Let λ :=∑n

j=1 xjdyj − yjdxj . We say Lt is exact if there exists some βtsuch that λ|Lt = dβt. If such a β exists then

d

dtβt = ∆βt − 2θt.

Consequently αt := βt + 2tθt satisfies

d

dtαt = ∆αt,

and ∇αt = J(x⊥ − 2t ~H)

Monotonicity

Because αt solves the heat equation, we can combine this fact with themonotonicity formula to get

d

dt

∫Lt

α2tφρdHn ≤ −

∫Lt

|2t ~H − x⊥|2φρdHn + C

∫Lt∩(B3\B2)

α2t dHn

Integrating this we can show

Theorem

Let a > 1 and η > 0. There exists δ such that, if and s ≤ T ≤ δ then

1

(a− 1)T

∫ aT

T

∫Lst∩BR

| ~H − x⊥|2dHndt ≤ η.

Completing the proof (sketch)

Because the Ls have max |A| ≈ s−1/2, we have Θ(x, t, r) ≤ 1 + ε0 forr2, t ≤ s.

Let Ts be the supremum of times T for which Θ(x, t, r) ≤ 1 + ε0 for allr2, t ≤ T .

If Ts ≤ δ from the monotonicity theorem, we let T := Ts/a. ApplyingWhite’s theorem we get

|A| ≤ C0√t− T

.

We also know from the monotonicity theorem that we may pick a timet0 ∈ [(T + Ts)/2, Ts) where Lst is L2-close to a self-expander.

Together with the curvature bound and a bit of extra work, we can applythe stability result to see that Lst0 is close to a self-expander in C1,α. Thisimplies Gaussian density ratio controls for a fixed time interval whichexceeds Ts, a contradiction. Hence Ts ≥ δ.

Completing the proof (sketch)

Because the Ls have max |A| ≈ s−1/2, we have Θ(x, t, r) ≤ 1 + ε0 forr2, t ≤ s.

Let Ts be the supremum of times T for which Θ(x, t, r) ≤ 1 + ε0 for allr2, t ≤ T .

If Ts ≤ δ from the monotonicity theorem, we let T := Ts/a. ApplyingWhite’s theorem we get

|A| ≤ C0√t− T

.

We also know from the monotonicity theorem that we may pick a timet0 ∈ [(T + Ts)/2, Ts) where Lst is L2-close to a self-expander.

Together with the curvature bound and a bit of extra work, we can applythe stability result to see that Lst0 is close to a self-expander in C1,α. Thisimplies Gaussian density ratio controls for a fixed time interval whichexceeds Ts, a contradiction. Hence Ts ≥ δ.

Completing the proof (sketch)

Because the Ls have max |A| ≈ s−1/2, we have Θ(x, t, r) ≤ 1 + ε0 forr2, t ≤ s.

Let Ts be the supremum of times T for which Θ(x, t, r) ≤ 1 + ε0 for allr2, t ≤ T .

If Ts ≤ δ from the monotonicity theorem, we let T := Ts/a. ApplyingWhite’s theorem we get

|A| ≤ C0√t− T

.

We also know from the monotonicity theorem that we may pick a timet0 ∈ [(T + Ts)/2, Ts) where Lst is L2-close to a self-expander.

Together with the curvature bound and a bit of extra work, we can applythe stability result to see that Lst0 is close to a self-expander in C1,α. Thisimplies Gaussian density ratio controls for a fixed time interval whichexceeds Ts, a contradiction. Hence Ts ≥ δ.

Completing the proof (sketch)

Because the Ls have max |A| ≈ s−1/2, we have Θ(x, t, r) ≤ 1 + ε0 forr2, t ≤ s.

Let Ts be the supremum of times T for which Θ(x, t, r) ≤ 1 + ε0 for allr2, t ≤ T .

If Ts ≤ δ from the monotonicity theorem, we let T := Ts/a. ApplyingWhite’s theorem we get

|A| ≤ C0√t− T

.

We also know from the monotonicity theorem that we may pick a timet0 ∈ [(T + Ts)/2, Ts) where Lst is L2-close to a self-expander.

Together with the curvature bound and a bit of extra work, we can applythe stability result to see that Lst0 is close to a self-expander in C1,α. Thisimplies Gaussian density ratio controls for a fixed time interval whichexceeds Ts, a contradiction. Hence Ts ≥ δ.

Completing the proof (sketch)

Because the Ls have max |A| ≈ s−1/2, we have Θ(x, t, r) ≤ 1 + ε0 forr2, t ≤ s.

Let Ts be the supremum of times T for which Θ(x, t, r) ≤ 1 + ε0 for allr2, t ≤ T .

If Ts ≤ δ from the monotonicity theorem, we let T := Ts/a. ApplyingWhite’s theorem we get

|A| ≤ C0√t− T

.

We also know from the monotonicity theorem that we may pick a timet0 ∈ [(T + Ts)/2, Ts) where Lst is L2-close to a self-expander.

Together with the curvature bound and a bit of extra work, we can applythe stability result to see that Lst0 is close to a self-expander in C1,α. Thisimplies Gaussian density ratio controls for a fixed time interval whichexceeds Ts, a contradiction. Hence Ts ≥ δ.

The result

We can pass to a limit of flows, and use curvature estimates to prove thefollowing

Theorem (B.-Moore)

Suppose that L ⊂ Cn is a compact Lagrangian submanifold of Cn with afinite number of singularities, each of which is asymptotic to a pair oftransversally intersecting planes P1 + P2 where neither P1 + P2 norP1 − P2 are area minimizing. Then there exists T > 0 and a Lagrangianmean curvature flow (Lt)0<t<T such that as t 0, Lt → L as varifolds,and in C∞loc away from the singularities.

See our preprint on arXiv for full details.

Thanks for listening.

The result

We can pass to a limit of flows, and use curvature estimates to prove thefollowing

Theorem (B.-Moore)

Suppose that L ⊂ Cn is a compact Lagrangian submanifold of Cn with afinite number of singularities, each of which is asymptotic to a pair oftransversally intersecting planes P1 + P2 where neither P1 + P2 norP1 − P2 are area minimizing. Then there exists T > 0 and a Lagrangianmean curvature flow (Lt)0<t<T such that as t 0, Lt → L as varifolds,and in C∞loc away from the singularities.

See our preprint on arXiv for full details.

Thanks for listening.