Singularity of Mean Curvature Flow of Lagrangian Submanifolds

26 Pages·0.23 MB·English

by  Jingyi Chen

Singularity of Mean Curvature Flow of Lagrangian
Submanifolds
3
0
0
2
Jingyi Chen Jiayu Li
n ∗ †
a Department of Mathematics Institute of Mathematics
J

4 University of British Columbia Fudan University, Shanghai
2
Vancouver, B.C. V6T 1Z2 Academia Sinica, Beijing

]
G Canada P.R. China
D [email protected] [email protected]
.
h
t
a
m
[ Abstract

1
In this article we study the tangent cones at first time singularity of a
v
1 Lagrangian mean curvature flow. If the initial compact submanifold Σ is
0
8 Lagrangian and almost calibrated by ReΩ in a Calabi-Yau n-fold (M,Ω), and
2
T > 0 is the first blow-up time of the mean curvature flow, then the tangent
1
0 cone of the mean curvature flow at a singular point (X ,T) is a stationary
0
3 Lagrangian integer multiplicity current in R2n with volume density greater
0
/ than one at X0. When n = 2, the tangent cone consists of a finite union of
h
more than one 2-planes in R4 which are complex in a complex structure on
t
a R4.
m
:
v
1 Introduction
i
X
r
a Let M be a compact Calabi-Yau manifold of complex dimension n with a
Ka¨hler form ω, a complex structure J, a Ka¨hler metric g and a parallel holo-
morphic (n,0)-form Ω of unit length. An immersed submanifold Σ in M is
Lagrangian if ω = 0. The induced volume form dµ on a Lagrangian sub-
Σ Σ
|
manifold Σ from the Ricci-flat metric g is related to Ω by
Ω = eiθdµ = cosθdµ +isinθdµ , (1)
Σ Σ Σ Σ
|
∗Chen is supported partially by a Sloan fellowship and a NSERC grant.
†Li is partially supported by the National Science Foundation of China and by the Partner
Group of MPI for Mathematics.
1
wherethephasefunctionθ ismulti-valued andiswell-defineduptoanadditive
constant 2kπ,k Z. Nevertheless, cosθ and sinθ are single valued function
∈
on Σ. For any tangent vector X to M a straightforward calculation shows
Xθ = g(H,JX) (2)
−
whereHisthemeancurvaturevector ofΣinM (cf. [HL],[TY]).Equivalently,
H = J θ. The Lagrangian submanifold Σ is special, i.e. it is a minimal
∇
submanifold, if and only if θ is constant. When θ is constant on a Lagrangian
submanifold Σ, the real part of e iθΩ is a calibration of M with comass one
−
and Σ is a volume minimizer in its homology class [HL]. Let ReΩ be the real
part of Ω. A Lagrangian submanifold is called almost calibrated by ReΩ if
cosθ > 0.
Constructing minimal Lagrangian submanifolds is an important but very
challenging task. In a compact Ka¨hler-Einstein surface, Schoen and Wolfson
[ScW] have shown the existence of a branched surface which minimizes area
among Lagrangian competitors in each Lagrangian homology class, by varia-
tional method.
For a one-parameter family of immersions F = F(,t) : Σ M, we denote
t
· →
the image submanifolds by Σ = F (Σ). If Σ evolves along the gradient flow
t t t
of the volume functional, the first variation of the volume functional asserts
that Σ satisfy a mean curvature flow equation:
t
d
F(x,t) = H(x,t)
dt (3)

F(x,0) = F (x),
 0
When Σ is compact the mean curvature flow (3) has a smooth solution
for short time [0,T) by the standard parabolic theory. If Σ is Lagrangian
0
in a Ka¨hler-Einstein ambient space M, Smoczyk has shown that Σ remains
t
Lagrangian for t < T and the phase function θ evolves by
dθ
= ∆θ (4)
dt
where ∆ is the Laplacian of the induced metric on Σ ([Sm1-3], also see [TY]
t
for a derivation of (4)). It then follows that
∂cosθ
= ∆cosθ+ H 2cosθ. (5)
∂t | |
If the initial Lagrangian submanifold Σ is almost calibrated, Σ is almost
0 t
calibrated, i.e. cosθ > 0, alonga smoothmean curvatureflow by theparabolic
maximum principle.
It is well-known that if A 2, where A is the second fundamental form on
| |
Σ , is boundeduniformly as t T > 0 then (3) admits a smooth solution over
t
→
[0,T +ǫ) for some ǫ > 0. When max A 2 becomes unbounded as t T, we
Σt| | →
2
say that the mean curvature flow develops a singularity at T. A lot of work
has been devoted to understand these singularities (cf. [CL1-2], [E1-2], [H1-3],
[HS1-2], [I1], [Wa], [Wh1-3].)
In this paper, we shall study the tangent cones at singularities of the mean
curvature flow of a compact Lagrangian submanifold in a compact Calabi-
Yau manifold. Especially, we shall focus on the structure of tangent cones of
the mean curvature flow where a singularity occurs at the first singular time
T < .
∞
To describe the tangent cones, suppose that (X ,T) is a singular point of
0
the flow (3), i.e. A(x,t) becomes unbounded when (x,t) (X ,T). For an
0
| | →
arbitrary sequence of numbers λ and any t < 0, if T +λ 2t > 0 we set
−
→ ∞
F (x,t) = λ(F(x,T +λ 2t) X ).
λ − 0
−
We denote the scaled submanifold by (Σλ,dµλ). If the initial submanifold is
t t
Lagrangian and almost calibrated by ReΩ, it is proved in Proposition 2.3 that
there is a subsequence λ such that for any t < 0, (Σλi,dµλi) converges
i → ∞ t t
to (Σ ,dµ ) in the sense of measures; the limit Σ is called a tangent cone
∞ ∞ ∞
arising from the rescaling λ, or simply a λ tangent cone at (X ,T). This
0
tangent cone is independent of t as shown in Proposition 2.3.
There is also a time dependent scaling which we would like to consider
1
F(,s) = F(,t), (6)
· 2(T t) ·
−
where s = 1log(T t), ce s < p. Here we have chosen the coordinates so
−2 − 0 ≤ ∞
thatX = 0. Rescalingofthistypearisesnaturallyinclassification ofsingular-
0
ities of mean curvature flows [H2]: assume lim max A 2 = , if there
exists a positive constant C such that limsup t→T− (T Σtt)|m|ax ∞A 2 C,
t→T− − Σt| | ≤
the mean curvature flow F has a Type I singularity at T; otherwise it has a
(cid:0) (cid:1)
Type II singularity at T. Denote Σ the rescaled submanifold by F(,s). If a
s
·
subsequenceof Σ converges in measures to alimit Σ , then the limit is called
s
a tangent cone arising from the tieme dependent scal∞ing at (X ,T),eor simply
0
a t tangent conee. In this paper, a tangent cone of tehe mean curvature flow at
(X ,T) means either a λ tangent cone or a t tangent cone at (X ,T).
0 0
The main result of this paper is
Theorem 1.1 Let (M,Ω) be a compact Calabi-Yau manifold of complex di-
mension n. If the initial compact submanifold Σ is Lagrangian and almost
0
calibrated by ReΩ, and T > 0 is the first blow-up time of the mean curvature
flow (3), and (X ,T) is a singular point, then the tangent cone of the mean
0
curvature flow at (X ,T) is a stationary Lagrangian integer multiplicity cur-
0
rent in R2n with volume density greater than one at X . When n = 2, the
0
tangent cone consists of a finite union of more than one 2-planes in R4 which
are complex in a complex structure on R4.
3
For symplectic mean curvature flow in Ka¨hler-Einstein surfaces, results
similar to Theorem 1.1 was obtained in [CL1]. The authors are grateful to
Professor Gang Tian for stimulating conversations.
λ
2 Existence of tangent cones
This section contains basic formulas and estimates which are essential for this
article. First, we will derive a monotonicity formula which has a weight func-
tion introduced by the n-form ReΩ. Second, we use the monotonicity formula
to derive three integral estimates, which roughly say that when averaged over
any time interval the mean curvature vector H and the phase function cosθ
λ λ
both tend to 0 in L2 norm over a fixed ball near the singularity, as λ .
→ ∞
Another direct consequence of the monotonicity formula is that there is an
upper bound of the volume density of the rescaled submanifolds Σλ, which
t
allows us to extract converging subsequence in measure.
2.1 A weighted monotonicity formula
Let H(X,X ,t ,t) be the backward heat kernel on Rk. Let N be a smooth
0 0 t
family of submanifolds of dimension n in Rk defined by F :N Rk. Define
t
→
ρ(X,t) = (4π(t t))(k n)/2H(X,X ,t ,t)
0 − 0 0
−
1 X X 2
0
= exp | − | (7)
(4π(t0 t))n/2 − 4(t0 t) !
− −
for t < t .
0
A straightforward calculation (cf. [CL1], [H1], [Wa]) shows
∂ n H (X X ) X X 2
0 0
ρ = · − | − | ρ
∂t 2(t0 t) − 2(t0 t) − 4(t0 t)2 !
− − −
and along N
t
X X , X 2 X X ,∆X X 2
0 0
∆ρ= |h − ∇ i| h − i |∇ | ρ
4(t0 t)2 − 2(t0 t) − 2(t0 t)!
− − −
where ∆, are on N in the induced metric. Let N = Σ be a smooth
t t t
∇
1-parameter family of compact Lagrangian submanifolds in a compact Calabi-
Yau manifold (M,Ω) of complex dimensionn. Note that intheinducedmetric
on Σ
t
F 2 = n and ∆F = H.
|∇ |
Therefore
2
∂ (F X )
+∆ ρ = H+ − 0 ⊥ H 2 ρ. (8)
(cid:18)∂t (cid:19) −(cid:12)(cid:12) 2(t0−t) (cid:12)(cid:12) −| | 
(cid:12) (cid:12)
(cid:12) (cid:12) 
(cid:12) (cid:12)
4
On Σ we set
t
v = cosθ.
Denote the injectivity radius of (M,g) by i . For X M, take a normal
M 0
∈
coordinate neighborhood U and let φ C (B (X )) be a cut-off function
∈ 0∞ 2r 0
with φ 1 in B (X )), 0 < 2r < i . Using the local coordinates in U we may
r 0 M
≡
regard F(x,t) as a point in R2n whenever F(x,t) lies in U. We define
1
Ψ(X ,t ,t) = φ(F)ρ(F,X ,t,t )dµ
0 0 0 0 t
v
ZΣt
where ρ is defined by (7) by taking k = 2n.
Proposition 2.1 Let F : Σ M be a smooth mean curvature flow of a
t
→
compact Lagrangian submanifold Σ in a compact Calabi-Yau manifold M of
0
complex dimension n. Suppose that Σ is almost calibrated byReΩ. Then there
0
are positive constants c and c depending only on M, F and r which is the
1 2 0
constant in the definition of φ, such that
∂ 1
ec1√t0 t φρdµ
− t
∂t v
(cid:18) ZΣt (cid:19)
2
1 2 v 2 (F X ) H 2
ec1√t0−t φρ |∇ | + H+ − 0 ⊥ + | |
≤ − ZΣt v  v2 (cid:12)(cid:12) 2(t0−t) (cid:12)(cid:12) 2 
(cid:12) (cid:12)
+c ec1√t0 t.  (cid:12) (cid:12)  (9)
2 − (cid:12) (cid:12)
Proof. Notice that
∆F = H+gijΓαv
ij α
where v ,α = 1,...,n is a basis of T Σ , gij is the induced metric on Σ and
α ⊥ t t
Γα is the Christoffel symbol on M. Equation (8) reads as
ij
∂ +∆ ρ = H+ (F −X0)⊥ 2 H 2+ gijΓαijvα·(F −X0) ρ. (10)
(cid:18)∂t (cid:19) −(cid:12)(cid:12) 2(t0−t) (cid:12)(cid:12) −| | t0−t 
(cid:12) (cid:12)
(cid:12) (cid:12) 
¿From (5) we have (cid:12) (cid:12)
∂ 1 1 H 2 2 v 2
= ∆ | | |∇ |
∂tv v − v − v3
and
d
dµ = H 2dµ .
t t
dt −| |
Moreover
∂φ(F)
= φ H.
∂t ∇ ·
5
Now we have
d 1
φρ
dt v
ZΣt
1 H 2 2 1
= φρ∆ | | + v 2 φρ+ φ Hρ
ZΣt v −ZΣt v v3|∇ | ! ZΣt v∇ ·
1φ ∆ρ+ H+ (F −X0)⊥ 2 H 2+ gijΓαijvα·(F −X0) ρ
−ZΣt v  (cid:12)(cid:12) 2(t0−t) (cid:12)(cid:12) −| | t0−t  
(cid:12) (cid:12)
1  (cid:12) (cid:12)  
φρH 2 (cid:12) (cid:12)
− v | |
ZΣt
2
2 1 (F X ) H 2
φρ v 2 + H+ − 0 ⊥ + | |
≤ −ZΣt v3|∇ | v (cid:12)(cid:12) 2(t0−t) (cid:12)(cid:12) v 
(cid:12) (cid:12)
+ φρ∆1 1φ∆ρ (cid:12)(cid:12) 1φρgijΓαij(cid:12)(cid:12)vα·(F −X0)
v − v − v t t
ZΣt(cid:18) (cid:19) ZΣt 0−
1 1 φ2
+ ρ ǫ2φH 2+ |∇ | (11)
ZΣt v | | 4ǫ2 φ !
where we used Cauchy-Schwartz inequality for φ H. By Stokes formula
∇ ·
1 1 1 1
φρ∆ φ∆ρ = 2 φ ρ+ ρ∆φ.
v − v v∇ ∇ v
ZΣt(cid:18) (cid:19) ZΣt ZΣt
Since φ C (B (X ),R+), we have (cf. Lemma 6.6 in [Il])
∈ 0∞ 2r 0
φ2
|∇ | 2max 2φ.
φ ≤ φ>0 |∇ |
Note that φ 0 in B (X ), so ρ∆φ and φ ρ are bounded in B (X ).
r 0 2r 0
∇ ≡ | | |∇ ·∇ |
Hence
1 1 1 C
ρ∆φ + φ ρ C dµ vol(Σ ) (12)
t 0
v v∇ ·∇ ≤ v ≤ min v
ZΣt(cid:12) (cid:12) ZΣt(cid:12) (cid:12) ZΣt Σ0
(cid:12) (cid:12) (cid:12) (cid:12)
where C dep(cid:12)(cid:12)ends o(cid:12)(cid:12)nly on(cid:12)(cid:12)r,max( 2(cid:12)(cid:12)φ + φ).
|∇ | |∇ |
Since Γα(X )= 0, we may choose r sufficiently small such that
ij 0
gijΓα(F) C F X
| ij | ≤ | − 0|
in B (X ) for some constant C depending on M. We claim
2r 0
|gijΓαijvα·(F −X0)|ρ(F,t) c ρ(F,t) +C. (13)
1
t t ≤ √t t
0 0
− −
In fact it suffices to show for any x and s > 0
x2e x2/s 1 e x2/s
− −
C 1+ .
s sn/2 ≤ s1/2 sn/2 !
6
To see this, let y = x2/s and then it is easy to verify that
1
y C sn/2ey +
≤ s1/2
(cid:18) (cid:19)
holds trivially if y 1/s1/2 and follows from yn+1 Cey if y > 1/s1/2 for
≤ ≤
some C. So (13) is established.
Letting ǫ2 = 1/2 in (11) and applying (12), (13) to (11) we have
2
∂ 1 2 v 2 (F X ) H 2 c
0 ⊥ 1
Ψ φρ |∇ | + H+ − + | | + Ψ+c .
∂t ≤ −ZΣt v  v2 (cid:12)(cid:12) 2(t0 −t) (cid:12)(cid:12) 2  √t0−t 2
(cid:12) (cid:12)
 (cid:12) (cid:12) 
The proposition follows. (cid:12) (cid:12) Q.E.D.
Suppose that (X ,T) is a singular point of the mean curvature flow (3).
0
We now describe the rescaling process around (X ,T). For any t < 0, we set
0
F (x,t) = λ(F(x,T +λ 2t) X ) (14)
λ − 0
−
where λ are positive constants which go to infinity. The scaled submanifold
is denoted by Σλ = F (Σ,t) on which dµλ is the area element obtained from
t λ t
dµ . If gλ is the metric on Σλ, it is clear that
t t
gλ =λ2g , (gλ)ij = λ 2gij.
ij ij −
We therefore have
∂F ∂F
λ = λ 1
−
∂t ∂t
H = λ 1H
λ −
A 2 = λ 2 A 2.
λ −
| | | |
It follows that the scaled submanifold also evolves by a mean curvature flow
∂F
λ
= H . (15)
λ
∂t
Moreover, since
dµλ(F (x,t)) = λndµ (F(x,T +λ 2t))
t λ t −
Ω (F (x,t)) = λnΩ (F(x,T +λ 2t))
|Σλt λ |Σt −
we have
cosθ (F (x,t)) = cosθ(F(x,T +λ 2t)).
λ λ −
7
2.2 Integral estimates
Proposition 2.2 Let (M,Ω) be a Calabi-Yau manifold of complex dimension
n. If the initial compact submanifold is Lagrangian and is almost calibrated by
ReΩ, then for any R > 0 and any < s < s < 0, we have
1 2
−∞
s2
cosθ 2dµλdt 0 as λ , (16)
Zs1 ZΣλt∩BR(0)|∇ λ| t → → ∞
s2
H 2dµλdt 0 as λ , (17)
Zs1 ZΣλt∩BR(0)| λ| t → → ∞
and
s2
F 2dµλdt 0 as λ . (18)
Zs1 ZΣλt∩BR(0)| λ⊥| t → → ∞
Proof: For any R > 0, we choose a cut-off function φ C (B (0)) with
R ∈ 0∞ 2R
φ 1 in B (0), where B (0) is the metric ball centered at 0 with radius r in
R R r
≡
R2n. For any fixed t < 0, the mean curvature flow (3) has a smooth solution
near T+λ 2t < T for sufficiently large λ, since T > 0 is the firstblow-up time
−
of the flow. Let v = cosθ . It is clear
λ λ
1 1 F 2
φ (F )exp | λ| dµλ
ZΣλt vλ(0−t)n/2 R λ −4(0−t)! t
1 1 F(x,T +λ 2t) X 2
− 0
= φ(F ) exp | − | dµ ,
ZΣT+λ−2t vλ λ (T −(T +λ−2t))n/2 − 4(T −(T +λ−2t)) ! t
where φ is the function defined in the definition of Φ. Note that T+λ 2t T
−
→
for any fixed t as λ . By the weighted monotonicity formula (9),
→ ∞
∂
ec1√t0 tΨ c ec1√t0 t,
− 2 −
∂t ≤
(cid:16) (cid:17)
anditthenfollowsthatlim ec1√t0 tΨexists. Thisimplies,bytakingt = T
t t0 − 0
and t = T +λ 2s, that for→any fixed s and s with < s < s <0,
− 1 2 1 2
−∞
1 1 F 2
ec1√T−(T+λ−2s2)ZΣλs2 vλφR(0−s2)n/2 exp −4(|0−λ|s2)!dµλs2
1 1 F 2
−ec1√T−(T+λ−2s1)ZΣλs1 vλφR(0−s1)n/2 exp −4(0| −λ|s1)!dµλs1
0 as λ . (19)
→ → ∞
Integrating (9) from T +λ 2s to T +λ 2s , and letting T +λ 2s = t, we get
− 1 − 2 −
−ec1√−λ−2s2ZΣλs2 v1λφR(0−1s2)n/2 exp −4(|0F−λ|2s2)!dµλs2
8
+ec1√−λ−2s1ZΣλs1 v1λφR(0−1s1)n/2 exp −4(0|F−λ|2s1)!dµλs1
≥ ZTT++λλ−−2s21s2ec1√T−tZΣt v1φρ2|∇v2v|2 +(cid:12)(cid:12)H+ (F2(−T X−0t))⊥(cid:12)(cid:12)2+ |H2|2dµt
(cid:12) (cid:12)
c λ 2(s s )  (cid:12) (cid:12) 
− 2 − 2− 1 (cid:12) (cid:12)
2
≥ Zs1s2ec1√−λ−2sZΣλs v1λφRρ(Fλ,s)(cid:12)(cid:12)Hλ+ (2F(−λ)s⊥)(cid:12)(cid:12) dµλs
+Zs1s2ec1√−λ−2sZΣλs v1λφRρ(Fλ,s(cid:12)(cid:12)(cid:12))|H2λ|2dµλs (cid:12)(cid:12)(cid:12)
+Zs1s2ec1√−λ−2sZΣλs v2λ3|∇vλ|2φRρ(Fλ,s)dµλs
c λ 2(s s ). (20)
2 − 2 1
− −
From (19) and (20) the proposition follows. Q.E.D.
2.3 Upper bound on volume density
Now we show the existence of the λ tangent cones by deriving an finite upper
bound for the volume density. These cones are independent of t, but may
depend on the blowing up sequence λ.
Proposition 2.3 Suppose that Σ evolves along mean curvature flow and Σ
t 0
is a compact Lagrangian submanifold in (M,Ω) and is almost calibrated by
ReΩ. For any λ,R > 0 and any t < 0,
µλ(Σλ B (0)) CRn, (21)
t t ∩ R ≤
where B (0) is a metric ball in R2n and C > 0 is independent of λ. For any
R
sequence λ , there is a subsequence λ such that (Σλk,µλk)
i → ∞ k → ∞ t t →
(Σ ,µ ) in the sense of measure, for any fixed t < 0, where (Σ ,µ ) is
∞ ∞ ∞ ∞
independent of t. The multiplicity of Σ is finite.
∞
Proof: We shall first prove the inequality (21). We shall use C below for
uniformpositive constants which are independentofR and λ. Straightforward
computation shows
µλ(Σλ B (0)) = λn dµ
t t ∩ R t
ZΣT+λ−2t∩Bλ−1R(X0)
= Rn(λ 1R) n dµ
− − t
ZΣT+λ−2t∩Bλ−1R(X0)
≤ CRn v1 (4π)n/21(λ 1R)ne−4|X(λ−−X1R0|)22dµt
ZΣT+λ−2t∩Bλ−1R(X0) λ −
= CRnΨ(X ,T +(λ 1R)2+λ 2t,T +λ 2t).
0 − − −
9
By the weighted monotonicity inequality (9), we have
µλ(Σλ B (0)) CRnΨ(X ,T +(λ 1R)2+λ 2t,T/2)+CRn
t t ∩ R ≤ 0 − −
µ (Σ )
T/2 T/2 CRn+CRn.
≤ Tn/2min v
Σ0
Since volume is non-increasing along mean curvature flow:
∂
µ (Σ )= H 2dµ ,
t t t
∂t − | |
ZΣt
we have therefore established (21):
µλ(Σλ B (0)) CRn.
t t ∩ R ≤
By (21), the compactness theorem of the measures (c.f. [Si1], 4.4) and a
diagonal subsequenceargument, weconcludethatthereisasubsequenceλ
k
suchthat(Σλk,µλk) (Σ ,µ )inthesenseofmeasuresforafixedt <→0.
∞ t0 t0 → ∞t0 ∞t0 0
We now show that, for any t < 0, the subsequence λ which we have
k
chosen above satisfies (Σλk,µλk) (Σ ,µ ) in the sense of measure. And
t t → ∞t0 ∞t0
consequently the limiting submanifold (Σ ,µ ) is independent of t . Recall
∞t0 ∞t0 0
that the following standard formula for mean curvature flows
d
φdµλ = φH 2+ φ H dµλ (22)
dt ZΣλt t −ZΣλt (cid:16) | λ| ∇ · λ(cid:17) t
is valid for any test function φ C (M) (cf. (1) in Section 6 in [I2] and [B]
∈ 0∞
in the varifold setting).
Then for any given t < 0 integrating (22) yields
φdµλk φdµλk = t0 φH 2+ φ H dµλkdt
ZΣλtk t −ZΣλt0k t0 Zt ZΣλtk (cid:16) | λk| ∇ · λk(cid:17) t
0 as k by (17).
→ → ∞
So, for any fixed t < 0, (Σλk,µλk) (Σ ,µ ) in the sense of measures as
t t → ∞t0 ∞t0
k . We denote (Σ ,µ ) by (Σ ,µ ), which is independent of t .
→ ∞ ∞t0 ∞t0 ∞ ∞ 0
Theinequality (21)yieldsauniformupperboundonR nµλk(Σλk B (0)),
− t t ∩ R
which yields finiteness of the multiplicity of Σ . Q.E.D.
∞
Definition 2.4 Let (X ,T) be a singular point of the mean curvature flow of
0
a compact Lagrangian submanifold Σ in a compact Calabi-Yau manifold M.
0
We call (Σ ,dµ ) obtained in Proposition 2.3 a λ tangent cone of the mean
∞ ∞
curvature flow Σ at (X ,T).
t 0
10