A rigidity theorem on the second fundamental form for self-shrinkers

0.19 MB·

by  Qi Ding

A RIGIDITY THEOREM ON THE SECOND FUNDAMENTAL FORM
FOR SELF-SHRINKERS
QI DING
7
1
0
2 Abstract. In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under
the integral condition on the norm of the second fundamental form. In this paper, we
n
a relax thesuch boundto any finite constant (see Theorem 4.4 for details).
J

7
2

]
G 1. Introduction
D
Self-similar solutions for mean curvature flow play a key role in the understanding
.
h
the possible singularities that the flow goes through. Self-shrinkers are type I singularity
t
a modelsoftheflow. Huiskenmadeapioneerworkonself-shrinkingsolutionsoftheflow[22,
m
23]. ColdingandMinicozzi [8] gave acomprehensive studyforself-shrinkinghypersurfaces
[
and solve a long-standing conjecture raised by Huisken.

1
v Colding-Ilmanen-Minicozzi [9] showed that cylindrical self-shrinkers are rigid in a very
5 strong sense. Namely, any other shrinker that is sufficiently close to one of them on
4
a large, but compact set must itself be a round cylinder. See [25] by Guang-Zhu for
9
further results. Lu Wang in [37,38] proved strong uniqueness theorems for self-shrinkers
7
0 asymptotic to regular cones or generalized cylinders of infinite order.
.
1
For Bernstein type theorems, Ecker-Huisken [17] and Wang [36] showed the nonexis-
0
7 tence of nontrivial graphic self-shrinkinghypersurfaces in Euclidean space. For 2 n 6,
1 Guang-Zhu showed that any smooth complete self-shrinker in Rn+1 which is grap≤hica≤l in-
:
v sidealarge, butcompact, setmustbeahyperplane. Ding-Xin-Yang [14] studiedthesharp
Xi rigidity theoremswiththeconditiononGaussmapofself-shrinkers. Inhighcodimensions,
see [2,3,10,13,26] for more Bernstein type theorems.
r
a
Le-Sesum [30] showed that any complete embedded self-shrinking hypersurface with
polynomial volume growth mustbeahyperplaneprovided thesquared normof thesecond
fundamental form B 2 < 1. Cao-Li [1] showed that any complete self-shrinker (with high
| | 2
codimension) with polynomial volume growth must be a generalized cylinder provided
B 2 1. Later, Cheng-Peng [5] removed the condition of polynomial volume growth in
| | ≤ 2
the case of B 2 < 1 (See [4,6,12,42] for more results on the gap theorems of the norm of
| | 2
the second fundamental form). In [12], Ding-Xin proved a rigidity result for self-shrinkers
if the integration of B n is small. In this paper, we improve the small constant to any
| |
finite constant.
For a complete properly immersed self-shrinker Σn Rn+1, Ilmanen showed that there
existsacone Rn+1 withthecrosssection beingaco⊂mpactsetinSn suchthatλΣn
C ⊂ → C
as λ 0 locally in the Hausdorff metric on closed sets (see [28] Lecture 2, B, remark
+
→
The author would like to thank Yuanlong Xin for his interest in this work. He is supported partially
by NSFC.
1
2 QIDING
on p.8). In [35], Song gave a simple proof by ”maximum principle for self-shrinkers”. For
high codimensions, with backward heat kernel (see [8]) we show the uniqueness of tangent
cones at infinity for self-shrinkers with Euclidean volume growth in the current sense with
the condition on mean curvature(see Theorem 3.3).
ǫ-regularity theorems for the mean curvature flow have been studied by Ecker [15,16],
Han-Sun [19], Ilmanen [27], Le-Sesum [29]. Now we use the one showed by Ecker [16]
starting from self-similar solutions, and obtain the curvature estimates for self-shrinkers,
see Theorem 4.2. Combining Theorem 3.3, Theorem 4.2 and backward uniqueness for
parabolic operators [21], we can show that self-shrinkers with finite integration on B n
| |
must be planes, which improves a previous rigidity theorem in [12]. A litter more, we
obtain the following Theorem.
Theorem 1.1. Let M be an n-dimensional properly non-compact self-shrinker with com-
pact boundary in Rn+m, B denote the second fundamental form of M. If
(1.1) lim B ndµ = 0,
r | |
→∞ZM B2r Br
∩ \
then M must be an n-plane through the origin.
2. Preliminary
Let M be an n-dimensional C2-submanifold in Rn+m with the induced metric. Let
and betheLevi-Civita connections on M and Rn+m, respectively. We definethesecon∇d
∇
fundamental form B of M by
B(V,W)= ( W)N = W W
V V V
∇ ∇ −∇
for any V,W Γ(TM), where the mean curvature vector H of M is given by H =
trace(B) = n∈ B(e ,e ), where e is a local orthonormal frame field of M.
i=1 i i { i}
In this pPaper, Mn is said to be a self-shrinker in Rn+m if its mean curvature vector
satisfies
XN
(2.1) H = ,
− 2
where X = (x , ,x ) Rn+m is the position vector of M in Rn+m, and ( )N
1 n+m
··· ∈ ···
stands for the orthogonal projection into the normal bundle NM. Let ( )T denote the
···
orthogonal projection into the tangent bundle TM.
We define a second order differential operator as in [8] by
L
X2 X2 1
f = e| 4| div e−| 4| f = ∆f X, f
L ∇ − 2h ∇ i
(cid:18) (cid:19)
for any f C2(M). Let ∆ be the Laplacian of M, then for self-shrinkers,
∈
(2.2) ∆ X 2 = 2 X,∆X +2 X 2 = 2 X,H +2n = XN 2+2n.
| | h i |∇ | h i −| |
In [8], Colding and Minicozzi defined a function F for self-shrinking hypersurfaces
X0,t0
inEuclideanspace. Obviously, hypersurfacescan begeneralized tosubmanifoldsnaturally
in this definition. Set Φ C (Rn+m) for any t > 0 by
t ∞
∈
1 X2
Φt(X) = e−|4t| .
(4πt)n/2
A RIGIDITY THEOREM ON THE SECOND FUNDAMENTAL FORM FOR SELF-SHRINKERS 3
For an n-complete submanifold M in Rn+m, we define a functional F on M by
t
1 X2
Ft(M) = Φtdµ = e−|4t| dµ for t > 0,
(4πt)n/2
ZM ZM
where dµ is the volume element of M. Sometimes, we write F for simplicity if no ambigu-
t
ous in the text. If a self-shrinker is proper, then it is equivalent to that it has Euclidean
volume growth at most by [7] and [11]. We shall only consider proper self-shrinkers in the
following text.
Now we use the backward heat kernel to give a monotonicity formula for self-shrinkers
with arbitrary codimensions, which is essentially same as self-shrinking hypersurfaces es-
tablished by Colding-Minicozzi in [8].
Lemma 2.1. For any 0 < t t , each complete immersed self-shrinker Mn with
1 2
boundary ∂M (may be empty)≤in Rn≤+m∞satisfies
t2 Φ (X)
F (M) F (M) = XT,ν s ds
t2 − t1 − h ∂Mi 2s
(2.3) Zt1 (cid:18)Z∂M (cid:19)
t2 1 1
+ 1 XN 2Φ (X)dµ ds.
s
4s − s | |
Zt1 (cid:18) (cid:19)(cid:18)ZM (cid:19)
Proof. We differential F (M) with respect to t,
t
(2.4) Ft′ = (4π)−n2t−(n2+1) −n2 + |X4t|2 e−|X4t|2dµ.
ZM (cid:18) (cid:19)
A straightforward calculation shows (see also [11])
e|X4t|2div e−|X4t|2 X 2 = ∆ X 2+ 1 X 2 X 2
− ∇| | − | | 4t∇| | ·∇| |
(cid:18) (cid:19)
1
= 2 H,X 2n+ XT 2
− h i− t| |
(2.5)
XT 2
= XN 2+ | | 2n
| | t −
1 X 2
= 1 XN 2+ | | 2n,
− t | | t −
(cid:18) (cid:19)
where the third equality above uses the self-shrinkers’ equation (2.1). Then
Ft′ =(4π)−n2t−(n2+1) − 41div e−|X4t|2∇|X|2 − 14 1− 1t |XN|2e−|X4t|2 dµ
ZM (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
(2.6) =1(4π)−n2t−(n2+1) 2 XT,ν∂M e−|X4t|2 1 1 XN 2e−|X4t|2dµ
4 − h i − − t | |
(cid:18) Z∂M (cid:18) (cid:19)ZM (cid:19)
1 1 1
= XT,ν Φ (X) 1 XN 2Φ (X)dµ,
∂M t t
− 2t h i − 4t − t | |
Z∂M (cid:18) (cid:19)ZM
where ν is the normal vector of ∂M in Γ(TM). Then we complete the proof by inte-
∂M
gration from t to t . (cid:3)
1 2
Denote
1 1 1
(2.7) G (M) , F (M)+ XT,ν Φ (X) = 1 XN 2Φ (X)dµ.
t t′ 2t h ∂Mi t −4t − t | | t
Z∂M (cid:18) (cid:19)ZM
4 QIDING
The above Lemma implies G (M) 0 for each self-shrinker and t 1. If ∂M is bounded
t
≤ ≥
and has finite (n 1)-dimensional Hausdorff measure, then the limit
−
t
lim G (M)ds
s
t
→∞(cid:18)Z1 (cid:19)
always exists, and is a finite negative number. Hence, it’s clear that lim F (M) exists.
t t
→∞
3. Uniqueness of tangent cones at infinity for self-shrinkers
For any n-rectifiable varifold V Rn+m with multiplicity one, we define a functional
⊂
Ξ by
t
1 X2
Ξt(V,f) = (4πt)n/2 fe−|4t| dµV
ZsptV
for any t > 0, where µ is a measure on Rn+m associated with the Radon measure of V
V
in Rn+m G(n,n+m).
×
We suppose that M is a self-shrinker in Rn+m B with boundary ∂M ∂B for
R R
some R 1 and n 1(∂M) < . Let φ C1(Rn+m\ 0 ) be a homogeneous⊂function of
−
degree z≥ero. NamHely, for any 0∞= X Rn∈+m, \{ }
6 ∈
φ(X) = φ(X ξ) = φ(ξ)
| |
with ξ = X . Then
X
| |
δ x x
ij i j
(3.1) ∂xiφ = X − X 3 ∂ξjφ,
j (cid:18)| | | | (cid:19)
X
and
δ x x
(3.2) |∇φ|2 = Xjk2 − Xj 4k ∂ξjφ∂ξkφ≤ |X|−2 ∂ξjφ 2 , |X|−2|φ|21.
j,k (cid:18)| | | | (cid:19) j
X X(cid:0) (cid:1)
Taking the derivative of Ξ (M,φ) on t gets
t
∂tΞt(M,φ) = (4π)−n2t−(n2+1) n + |X|2 φe−|X4t|2dµ
−2 4t
(3.3) ZM (cid:18) (cid:19)
=(4π)−n2t−(n2+1) φdiv e−|X4t|2 X 2 φ 1 1 XN 2e−|X4t|2 dµ.
−4 ∇| | − 4 − t | |
ZM (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
Combining X φ =0, we have
·∇
φdiv e−|X4t|2 X 2 dµ
−4 ∇| |
ZM (cid:18) (cid:19)
= 1div φe−|X4t|2 X 2 dµ+ 1 φ X 2e−|X4t|2dµ
−4 ∇| | 4∇ ·∇| |
(3.4) ZM (cid:18) (cid:19) ZM
= 1 φ XT,ν∂M e−|X4t|2 + 1X φe−|X4t|2dµ
− 2 h i 2 ·∇
Z∂M ZM
= 1 φ XT,ν∂M e−R4t2 1 XN φe−|X4t|2dµ.
− 2 h i − 2 ·∇
Z∂M ZM
A RIGIDITY THEOREM ON THE SECOND FUNDAMENTAL FORM FOR SELF-SHRINKERS 5
Set cR = 2−1(4π)−n2R n−1(∂M). Substituting (3.2) and (3.4) into (3.3) gets
·H
∂tΞt(M,φ) 2−1(4π)−n2t−(n2+1) XN φe−|X4t|2dµ
| | ≤ | |·|∇ |
(cid:18)ZM
+ φ0Re−R4t2 n−1(∂M) + φ0 Gt(M)
| | H | | | |
(cid:19)
2−1(4π)−n2t−(n2+1) |XN| φ1e−|X4t|2dµ+ φ0 Gt(M) +cRt−(n2+1)
≤ X | | | | | |
ZM | | (cid:16) (cid:17)
(3.5) φ0 Gt(M) +cRt−(n2+1)
≤| | | |
(cid:16) (cid:17) 1/2 1/2
+2−1(4π)−n2t−(n2+1) φ1 XN 2e−|X4t|2dµ X −2e−|X4t|2dµ
| | | | | |
(cid:18)ZM (cid:19) (cid:18)ZM (cid:19)
φ0 Gt(M) +cRt−(n2+1)
≤| | | |
+ φ(cid:16)1 Gt(M) 1/2 t (cid:17)(4π)−n2t−(n2+2) X −2e−|X4t|2dµ 1/2.
| | | | t 1 | |
r − (cid:18) ZM (cid:19)
Put D = M B for every r > 0. There is a constant c > 0 depending only on M such
r r 0
∩
that for all r > 0
1dµ < c rn.
0
ZDr
Note M Rn+m B . Then for n 2, t R2, one has
R
⊂ \ ≥ ≥
n t X2 n ∞ t X2
t−2 e−|4t| dµ t−2 e−|4t| dµ
X 2 ≤ X 2
ZM | | k= 1 X[log(tR−2)]ZD2k+1√t\D2k√t | |
− − 2log2
t−n2 ∞ 1 e−4k−1 1dµ
≤ 4k
k= 1 X[log(tR−2)] ZD2k+1√t\D2k√t
− − 2log2
1
(3.6) c ∞ 4 ke 4k2(k+1)n +c − 4 k2(k+1)n
0 − − 0 −
≤
Xk=0 k= 1 X[log(tR−2)]
− − 2log2
1+[log(tR−2)]
2log2
c0 ∞ 2k(n−2)+ne−4k−1 +c0 2−k(n−2)+n
≤
k=0 k=1
X X
n
(4π)2c1(1+logt 2logR),
≤ −
where c is a constant depending only on n,c . Therefore
1 0
√1+logt t 1/2
∂tΞt(M,φ) √c1 φ1 Gt(M) + φ0 Gt(M) +cRt−(n2+1)
| | ≤ t | | t 1 | | | |
(3.7) 1+logt (cid:12)(cid:12) − (cid:12)(cid:12) (cid:16) (cid:17)
c1 φ1 +cR(cid:12)(cid:12) t−(n2+1) φ0(cid:12)(cid:12)+(φ0 + φ1) Gt(M).
≤ 4t(t 1)| | | | | | | | | |
−
Theorem 3.1. Let M be an n-dimensional self-shrinker in Rn+m with Euclidean volume
growth and boundary ∂M ∂B . If
R
⊂
(3.8) limsup r1 n H < ,
−
| | ∞
r→∞ (cid:18) ZM∩Br (cid:19)
6 QIDING
then there is a sequence t such that
i
→ ∞
M , t 1M = X Rn+m t X M
ti −i { ∈ | i ∈ }
converges to a cone C in Rn+m.
Proof. By co-area formula, we can choose R > 0 so that n 1(∂M) < with ∂M
′ −
H ∞ ⊂
∂B . Denote R by R for convenience. Let M , t 1M = X Rn+m tX M for
R ′ t −
′ { ∈ | ∈ }
any t > 0. Since M has Euclidean volume growth and (3.8) holds, then by compactness
of varifolds, there exists an n-rectifiable varifold T in Rn+m with integer multiplicity and
a sequence of t such that M = t 1M ⇀ T in the sense of Radon measure (See 42.7
i ti −i
Theorem of [34] for example).
Denote φ and Ξ (M,φ) as above. Set µ be the volume element of M . Since
t t t
1 X2 1 X2
(3.9) Ξt2(M,φ) = (4πt2)n/2 φe−|4t|2 dµ = (4π)n/2 φe−| 4| dµt = Ξ1(Mt,φ),
ZM ZMt
then for all R > 0
1 X2
(3.10) il→im∞Ξ1(MtiR,φ) = il→im∞ΞR2(Mti,φ) = (4πR2)n/2 ZT φ e−|4R|2 dµT = ΞR2(T,φ).
Note that G (M) does not change sign for t > 1. Fixing 0 < r < R < , from (3.7) we
t
∞
have
t2R2
i
Ξ (M,φ) Ξ (M,φ) ∂ Ξ (M,φ)ds
(cid:12) t2ir2 − t2iR2 (cid:12) ≤ Zt2ir2 | s s |
(3.11) (cid:12)(cid:12) t2iR2 c1 1+logs φ1+c(cid:12)(cid:12)R φ0s−(n2+1)+( φ0+ φ1) Gs(M) ds
≤ 4s(s 1)| | | | | | | | | |
Zt2ir2 (cid:18) − (cid:19)
c1 φ1 ∞ 1+logsds+ 2 (tir)−n−2cR φ0+(φ0 + φ1) t2iR2Gs(M)ds
≤ 4 | | Zt2ir2 s(s−1) n | | | | | | (cid:12)(cid:12)Zt2ir2 (cid:12)(cid:12)
(cid:12) (cid:12)
for all ti with rti 2. Since (cid:12) (cid:12)
≥ (cid:12) (cid:12)
t2iR2 t2iR2 t2iR2 1
G (M)ds F (M)ds+ XT,ν Φ (X) ds
(cid:12)(cid:12)Zt2ir2 s (cid:12)(cid:12) ≤ (cid:12)(cid:12)Zt2ir2 t′ Zt2ir2 (cid:18)2s Z∂Mh ∂Mi s (cid:19) (cid:12)(cid:12)
(3.12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) t2iR2 R (cid:12)(cid:12)
≤(cid:12)(cid:12)Ft2ir2(M)−Ft2iR(cid:12)2(M(cid:12))(cid:12)+Zt2ir2 (cid:18)2sHn−1(∂M)(4πs)−n/2(cid:19)ds (cid:12)
(cid:12) (cid:12) R
=(cid:12)F (M) F (M)(cid:12)+ (4π) n/2 n 1(∂M)(t r) n
t2ir2 − t2iR2 n − H − i −
(cid:12) (cid:12)
and limt (cid:12) Ft exists, we obtain(cid:12)
→∞(cid:12) (cid:12)
(3.13) ilim Ξ1(Mtir,φ) = ilim Ξ1(MtiR,φ) = ΞR2(T,φ).
→∞ →∞
Hence
1 X2
(3.14) Ξt(T,φ) = φe−|4t| dµT
(4πt)n/2
ZT
is independent of t (0, ).
∈ ∞
Clearly,
0< n(T B ) c rn
r 2
H ∩ ≤
A RIGIDITY THEOREM ON THE SECOND FUNDAMENTAL FORM FOR SELF-SHRINKERS 7
for some constant c > 0 and all r > 0. By the following lemma for V(r) = φ dµ ,
2 T Br T
we conclude that ∩
R
(3.15) r−n φ dµT
ZT∩Br
is a constant independent of r. An analog argument as the proof of 19.3 in [34] implies
that T is a cone. (cid:3)
Lemma 3.2. Let V(r) be a monotone nondecreasing continuous function on [0, ) with
∞
V(0) = 0 and V(r) c rn for some constant c > 0. If the quantity
3 3
≤
1 ∞ r2
(3.16) e−4tdV(r)
(4πt)n/2
Z0
is a constant for any t > 0, then r nV(r) is a constant.
−
Proof. There are constants κ ,κ > 0 such that for all t > 0
0 1
(3.17) ∞e−rt2dV(r) = κ0tn/2 = κ1 ∞e−rt2drn,
Z0 Z0
namely,
(3.18) ∞e−rt2d(V(r) κ1rn)= 0.
−
Z0
Integrating by parts implies
(3.19) ∞(V(r) κ1rn)re−rt2dr = 0.
−
Z0
Suppose that there is a constant r > 0 such that V(r ) κ rn > 0 (Or else we complete
the proof by (3.19)). Then there is0a 0 < δ < r0 and ǫ0>−0 s1uc0h that V(r) κ rn ǫ for
2 − 1 ≥
all r (r δ,r +δ). Set t = 2r2, then in (0, ) the function
∈ 0− 0 p p 0 ∞
r2
rpe−tp
attains its maximal value at r = r .
0
Now we claim
(3.20) plim pr1p2+e1p2 r0+δrpe−rtp2dr = ∞ e−t2dt = √π.
→∞ 0 Zr0−δ Z−∞
In fact,
I(p) ,pr0p12+e1p2 Zr0r−0+δδrpe−rtp2dr = p21ep2 Z−rrδ0δ0(1+s)pe−p2(1+s)2ds
(3.21) = rδ0√p 1+ t pe−p2(cid:16)√2tp+tp2(cid:17)dt
Z−rδ0√p(cid:18) √p(cid:19)
= rδ0√p eplog(cid:16)1+√tp(cid:17)e−√pt−t22dt.
Z−rδ0√p
When 1 s < , a simple calculation implies
−2 ≤ ∞
8 s2 s3
min 0, s3 log(1+s) s+ .
3 ≤ − 2 ≤ 3
(cid:26) (cid:27)
8 QIDING
Combining the above inequality, we get
limsupI(p) limsup rδ0√p e−t2+3t√3pdt
(3.22) p→∞ ≤ p→∞ Z−rδ0√p
δ √p
= lim r0 e−t2(1−3√tp)dt = ∞ e−t2dt,
p→∞Z−rδ0√p Z−∞
and
liminfI(p) lim rδ0√pe−t2dt+liminf 0 e−t2+38√t3pdt
(3.23) p→∞ ≥p→∞Z0 p→∞ Z−rδ0√p
0
= ∞e−t2dt+ lim e−t2(cid:16)1−38√tp(cid:17)dt = ∞ e−t2dt.
Z0 p→∞Z−rδ0√p Z−∞
Hence we have shown (3.20).
For p > 1,
(3.24) pr021p+e1p2 Zr0∞+δrn+pe−rtp2dr = r0nZrδ0∞√pe(n+p)log(cid:16)1+√tp(cid:17)e−√pt−t22dt
≤r0nZrδ0∞√pe(n+p)√tpe−√pt−t22dt ≤ r0nZrδ0∞√pe√npt−t22dt.
Then
(3.25)
lipminf pr12p+e1p2 ∞(V(r)−κ1rn)rpe−rtp2dr
→∞ 0 Z0
≥lipminf pr12p+e1p2 ǫ r0+δrpe−rtp2dr−κ1 r0−δrn+pe−rtp2dr−κ1 ∞ rn+pe−rtp2dr
→∞ 0 (cid:18) Zr0 δ Z0 Zr0+δ (cid:19)
−
≥ǫ√π−κ1r0nlimp→s∞up pr012p+e1p2 Z0r0−δrpe−p2rr022dr+Zrδ0∞√pe√npt−t22dr!
=ǫ√π−κ1r0nlimp→s∞up Z−−√rpδ0√peplog(cid:16)1+√tp(cid:17)e−√pt−t22dt+Zrδ0∞√pe−t2(cid:16)12−√npt(cid:17)dr!
≥ǫ√π−κ1r0nlimsup −rδ0√pe√pte−√pt−t22dt = ǫ√π.
p→∞ Z−√p !
Taking the derivative of t in (3.19) yields
(3.26) ∞(V(r) κ1rn)r2k+1e−rt2dr = 0
−
Z0
for any t > 0 and k = 0,1,2 . If we choose p = 2k+1, r2 > e, t = 2r2 in (3.25), then
··· 0 p p 0
we get the contradiction provided k is sufficiently large. Hence V(r) κ rn 0. (cid:3)
1
− ≡
Theorem 3.3. Let M be an n dimensional smooth self-shrinker with Euclidean volume
growth and boundary ∂M ∂B in Rn+m. If (3.8) holds, then the limit lim r 1M
R r −
⊂ →∞
exists and is cone, namely, the tangent cone at infinity of M is a unique cone.
A RIGIDITY THEOREM ON THE SECOND FUNDAMENTAL FORM FOR SELF-SHRINKERS 9
Proof. We claim
(3.27) lim r n φdµ
−
r
→∞(cid:18) ZM∩Br (cid:19)
exists for every homogeneous function φ C1(Rn+m 0 ) with degree zero. Suppose
∈ \{ }
(3.28) limsupr−n φdµ > liminfr−n φdµ
r→∞ ZM∩Br r→∞ ZM∩Br
for some homogeneous function φ C1(Rn+m 0 ) with degree zero. Then there exist
∈ \{ }
two sequences p and q such that
i i
→ ∞ → ∞
(3.29) lim p−i n φdµ > lim qi−n φdµ.
i→∞ ZM∩Bpi i→∞ ZM∩Bqi
By compactness of varifolds and Theorem 3.1, there exist two cones C ,C in Rn+m with
1 2
integer multiplicities and subsequences p of p and q of q such that M ⇀ C and
ki i ki i pki 1
M ⇀ C in the sense of Radon measure. So we have
qki 2
φdµ = lim φdµ = lim p n φdµ
ZC1∩B1 C1 i→∞ZMpki∩B1 pki i→∞ −ki ZM∩Bpki
(3.30) >il→im∞qk−inZM∩Bqki φdµ = il→im∞ZMqki∩B1φdµqki
= φdµ ,
C2
ZC2∩B1
which implies
X2 X2
(3.31) φe−| 4| dµC1 > φe−| 4| dµC2
ZC1 ZC2
by co-area formula.
From the previous argument, the limit
1 X2
(3.32) lim Ξt(M,φ) = lim φe−|4t| dµ
t t (4πt)n/2
→∞ →∞ ZM
exists. It infers that
X2 X2 1 X2
(3.33) ZC1φe−| 4| dµC1 =il→im∞ZMpki φe−| 4| = tl→im∞ tn/2 ZM φe−|4t| dµ
X2 X2
=il→im∞ZMqki φe−| 4| = ZC2φe−| 4| dµC2.
However, (3.33) contradicts (3.31). Hence, the claim (3.27) holds.
Iflim r 1M ⇀ C+, lim s 1M ⇀ C andC+ = C arecones, thenfrom(3.33)
i→∞ i− i→∞ −i − 6 −
one has
X2 X2
(3.34) φe−| 4| dµC+ = φe−| 4| dµC
−
ZC+ ZC−
for every homogeneous function φ C1(Rn+m 0 ) with degree zero. It’s clear that
∈ \{ }
(3.35) φ = φ.
ZC+∩∂B1 ZC−∩∂B1
10 QIDING
Arbitrariness of φ implies C+ = C . Therefore, the tangent cone at infinity of M is a
−
unique cone. (cid:3)
4. A rigidity theorem for self-shrinkers
Let us recall an ǫ-regularity theorem for mean curvature flow showed by Ecker (A litter
different from Theorem 1.8 in [16]).
Theorem 4.1. For p [n,n+2], there exists a constant ǫ > 0 such that for any smooth
0
properly immersed solu∈tion = ( ) of mean curvature flow in Rn+m, every X
t t ( 4,0) 0
which the solution reaches aMt timeMt ∈[ −1,0), the assumption
0
∈ −
1 ρ2
(4.1) I , sup − B p ǫ
X0,t0 √−t0≤ρ<ρ′≤2(ρ′2−ρ2)n+22−p Z−ρ′2 ZMt∩B2(X0)| | ≤ 0
implies
2
(4.2) sup σ2 sup sup |B|2 ≤ ǫ−01IX0,t0 p .
σ∈[0,1] t∈(t0−(1−σ)2,t0) Mt∩B1−σ(X0) ! (cid:0) (cid:1)
For completeness, we give the proof in appendix which is based on Ecker’s proof. Let
us consider the mean curvature flow in Theorem 4.1 which starts from a self-shrinker. Let
M be a self shrinker, then the one-parameter family = √ tM is a mean curvature
t
M −
flow for 4 t < 0. In this case,
− ≤
IX0,t0 = sup ρ′2−ρ2 −n+22−p −ρ2 |B|p dt
√−t0≤ρ<ρ′≤2(cid:0) (cid:1) Z−ρ′2 Z√−tM∩B2(X0) !
1
(4.3) = sup ρ′2 ρ2 −n+22−p ρ B p 2 dr
√−t0≤ρ<ρ′≤2(cid:0) − (cid:1) Zρ1′ Zr1M∩B2(X0)| | ! r3
1
= sup 2 ρ′2 ρ2 −n+22−p ρ rp−n−3 B pdµ dr.
√−t0≤ρ<ρ′≤2 (cid:0) − (cid:1) Zρ1′ ZM∩B2r(rX0)| | !
For any 1 < t < 0 and X √ t M, I ǫ implies
−4 0 0 ∈ − 0 X0,t0 ≤ 0
1 2
sup sup B 2 ǫ 1I p .
(4.4) 4 | | ≤ −0 X0,t0
t∈(t0−14,t0) √−tM∩B12(X0) (cid:0) (cid:1)
Hence
2
(4.5) sup  sup |B|2 ≤ 4 ǫ−01IX0,t0 p .
t∈(2,(−t0)−1/2) 1tM∩B12(X0) (cid:0) (cid:1)
 
Now we have the following curvature estimates for self-shrinkers.
Theorem 4.2. Let M be an n dimensional proper self-shrinker in Rn+m. If for some
p [n,n+2) there is
∈
(4.6) lim B pdµ = 0,
R→∞ZM∩B2R\BR| |