on short time existence of lagrangian mean curvature flow · on short time existence of lagrangian...
TRANSCRIPT
![Page 1: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/1.jpg)
On Short Time Existence of Lagrangian Mean CurvatureFlow
Tom Begley
Joint work with Kim Moore
March 15, 2016
![Page 2: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/2.jpg)
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(·).
![Page 3: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/3.jpg)
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(·).
![Page 4: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/4.jpg)
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(·).
![Page 5: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/5.jpg)
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.
![Page 6: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/6.jpg)
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.
![Page 7: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/7.jpg)
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.
![Page 8: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/8.jpg)
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.
![Page 9: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/9.jpg)
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.
![Page 10: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/10.jpg)
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.
![Page 11: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/11.jpg)
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?
![Page 12: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/12.jpg)
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?
![Page 13: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/13.jpg)
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?
![Page 14: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/14.jpg)
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?
![Page 15: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/15.jpg)
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?
![Page 16: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/16.jpg)
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?
![Page 17: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/17.jpg)
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?
![Page 18: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/18.jpg)
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?
![Page 19: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/19.jpg)
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?
![Page 20: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/20.jpg)
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.
![Page 21: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/21.jpg)
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.
![Page 22: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/22.jpg)
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.
![Page 23: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/23.jpg)
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)
![Page 24: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/24.jpg)
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.
![Page 25: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/25.jpg)
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.
![Page 26: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/26.jpg)
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.
![Page 27: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/27.jpg)
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.
![Page 28: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/28.jpg)
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.
![Page 29: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/29.jpg)
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.
![Page 30: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/30.jpg)
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.
![Page 31: On Short Time Existence of Lagrangian Mean Curvature Flow · On Short Time Existence of Lagrangian Mean Curvature Flow Tom Begley Joint work with Kim Moore March 15, 2016. ... Then](https://reader031.vdocuments.us/reader031/viewer/2022021807/5bdc1a7609d3f2bc1c8d3fec/html5/thumbnails/31.jpg)
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.