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Willmore minimizers withprescribed isoperimetric ratio
Johannes Schygulla*Mathematisches Institut der
Albert-Ludwigs-Universitt Freiburg
Eckerstrae 1, D-79104 Freiburg, Germanyemail:
[email protected]
Abstract: Motivated by a simple model for elastic cell
membranes,we minimize the Willmore functional among two-dimensional
spheresembedded in 3 with prescribed isoperimetric ratio.
Key words: Willmore functional, geometric measure theory.
MSC: 53 A 05, 49 Q 20, 49 Q 15, 74 G 65
1 Introduction
In the spontaneous curvature model for lipid bilayers due to
Helfrich [8], the mem-brane of a vesicle is described as a
two-dimensional, embedded surface 3,whose energy is given by
E() =
(H C0)2 d + G
K d,
where H, K denote the mean curvature and Gauss curvature, is the
induced areameasure and and G are constant bending
coefficients.
Restricting to surfaces of the type of the sphere, the second
term reduces to theconstant 4piG by the Gauss-Bonnet theorem.
Reducing further to the simplest caseof spontaneous curvature C0 =
0, the energy becomes up to a factor the Willmoreenergy
W() = 14
| ~H|2 d. (1.1)
According to [8], the shapes of the vesicles should be
minimizers of the elasticenergy E subject to prescribed area and
enclosed volume. Since the Willmoreenergy is scaling invariant, the
two constraints actually reduce to the condition thatthe
isoperimetric ratio of the surface , given by
I() =(6pi) 1
3 V()13
A() 12, (1.2)
is prescribed. Here A() denotes the area of and V() the volume
enclosed by, i.e. the volume of the bounded component of 3 \. The
normalizing constant
*J.Schygulla was supported by the DFG Collaborative Research
Center SFB/Transregio 71.
1
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(6pi ) 13 is chosen such that I() (0, 1], in particular I(2) =
1.For given (0, 1], we denote by M the class of smoothly embedded
surfaces 3 with the type of 2 and with I() = , and we introduce the
function
: (0, 1] +, () = infM
W().
We have M1 ={round spheres 3
}and (1) = 4pi.
Here we prove the following result.Theorem 1.1 For every (0, 1)
there exists a surface M such that
W() = ().Moreover the function is continuous, strictly
decreasing and satisfies
lim0
() = 8pi.
Assuming axial symmetry, several authors computed possible
candidates for mini-mizers by solving numerically the
Euler-Lagrange equations (see [1], [5]). In [12]the authors prove
existence of a one-parameter family of critical points
bifurcatingfrom the sphere. It appears that so far no global
existence results for the Helfrichmodel have been obtained. In
order to prove Theorem 1.1 we adopt the methodsof L. Simon in [14],
where he proved existence of Willmore minimizers for fixedgenus p =
1.
Moreover we show the following result.Theorem 1.2 Let {k}k (0,
1) such that k 0 and k Mk such thatW(k) = (k). After translation
and scaling (such that 0 k and H2(k) = 1),there exists a
subsequence k which converges to a double sphere in the sense
ofmeasures, namely
k in C0c (3),where k = H2xk and = 2H2xBr(a) for some r > 0
and a 3.We now briefly outline the content of the paper. In section
2 we prove that isdecreasing and () < 8pi for all (0, 1]. In
section 3 we prove Theorem 1.1.Section 4 is dedicated to the proof
of Theorem 1.2 using similar techniques as in theproof of Theorem
1.1. Finally in the appendix we collect some important resultswe
need during the proofs, as for example the graphical decomposition
lemma andthe Monotonicity formula proved by Simon in [14].This work
was done within the framework of project B.3 of the DFG
CollaborativeResearch Center SFB/Transregio 71. I would like to
thank my advisor Prof. ErnstKuwert for his support. I also would
like to express my gratitude for the support Ireceived from the DFG
Collaborative Research Center SFB/Transregio 71.
2 Upper bound for the InfimumIn this section we prove an upper
bound for the infimum of the Willmore energyin the class M . The
proof is based on the inversion of a catenoid at a spheretogether
with an argument involving the Willmore flow and its properties. A
ref-erence where the authors also analyze inverted catenoids and
their relation to theWillmore energy is [4]. For the part
concerning the Willmore flow see [10].
2
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Lemma 2.1 The function is decreasing and
() = infM
W() < 8pi for all (0, 1].
Proof: Define the (scaled) catenoid in 3 as the image of ga :
[0, 2pi) 3given by
ga(s, ) =(a cosh s
acos , a cosh s
asin , s
),
where a > 0 is a positive constant. Next we invert this
catenoid at the sphereB1(e3) to get the function fa = I ga, where
I(x) = e3 + xe3|xe3 |2 describes theinversion at the sphere. Define
the set a 3 by
a = fa([0, 2pi)
) {e3}.
graph u+graph u
U
Figure 1: fa(({0}) ({pi})) for a = 0.1. a results from
rotation.
First of all a is smooth away from e3. Because of the inverse
function theoremand by explicit calculation there exists an open
neighborhood U of e3 in whicha can be written as graph u+ graph u,
where u C1,(BR(0)) W2,p(BR(0))for all (0, 1), p 1 and some R >
0, and which are smooth away from theorigin. Moreover direct
calculation yields W(a) = 8pi. Since variations of aaway from e3
correspond to variations of the catenoid away from infinity and
sincethe catenoid is a minimal surface, the L2-gradient ~W( fa) of
the Willmore energyof fa satisfies
~W( fa) = 0 on (,) [0, 2pi).Since u C1,(BR(0)) W2,p(BR(0)) for
all (0, 1) and p 1, it follows thatthe L2-gradient of the Willmore
energy of graph u satisfies
~W(F) = 0 on BR(0) \ {0},
where F(x, y) = (x, y, u(x, y)). Let Cc (BR(0)) and define the
functionFt(x, y) = (x, y, u(x, y) + t(x, y)). For BR(0) denote by
W(Ft,) the Will-more energy of Ft restricted to . Because of the
given regularity of u and since
3
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spt BR(0) it follows thatddt W(F
t)|t=0 =
ddt W(F
t, BR(0))|t=0 = lim
0ddt W(F
t, BR(0) \ B(0))|t=0 .
Since ~W(F) = 0 on BR(0) \ B(0), it follows from the first
variation formula forthe Willmore energy that only a boundary term
remains. Exploiting this boundaryintegral yields
lim0
ddt W(F
t, BR(0) \ B(0))|t=0 = c(0),
where c > 0 is a positive constant. This shows that the first
variation of the Will-more energy of graph u+ is negative for
variations in the direction e3 and that thefirst variation of the
Willmore energy of graph u is negative for variations in
thedirection e3. Now notice that the isoperimetric ratio I(a) 0 as
a 0 andthat a can be parametrized over 2. After approximation by
smooth surfaces wehave therefore shown that for every > 0 there
exists a smooth, embedded surface 3 of the type of 2, with
isoperimetric ratio I() < and W() < 8pi. UsingTheorem 5.2 in
[10], the Willmore flow t with initial data exists smoothly forall
times and converges to a round sphere such that W(t) is decreasing
in t. Thisshows () < 8pi. In order to prove the monotonicity let
0 (0, 1) and > 0such that (0) + < 8pi. Let 0 M0 such that
W(0) (0) + . Again theWillmore flow t with initial data 0 exists
smoothly for all times, converges to around sphere and W(t) is
decreasing in t. Therefore for every (0, 1] thereexists a surface M
with W() W(0) (0) + and the lemma followsby letting 0.
3 Proof of Theorem 1.1For (0, 1) let {k}k M be a minimizing
sequence. Since the Willmoreenergy is invariant under translations
and scalings and in view of Lemma 2.1 wemay assume that for some 0
> 0
H2(k) = 1 , 0 k , W(k) 8pi 0. (3.1)
Using Lemma 1.1 in [14] we get an uniformly diameter bound for k
and therefore
k BR(0) for some R < . (3.2)
Define the integral, rectifiable 2-varifolds k in 3 by
k = H2xk. (3.3)
By a compactness result for varifolds (see [15]), there exists
an integral, rectifiable2-varifold in 3 with density (, ) 1 -a.e.
and weak mean curvature vector~H L2(), such that (after passing to
a subsequence) k in C0c (3) and
limk
X, ~Hk
dk = X, ~H d for all X C1c (3,3), (3.4)14
U| ~H|2 d lim inf
k14
U| ~Hk |2 dk 8pi 0 for all open U 3 . (3.5)
4
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Theorem A.1 and (3.5) applied to U = 3 yield
2(, x) [1, 2 0
4pi
]for all x spt . (3.6)
Since is integral we also get that
2(, x) = 1 for -a.e. x spt . (3.7)Our candidate for a minimizer
is given by
= spt . (3.8)Using Theorem A.1 we get (up to subsequences) as in
[14], page 310, that
k in the Hausdorff distance sense. (3.9)Therefore (3.2) and the
varifold convergence yield that BR(0) and (3) = 1.In order to prove
regularity we would like to apply Simons graphical decompo-sition
lemma Theorem B.1 to k simultaneously for infinitely many k .
Butthe most important assumption in the graphical decomposition
lemma is that theL2-norm of the second fundamental form is locally
small, which we will need si-multaneously for infinitely many k .
Therefore we define the so called badpoints with respect to a given
> 0 in the following way: Define the radon mea-sures k on
3 byk = kx|Ak|2.
From the Gauss-Bonnet formula and (3.1) it follows that k(3)
24pi. By com-pactness there exists a radon measure on 3 such that
(after passing to a subse-quence) k in C0c (3). It follows that spt
and (3) 24pi. Now wedefine the bad points with respect to > 0
by
B ={
({}) > 2} . (3.10)Since (3) c, there exist only finitely many
bad points. Moreover for 0 \Bthere exists a 0 < 0 = 0(0, ) 1
such that (B0(0)) < 22, and since k weakly as measures we
get
kB0 (0)|Ak|2 dH2 22 for k sufficiently large. (3.11)
Now fix 0 \ B and let 0 as in (3.11). Let B 02
(0). We want toapply Simons graphical decomposition lemma to
show that the surfaces k can bewritten as a graph with small
Lipschitz norm together with some "pimples" withsmall diameter in a
neighborhood around the point . This is done in exactly thesame way
Simon did in [14]. We just sketch this procedure. By (3.9) there
exists asequence k k such that k . In view of (3.11) and the
Monotonicity formulaapplied to k and k the assumptions of Simons
graphical decomposition lemma(see Theorem B.1 in the appendix) are
satisfied for 04 and infinitely manyk . Since W(k) 8pi 0, we can
apply Lemma 1.4 in [14] to deduce that for
(0, 12
)small enough,
(4 ,
2
)and infinitely many k only one of the discs
Dk,l appearing in the graphical decomposition lemma can
intersect the ball B 4 (k)(see Theorem B.1 for the notation).
Moreover, by a slight perturbation from k
to , we may assume that Lk for all k . Now Lk L in + G2(3),
andtherefore we may furthermore assume that the planes, on which
the graph functionsare defined, do not depend on k . After all we
get a graphical decomposition inthe following way.
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Lemma 3.1 For 0, 04 and infinitely many k there exist
pairwisedisjoint closed subsets Pk1, . . . , PkNk of k such
that
k B 8 () = Dk B 8 () =graph uk
n
Pkn
B 8 (),where Dk is a topological disc and where the following
holds:
1. The sets Pkn are topological discs disjoint from graph uk.2.
uk C(k, L), where L 3 is a 2-dim. plane such that L, andk =
(Bk() L
) \m dk,m. Here k > 4 and the sets dk,m L are
pairwisedisjoint closed discs.
3. The following inequalities hold:
m
diam dk,m +
n
diam Pkn c
kB2()|Ak|2 dH2
14
c 12 , (3.12)
||uk ||L(k) c16 + k where k 0, (3.13)
||D uk ||L(k) c16 + k where k 0. (3.14)
Now we leave the varifold context and define the functions
k = k BV(3),
where k 3 is the open, bounded set surrounded by k. We have
|| k ||L1(3) =3
6pi
and |D k |(U) = k(U) 1 for every open U 3 .
Therefore the sequence k is uniformly bounded in BV(3) and a
compactnessresult for BV-functions (see [6]) yields that (after
passing to a subsequence)
k in L1(3) and pointwise a.e.
for some function BV(3). Since the functions k are
characteristic functionswe may assume without loss of generality
that is the characteristic function of aset 3 with L3() = 36pi .
Because of the lower semicontinuity of the perime-ter on open sets
and the upper semicontinuity on compact sets under convergenceof
measures we get that
|D | as measures. (3.15)In the end we would like to have that =
is smooth. Therefore it is necessarythat |D | = as measures, which
actually holds.Lemma 3.2 In the above setting we have for 0 that |D
| = .
Proof: Let 0 \ B and 0 = 0(0, ) > 0 as in (3.11). Let 0 such
thatLemma 3.1 holds and let 04 . Let uk C1,1(Bk(0) L, L) be an
extension
6
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of uk to the whole disc Bk(0) L as in Lemma C.1, i.e. uk = uk in
k. From theL-bounds for the function uk and since k > 4 it
follows that
ukL(B 4
(0)L) c16 + k c,
D ukL(B 4
(0)L) c16 + k c.
Thus it follows that the sequence uk is equicontinuous and
uniformly bounded inC1(B
4(0) L, L) and W1,2(B 4 (0) L, L
). Therefore there exists a functionu C0,1(B
4(0) L, L) such that (after passing to a subsequence)
uk u in C0(B 4 (0) L, L),
uk u weakly in W1,2(B 4 (0) L, L),
1uL(B
4(0)L) + D uL(B
4(0)L) c
16 .
Let g C1c (B 8 (0),3) with |g| 1. It follows from the definition
of |D | that
|D |(B 8 (0)) div g = lim
k
k div g.
Lemma 3.1 yieldsk div g =
graph ukB 8 (0)
g, k dH2 +
n
PknB 8 (0)
g, k dH2,
where k denotes the outer normal to k = k. Because of the
Monotonicityformula and the diameter estimates for the sets Pkn we
can estimate the second termon the right hand side by
n
PknB 8 (0)
g, k dH2
n
H2(Pkn) c
n
(diam Pkn
)2 c2.Because of the diameter estimates for the sets dk,m and
the L-bounds for the func-tions uk the first term on the right hand
side can be estimated by
graph ukB 8 (0)
g, k dH2
graph ukB 8 (0)g, k dH2
c2,where k denotes the outer normal to graph uk. Using the
convergence stated abovetogether with the estimates for the limit
function u we get that
lim infk
graph ukB 8 (0)g, k dH2
(
8 c16
)2pi2
and therefore|D |(B 8 (0))
(
8 c 16
)2pi2 c2.
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In the same way, using that k in C0c (3), we get
(B 8 (0))
1 + c 13(
8 + c16
)2pi2 + c2.
Since the derivation D |D |(0) exists for -a.e. 0 (see [6]) we
get for -a.e.0 \ B that
D |D |(0) (1 c 16
)2pi c
1 + c 13(1 + c 16
)2pi + c
.
Letting = 1n 0 we get in view of (3.15)
D |D |(0) = 1 for -a.e. 0 \
n
B 1n.
Since each set B 1n
contains only finitely many points and since the
Monotonicityformula yields ({}) = 0 for every 3, we get that
D |D |(0) = 1 for -a.e. 0 3 .
Using the theorem of Radon-Nikodym the lemma follows from
(3.15).
Remark 3.3 Notice that the only thing we needed up to now was
the bound on theWillmore energy W(k) 8pi 0. We are now able to
prove that
lim0
() = 8pi. (3.16)
We already know that is decreasing and bounded by 8pi. Therefore
the limitexists. Let l 0 and assume (3.16) is false. After passing
to a subsequencethere exists a 0 > 0 such that (l) 8pi 0 for all
l . Let l Ml suchthat W(l) (l) + 02 8pi 02 and let l 3 be the open
set surroundedby l. Again after scaling and translation we may
assume that H2(l) = 1 and0 l, and that the radon measures l = H2xl
converge to a radon measure with (3) = 1. On the other hand we have
that the BV-functions l = l areuniformly bounded and therefore
converge (after passing to a subsequence) in L1to a BV-function .
Since I(l) 0 and H2(l) = 1 it follows that = 0. Finally,since W(l)
8pi 02 , we can do exactly the same as before to get = |D |,which
contradicts (3) = 1. Therefore (3.16) holds.
We continue with the proof of Theorem 1.1. The main idea to
prove regularityis to derive a power decay for the L2-norm of the
second fundamental form viaconstructing comparison surfaces by a
cut-and-paste procedure as done in [14].But this method cannot be
directly applied in our case, since the isoperimetricratio might
change by this procedure. In order to correct the isoperimetric
ratioof the generated surfaces, we will apply an appropriate
variation. But what is anappropriate variation in our case? Which
is the quantity we have to look at? Toanswer this question let : (,
) 3 3 be a C2-variation with compactsupport and define k,t = t(k),
k,t = k,t = t(k) and X(x) = tt(x)|t=0 . Itfollows that
ddt I(k,t)|t=0 =
I(k)3H2(k)
(32
X, ~Hk
dk + H2(k)L3(k)k div X
). (3.17)
8
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Because of (3.4) and since k , we get in view of Lemma 3.2
that
limk
ddt I(k,t)|t=0 =
3
X,
32~H 6
pi
3
d, (3.18)
where is given by the equation div g =
g,
d|D | = g, dfor g C1c (3,3). This follows from the Riesz
representation theorem applied toBV-functions and Lemma 3.2.
Now if there would exist a vector field X Cc (3,3) such that the
right hand sideof (3.18) is not equal to 0, we would have that the
first variation of the isoperimetricratio of k not equals 0 for k
sufficiently large, and in conclusion we would have achance to
correct the isoperimetric ratio of the generated surfaces. The next
lemmais concerned with the existence of such a vector field and
relies on the fact thateach surface M is not a round sphere.Lemma
3.4 There exists a R > 0 such that for every there exists a
point \ BR() such that for all > 0 there exists a vectorfield X
Cc (B(),3)such that
X,32~H 6
pi
3
d , 0.
Proof: Assume the statement is false. Then there exists a
sequence Rk 0 andk such that for all \ BRk(k) there exists a > 0
such that
X,32~H 6
pi
3
d = 0
for all X Cc (B(),3). Since is compact, it follows after passing
to a subse-quence that k , and since ({}) = 0, which follows from
Theorem A.1,we get after all that
~H(x) = 4pi
3(x) for -a.e. x . (3.19)
Now the idea of the proof is the following: We just have to show
that is smooth,because then would be a smooth surface with constant
mean curvature and Will-more energy smaller than 8pi. By a theorem
of Alexandroff would be a roundsphere which contradicts our choice
of (0, 1). To show that is smooth wejust have to show that 2(, x) =
1 for every x , because then Allards regularitytheorem would yield
(remember that ~H L() now) that can be written as aC1,-graph around
x that solves the constant mean curvature equation and is
there-fore smooth.
Let x0 . To prove that 2(, x0) = 1 notice that, since BV(3),
gener-ates an integer multiplicity, rectifiable 2-current M D2(3)
with M = 0.Denote by x0, the blow-ups of around x0. Now also the
blow-ups generateinteger multiplicity, rectifiable 2-currents Mx0 ,
with Mx0 , = 0. Moreover themass of Mx0 , of a set W 3 such that W
BR(0) is estimated in view of theMonotonicity formula by
MW(Mx0,) x0,(BR(0)) = 2(BR(x0)) cR2.
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By a compactness theorem for integer multiplicity, rectifiable
2-currents (see [15])there exists an integer multiplicity,
rectifiable 2-current Mx0 D2(3) such thatMx0 = 0 and (after passing
to a subsequence)
Mx0 , Mx0 for 0 weakly as currents.
Let x0 be the underlying varifold.
On the other hand there exists a stationary, integer
multiplicity, rectifiable 2-cone such that (after passing to a
subsequence)
x0, for 0 weakly as varifolds.
Now we get the following:1.) x0 : This follows from the lower
semicontinuity of the mass with re-
spect to weak convergence of currents and the upper
semicontinuity on com-pact sets with respect to weak convergence of
measures.
2.) 2(, ) 2 04pi everywhere: Since is a stationary 2-cone, the
Mono-tonicity formula yields for all z 3 and all B(0) such that
(B(0)) = 0
2(, z) 2(, 0) = (B(0))pi2
= lim inf0
x0 ,(B(0))pi2
.
Now since x0 ,(B(0))pi2
=(B(x0))pi()2 , it follows that
2(, z) 2(, x0), and theclaim follows from (3.6).
3.) 2(, ) = 1 -a.e.: This follows from 2.) since is integral.4.)
x0 = : Choose a point x 3 such that 2(, x) = 1. By Allards
regularity theorem there exists a neighborhood U(x) of x in
which canbe written as a C1,-graph, which is actually smooth since
is stationary.Moreover we get that the convergence x0 ,xU(x) xU(x)
is in C1,.Thus xU(x) x0 ,xU(x) x0xU(x), hence for U =
2(,x)=1 U(x)
we have that xU = x0xU. Since we already know that 2(, x) = 1
for-a.e. x 3 we get 4.).
From 4.) it follows that Mx0 is a stationary, integer
multiplicity, rectifiable 2-current with Mx0 = 0. Since moreover x0
= is a stationary, rectifiable2-cone we get for all > 0 and all
z 3 that
x0 (B(z))pi2
2(x0 , 0) 2 04pi
.
Letting we get that 2(x0 ,) 2 04pi . Using Theorem 2.1 in [9] it
followsthat Mx0 is a unit density plane or
x0 = = H2xP for some P G2(3).
Therefore we get for all balls B(0) such that (B(0)) = 0
2(, x0) = lim0
(B(x0))pi()2 = lim0
x0 ,(B(0))pi2
=(B(0))
pi2=H2xP(B(0))
pi2= 1
and the lemma is proved.
10
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In the next step we prove a power decay for the L2-norm of the
second fundamentalform on small balls around the good points \ B.
This will help us to showthat is actually C1, W2,2 away from the
bad points.Lemma 3.5 Let 0 \ B. There exists a 0 = 0(0, ) > 0
such that for all B 0
2(0) and all 04 we have
lim infk
kB 8 ()
|Ak|2 dH2 c,
where (0, 1) and c < are universal constants.
Proof: Choose according to Lemma 3.4 a R > 0 such that for
every thereexists a point \ BR() such that for all > 0 there
exists a vectorfieldX Cc (B(),3) such that
X,32~H 6
pi
3
d , 0. (3.20)
Let 0 \B, 0 > 0 as in (3.11). We may assume without loss of
generality that
0 0. (3.28)
12
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Notice that the constant c0 does not depend on , , or k .Let
C(3,3) be the flow of the vectorfield X, namely
t() = (t, ) C(3,3) is a diffeomorphism for all t ,(0, z) = z for
all z 3,t(t, z) = X((t, z)) for all (t, z) 3 .
Since spt X B R2() there exists a T0 = T0(X) > 0 such that
for all t (T0, T0)
t = Id in 3 \B R2().
Define the setstk = t( k) and tk = tk = t( k). (3.29)
Choosing T0 smaller if necessary (depending on X) it follows for
t (T0, T0) that
H2( tk) =
k
Jkt dH2 and L3( tk) =
k
det Dt.
By choosing T0 smaller if necessary (depending on X) and
estimating very roughlywe get that there exists a constant 0 < c
= c(X) < such that for all t (T0, T0)
(i) 1cH2( k) sup
t(T0,T0)H2( tk) cH2( k),
(ii) 1cL3( k) sup
t(T0,T0)L3( tk) cL3( k),
(iii) supt(T0,T0)
ddt H2( tk) + sup
t(T0 ,T0)
ddt L3( tk) c,
(iv) supt(T0,T0)
d2
dt2H2( tk)
+ supt(T0,T0) d
2
dt2L3( tk)
c,(v) sup
t(T0,T0)
ddt
tk
|Atk|2 dH2 c.
The last inequality can be proved by writing k locally as a
graph with small Lips-chitz norm and using a partition of
unity.
Now first of all it follows for the first variation of the
isoperimetric coefficient ofk, using that spt X B R
2() and k B R
2() = k B R
2(),
ddt I(
tk)|t=0 =I( k)
3H2( k)
k
X,
32~Hk H
2( k)L3( k)
k
dH2 .
Now it follows from (3.22)-(3.25) thatk
X,
(H2( k)L3( k)
6pi
3
)k
dH2
cH
2( k)L3( k)
H2(k)
L3(k)
H2(k) c,where c = c(X), and therefore (3.24) and (3.27)
yield
ddt I(
tk)|t=0 I( k)
3H2( k)
k
X,
32~Hk
6pi
3k
dH2 c.
13
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Sincek
X, 32 ~Hk
6pi
3k
dH2
X, 32 ~H 6pi
3
d c0, it follows from(3.24) and (3.27) that there exists a
constant 0 < c0 < independent of , , andk , such that for k
sufficiently large
ddt I(
tk)|t=0 c0 c. (3.30)
Moreover using the estimates (iii) and (iv) above it follows
that
supt(T0 ,T0)
d2
dt2I( tk)
c, (3.31)where c = c(X) < is a universal constant.Using
Taylors formula we get in view of (3.26) that for each k there
exists atk with |tk | c such that
I( tkk ) = .Therefore we get by construction that tkk M is a
comparison surface to k.Moreover it follows from (v) above that
tkk
|Atkk |2 dH2
k
|Ak|2 dH2 |tk | supt[tk ,tk]
ddt
tk
|Atk|2 dH2 c.
Since k is a minimizing sequence for the Willmore functional in
M and by theGauss-Bonnet theorem therefore a minimizing sequence
for the functional
|A|2,
we get k
|Ak|2 dH2
k
|Ak|2 dH2 +c + k with k 0.
Now by definition of k it follows thatDkC()
|Ak|2 dH2
graph wk|Ak|2 dH2 +c + k.
By definition of wk and the choice of we getgraph wk
|Ak|2 dH2 c
DkC 34
8
()\C16
()|Ak|2 H2 .
Since B 16 () C(), we get that (remember that Dk B 8 () = k B 8
())kB 16 ()
|Ak|2 dH2 ckB 8 ()\B 16 ()
|Ak|2 dH2 +c + k.
Now by adding c times the left hand side of this inequality to
both sides ("holefilling") we deduce the following:For 04 and
infinitely many k it follows that
kB 16 ()|Ak|2 dH2
kB 8 ()
|Ak|2 dH2 +c + k.
14
-
where = cc+1 (0, 1) is a fixed universal constant. If we let
g() = lim infk
kB 16 ()
|Ak|2 dH2
we get thatg() g(2) + c for all 0
4.
Now in view of Lemma C.2 it follows that
g() c for all 02
and the lemma is proved.
In the next step we want to do the same as in the proof of Lemma
3.2 where weconstructed a sequence of functions which converged
strongly in C0 and weakly inW1,2. But now with the estimate of
Lemma 3.5 we will get better control on thesequence.
So let B 02
(0). Define the quantity k() by
k() =kB2()
|Ak|2 dH2
and notice that by the choice of 0 and Lemma 3.5 we have
that
k() c2 and lim infk k() c for all 064 . (3.32)
Furthermore we get from Lemma 3.1 and the Monotonicity formula
thatm
diam dk,m ck()14 c 12 and
m
L2 (dk,m) ck() 12 2, (3.33)
n
diam Pkn ck()14 and
n
H2(Pkn
) ck()
12 2. (3.34)
Therefore for 0 we may apply the generalized Poincar inequality
Lemma C.3to the functions f = D j uk and = ck() 14 to get a
constant vector k with|k | c 16 + k c such that
k
|D uk k |2 c2k
|D2 uk |2 + ck() 14 2 supk
|D uk |2.
Sincek
|D2 uk |2 c
graph uk|Ak|2 dH2 c
kB2()
|Ak|2 dH2 ck(),
it follows for 0 that k
|D uk k|2 ck() 14 2. (3.35)
Let again uk C1,1(Bk ()L, L) be an extension of uk to the hole
disc Bk ()Las in Lemma C.1, i.e. uk = uk in k. We again have
that
ukL(Bk ()L) c16 + k c,
D ukL(Bk ()L) c16 + k c.
15
-
From the gradient estimates for the function uk, since |k | c,
from (3.33), (3.35)and the choice of 0 we get that
Bk ()L|D uk k|2 =
k
|D uk k |2 +
m
dk,m
|D uk k |2
ck()14 2
c 12 2,and therefore in view of (3.32) (we will always write
even if it might changefrom line to line) that
lim infk
Bk ()L
|D uk k |2 min{c2+, c
12 2
}for all 064 . (3.36)
Since k > 4 the sequence uk is therefore equicontinous and
uniformly bounded inC1(B
4() L, L) and W1,2(B
4() L, L) and we get the existence of a function
u C0,1(B 4 () L, L) such that (after passing to a
subsequence)
uk u in C0(B 4 () L, L),
uk u weakly in W1,2(B 4 () L, L),
1uL(B
4()L) + D uL(B
4()L) c
16 .
Remark: Be aware that the limit function depends on the point ,
since our se-quence comes (more or less) from the graphical
decomposition lemma (which is alocal statement) and therefore
depends on .Moreover we have that k with || c 16 . Since D uk D u
weakly inL2(B
4() L), it follows from lower semicontinuity and (3.36) that
B 4
()L|D u |2 c2+ c
12 2 for all 064 . (3.37)
In the next lemma we show that our limit varifold is given by a
graph around thegood points.
Lemma 3.6 For all B 02
(0) and all < 0512 we have that
xB() = H2x(graph u B()
),
where u C0,1(B 4 () L, L) is as above.
Proof: From the definition of uk it follows for 064 that
H2x(k B()
)= H2x
(Dk B()
)= H2x
(graph uk B()
)+H2x
(Dk \ graph uk B()
)H2x
(graph uk \ Dk B()
)= H2x
(graph uk B()
)+
k, (3.38)
16
-
where k is given by
k = H2x
(Dk \ graph uk B()
)H2x
(graph uk \ Dk B()
)= 1k 2k
is a signed measure. The total mass |k | of k, namely
1k (3) + 2k (3), can be
estimated in view of (3.32), (3.33) and (3.34) by
1k(3) + 2k (3)
n
H2(Pkn
)+
m
dk,m
1 + |D uk |2
ck()12 2
c2.
It follows from (3.32) thatlim inf
k
(1k (3) + 2k(3)
) c2+ c2. (3.39)
By taking limits in the measure theoretic sense we get that
xB() = H2x(graph u B()
)+ , (3.40)
where is a signed measure with total mass | | c2+ c 14 2. This
equationholds for all 064 such that
(B()
)= H2xgraph u
(B()
)= 0,
which holds for a.e. 064 .To prove (3.40) let U 3 open.1.) Let
064 such that
(B()
)= 0. Moreover assume that xB() (U) = 0.
Therefore ((U B()
))= 0 and we get k
(U B()
)
(U B()
). It
follows thatH2x
(k B()
)(U) xB()(U). (3.41)
2.) Let 064 such that H2xgraph u(B()
)= 0. Moreover assume that
H2x(graph u B()
)(U) = 0. We have that
H2x(graph uk B()
)(U) =
LUB()(x + uk(x))
1 + |D uk(x)|2.
It follows from the L-bounds for uk that
LUB()(x + uk(x))
1 + |D uk(x)|2
LUB()(x + u(x))
1 + |D u(x)|2
c
L
UB()(x + uk(x)) UB()(x + u(x))+
LUB()(x + u(x))
1 + |D uk(x)|2
1 + |D u(x)|2 .
Since uk u uniformly and H2xgraph u((U B()
))= 0 we first of all get
thatUB()(x + uk(x)) UB()(x + u(x)) for a.e. x L.
17
-
The dominated convergence theorem yieldsL
UB()(x + uk(x)) UB()(x + u(x)) 0.On the other hand we have
that
LUB()(x + u(x))
1 + |D uk(x)|2
1 + |D u(x)|2
c
LUB()(x + u(x)) |D uk(x) k | + c
LUB()(x + u(x)) |k |
+c
LUB()(x + u(x))
D u(x) .From the L-bound for u it follows that UB()(x+u(x)) = 0
if x < B(1c 16 )()Land we get that
(LUB()(x + u(x))
) 12
L2B(1c 16 )() L
12
c.
In view of (3.36), (3.37) and since k we get that
lim infk
LUB()(x + u(x))
1 + |D uk(x)|2
1 + |D u(x)|2 c2+ c 14 2,
and it follows after all that
H2x(graph uk B()
)(U) = H2x
(graph u B()
)(U) + k(U),
where k is a signed measure with lim infk |k | c2+ c
14 2. After passing
to a subsequence, the ks converge to some signed measure with
total mass| | c2+ c 14 2. Assume that (U) = 0. Then it follows that
k(U) (U)and therefore we get
limk
H2x(graph uk B()
)(U) = H2x
(graph u B()
)(U) + (U). (3.42)
3.) Since the ks were signed measures such that lim inf |k | c2+
c
14 2, they
converge in the weak sense (after passing to a subsequence) to a
signed measure with total mass | | c2+ c 14 2. Assuming that (U) =
0, we getk(U) (U).
Now by taking limits in (3.38) it follows that
xB()(U) = H2x(graph u B()
)(U) + (U), (3.43)
where = + is a signed measure with total mass | | c2+ c 14 2.
Noticethat this equation holds for every U 3 open such that
xB()(U) = H2x(graph u B()
)(U) = (U) = (U) = 0.
By choosing an appropriate exhaustion this equation holds for
arbitrary open setsU 3 and (3.40) follows.Now choose a radius
(
0128 ,
064
)such that (3.40) holds. We take a closer look
18
-
to two cases.
1.) Let x B 2(): Notice that by (3.5) and the choice of 0
W(, B 2(x)) lim inf
kW(k, B 2 (x)) c
kB()
|Ak|2 dH2 22.
Since 2(, ) 1 on spt , it follows for 0 from Theorem A.1
thatxB()
(B
2(x)
)=
(B
2(x)
) c2.
From (3.40), especially the bound on the total mass of , it
follows that
c2 H2(graph u B 2 (x)
)+ c
14 2.
Therefore H2(graph u B 2 (x)
)> 0 for 0 and thus x graph u.
2.) Let x graph u B 2 (): Write x = z + u(z). If y B 4 (z) L, it
follows fromthe estimates for u that y + u(y) B 2 (x) for 0.
Therefore we get that
H2xgraph u(B
2(x)
)B
2(x)(y + u(y)) c2.
As above it follows that x for 0.After all we get for 0
B() = graph u B() for all < 0256 . (3.44)
Moreover we get that the function u does not depend on the point
in the follow-ing sense: Let x B 0
2(0) and < 0256 . Then we have that
graph u (B() B(x)
)= graph ux
(B() B(x)
). (3.45)
In the next step choose (
0512 ,
0256
)such that
(B()
)= H2xgraph u
(B()
)= 0,
and that therefore due to (3.40)xB() = H2x
(graph u B()
)+ . (3.46)
Let x B() = graph u B() and > 0 such that B(x) B() and
suchthat (due to (3.40) for the point x)
xB(x) = H2x(graph ux B(x)) + x, (3.47)where x is a signed
measure with total mass smaller than c2+.
From (3.45), (3.46) and (3.47) it follows that (B(x)) = x
(B(x))
and we get a nice decay for the signed measure , namely
lim0
(B(x))2
= 0. (3.48)
19
-
Since we already know that 2(, ) 1 on , it follows from (3.46)
that
H2x(graph u B()
)(B(x))
xB() (B(x)) = 1 (B(x))
xB() (B(x)) ,
and by (3.48) the right hand side goes to 1 for 0. This shows
that
DxB()(H2x
(graph u B()
))(x) = 1
for all x B() = graph u B() and the lemma follows from the
theoremof Radon-Nikodym.
Now let 0 \ B. Since we already know that admits a generalized
meancurvature vector ~H L2(), it follows from the definition of the
weak mean cur-vature vector and by applying Lemma 3.6 to 0 (and
writing u for u0 ) that u isa weak solution of the mean curvature
equation, namely u is a weak solution inW1,20
(B 0512 (0) L, L
)
of
2i, j=1
j(
det g gi jiF)=
det g ~H F, (3.49)
where F(x) = x + u(x) and gi j = i j + iu ju.Since the norm of
the mean curvature vector can be estimated by the norm of thesecond
fundamental form, it follows from Lemma 3.5 and (3.5) applied to
B()that for all B 0
2(0) and all 0128
B()| ~H|2 d c.
Lemma 3.6 yields xB() = H2x(graph u B()
)for all points B 01024 (0)
and all 01024 and thereforegraph uB()
| ~H|2 dH2 c (3.50)
for all B 01024 (0) and all 0
1024 .
Using a standard difference quotient argument (as for example in
[7], Theorem8.8), it follows from (3.49) and ~H L2() that
u W2,2loc(B 01024 (0) L, L
).
Now let = , where L and W1,20(B 01024 (0) L
)is of the form
= f 2lu
?B(x)\B
2(x)L
lu
,where x B 02048 (0) L,
02048 and f Cc (B(x) L) such that 0 f 1,
f 1 on B 2(x) L and |D f | c .
20
-
By applying (3.49) to this test functions we get in view of
(3.50) thatB(x)L
|D2 u|2 c for all x B 02048 (0) L and all 0
2048 . (3.51)
From Morreys lemma (see [7], Theorem 7.19) it follows that
u C1,(B 02048 (0) L, L
) W2,2
(B 02048 (0) L, L
). (3.52)
Thus we have shown that our limit varifold can be written as a
C1,W2,2-graphaway from the bad points.
Now we will handle the bad points B and prove a similar power
decay as inLemma 3.5 for balls around the bad points. Since the bad
points are discrete andsince we want to prove a local decay, we
assume that there is only one bad point .
As mentioned in the definition of the bad points (see (3.10)),
the radon measuresk = kx|Ak|2 converge weakly to a radon measure ,
and it follows for all z 3that (B(z) \ {z}) 0 for 0. Therefore for
given > 0 there exists a 0 > 0such that
(B() \ {}) < 2 for all 0.Since k in C0c (3), it follows that
for < 0 and k sufficiently large
kB()\B 2
()|Ak|2 dH2 < 2. (3.53)
Moreover it follows from Theorem A.1 applied to our minimizing
sequence k and(3.2) that for all > 0
3 \B()
14 ~Hk(x) + (x )
|x |22
dk(x) 14piW(k) 2(k, ) c,
where c is an universal constant independent of k and . Here
denotes theprojection onto Txk. Rewriting the left hand side and
using Cauchy-Schwarz weget
3 \B()|(x )|2|x |4 dk(x) c,
where again c is an universal constant independent of k and .
Now we can use themonotone convergence theorem to get for 0 that
the integral |(x )|2
|x |4 dk(x)
exists for all k and is bounded by a uniform, universal constant
c independent of k.Moreover the function
fk = |(x )|2
|x |4 L1(k).
Now define the radon measures
k = fkxk.
21
-
It follows that k(3) c and therefore (after passing to a
subsequence) there existsa radon measure such that k in C0c (3).
Moreover (B() \ {}) 0 for 0. Therefore there exists a 0 such
that
(B() \ {}) < 4 for all 0.
Let < 0 and g C0c (B0() \ {}) such that 0 g 1 and g B()\B
2
(). Itfollows that
B()\B 2
()|(x )|2|x |4 dk(x)
g dk
g d (B0() \ {}) < 4.
Thus we get for k sufficiently large thatkB()\B
2()
|(x )|2|x |4 dH
2(x) 4.
Now let Bk ={
x k B0() |(x) ||x| >
}. It follows for < 0 and k
sufficiently large that
H2(k B() \ B 2 () Bk
) c22. (3.54)
Moreover by choosing 0 238pi we also get for < 0 and for k
large that
k B 34
() , . (3.55)
To prove this notice that due to the diameter estimate in Lemma
1.1 in [14] we have
diam k H2(k)W(k)
1
8pi.
Let k k such that k . It follows that k B 34
() , for k sufficientlylarge. Now suppose that k B 3
4() = . Since k is connected, we get that
k B 34
() and therefore diam k 32 < 320 18pi , a contradiction.
After all according to (3.53)-(3.55) the following is shown: For
< 0 and k sufficiently large we have that
(i)kB()\B
2()|Ak|2 dH2 < 2,
(ii) |(x )|
|x | for all x (k B() \ B 2 ()
)\ Bk,
where Bk k B0() with H2(k B() \ B 2 () Bk
) c2
and (x ) = (x ) PTxk (x ),(iii) k B 3
4() , .
Let zk k B 34
(). It follows thatkB
8(zk)|Ak|2 dH2
kB()\B
2()|Ak|2 dH2 2.
22
-
The Monotonicity formula applied to zk and k yields that we may
apply the graph-ical decomposition lemma to k, zk and infinitely
many k as well as Lemma1.4 in [14] to get as in Lemma 3.1 that
there exists a
(0, 12
)(independent of
j {1, . . . , P} and k ) and pairwise disjoint subsets Pk1, . .
. , PkNk k such that
k B 32 (zk) =graph uk
n
Pkn
B 32 (zk),where the following holds:
1. The sets Pkn are closed topological discs disjoint from graph
uk.2. uk C(k, Lk ), where Lk 3 is a 2-dim. plane such that zk Lk
andk =
(Bk(zk) Lk
) \ m dk,m, where k > 16 and where the sets dk,m arepairwise
disjoint closed discs in Lk.
3. The following inequalities hold:
m
diam dk,m +
n
diam Pkn c12 , (3.56)
ukL(k) c16 + k where limk k = 0, (3.57)
D ukL(k) c16 + k where limk k = 0. (3.58)
In the next step we show that
dist (, Lk) c(
16 + k
). (3.59)
To prove this notice first of all that it follows from Theorem
A.1 applied to zk, kand (i) above that for 0
H2(k B 32 (zk)) c2 with c independent of k. (3.60)
Now to prove (3.59) notice that(graph uk B 32 (zk)
)\ Bk , ,
where Bk k B0() is the set in (ii) above. This follows from the
graphicaldecomposition above, the diameter estimates for the sets
Pkn, the area estimate con-cerning the set Bk in (ii) and
(3.60).Let z
(graph uk B 32 (zk)
)\ Bk
(k B() \ B 2 ()
)\ Bk. It follows from (ii)
that| pi(z+Tzk)()| |z | (|z zk | + |zk |) c.
Define the perturbed 2-dim. plane Lk by Lk = Lk + (z piLk(z)),
where we have thatdist( Lk, Lk) = |z piLk (z)| c
16 (since z graph uk B 32 (zk)). Now it follows
from Pythagoras that |z piLk(pi(z+Tzk)())|2 |z pi(z+Tzk)()|2 |z
|2 c2.
Since z + Tzk can be parametrized in terms of D uk(z) over Lk,
we get that
|pi(z+Tzk)() pi Lk(pi(z+Tzk)())| D ukL |z pi Lk (pi(z+Tzk)())|
c(
16 + k
).
23
-
Therefore we finally get that
dist(, Lk) = piLk()
piLk(pi(z+Tzk)())
pi(z+Tzk)() + pi(z+Tzk)() pi Lk (pi(z+Tzk)())+pi
Lk(pi(z+Tzk)()) piLk (pi(z+Tzk)())
c(
16 + k
),
and (3.59) is shown.Since dist(, Lk) c
(
16 + k
), we may assume (after translation) that Lk
for all k and keeping the estimates for uk. Moreover we again
have thatLk L = 2-dim. plane with L. Therefore for k sufficiently
large we mayassume that Lk is a fixed 2-dim. plane L.
Define the set
Tk =
(
64 ,
2 32
) B(zk) m
dk,m = .
It follows from the diameter estimates and the selection
principle in [14] that for 0 there exists a
(64 ,
232
)such that Tk for infinitely many k .
Since L it follows from the choice of that for 0
B 34
() B(zk) L = {p1,k, p2,k} ,where p1,k, p2,k
(B
232(zk) L
)\m dk,m are distinct.
Define the image points zi,k graph uk by
zi,k = pi,k + uk(pi,k).
Using the L-estimates for uk we get for 0 that 58 < |zi,k |
< 78 andtherefore
kB 8
(zi,k)|Ak|2 dH2
kB()\B
2()|Ak|2 dH2 < 2.
Therefore we can again apply the graphical decomposition lemma
to the points zi,k.Thus there exist pairwise disjoint subsets Pi,k1
, . . . , Pi,kNi,k k such that
k B 32 (zi,k) =graph ui,k
n
Pi,kn
B 32 (zi,k),where the following holds:
1. The sets Pi,kn are closed topological discs disjoint from
graph ui,k.2. ui,k C(i,k, Li,k), where Li,k 3 is a 2-dim. plane
such that zi,k Li,k andi,k =
(Bi,k (zi,k) Li,k
)\m di,k,m, where i,k > 16 and where the sets di,k,m
are pairwise disjoint closed discs in Li,k.
24
-
3. The following inequalities hold:
m
diam di,k,m +
n
diam Pi,kn c12 , (3.61)
ui,kL(i,k) c16 + i,k where limk i,k = 0, (3.62)
D ui,kL(i,k) c16 + i,k where limk i,k = 0. (3.63)
Since dist(zi,k, L) c 16 + k (this follows since zi,k graph uk)
and since theL-norms of uk and ui,k are small, we may assume (after
translation and rotationas done before) that Li,k = L.By continuing
with this procedure we get after a finite number of steps,
dependingnot on and k , an open cover of B 3
4() L which also covers the set
B ={
x L dist (x, B 3
4() L
)<
2 64
}
and which include finitely many, closed discs dk,m withm
diam dk,m c12 .
We may assume that these discs are pairwise disjoint since
otherwise we can ex-change two intersecting discs by one disc whose
diameter is smaller than the sumof the diameters of the
intersecting discs.
Because of the diameter estimate and again the selection
principle there exists a
(
128 ,
264
)such that
{x L
dist (x, B 34
() L)=
}m
dk,m = .
Finally we get the following: There exist pairwise disjoint
subsets Pk1, . . . , PkNk ksuch that
k A() =graph uk
n
Pkn
A(),where the following holds:
1. The sets Pkn are closed topological discs disjoint from graph
uk.2. uk C(Ak(), L), where L 3 is a 2-dim. plane with L.3. The set
Ak() is given by
Ak() ={
x L dist (x, B 3
4() L
)<
}\m
dk,m,
where (
128 ,
264
)and where the sets dk,m are pairwise disjoint closed
discs in L which do not intersect{x L
dist (x, B 34
() L)=
}.
25
-
4. The set A() is given by
A() ={
x + y 3 x L, dist (x, B 3
4() L
)< , y L, |y| < 64
}.
5. The following inequalities hold:m
diam dk,m +
n
diam Pkn c12 , (3.64)
ukL(Ak()) c16 + k where limk k = 0, (3.65)
D ukL(Ak()) c16 + k where limk k = 0. (3.66)
From the estimates for the function uk and the diameter
estimates for the sets Pknwe also get for 0 and k sufficiently
large that
k A() {
x + y 3 x L, dist (x, B 3
4() L
)< , y L, |y| <
128
}.
Since k in the Hausdorff distance sense it follows that
, A() {x + y 3
x L, dist (x, B 34
())< , y Lk , |y| <
128
}.
Now we show that for all < 0 (after choosing 0 smaller if
necessary) A() B( 34+ 256 )() \ B( 34 256 )() = B( 34+ 256 )() \ B(
34 256 )().
To prove this notice that due to Theorem A.1
2(, x) 14pi
W() 2 04pi
for all x 3 .
Now assume that our claim is false, i.e. there exists a sequence
l 0 such that( B( 34+ 256 )l() \ B( 34 256 )l()
)\ A(l) , for all l.
Since we already know that can locally be written as a C1,
W2,2-graph awayfrom the bad point we get that B 3
41() contains two components 1 and 2
such that 1 2 = {}. Since i can locally be written as a C1,
W2,2-graph inB 3
41() \ {}, we get that 2(i, x) = 1 for all x , , and by upper
semicontinuity
that 2(i, ) 1. Therefore it follows that 2(, ) 2(1, ) + 2(2, )
2, acontradiction and the claim follows.
From this and k we get for < 0 and k sufficiently large
that
k A() B( 34+ 512 )() \ B( 34 512 )() = k B( 34+ 512 )() \ B( 34
512 )().Define the set
Ck =s
(0,
1024
) B 34+s() L m
dk,m = .
The diameter estimates for the discs dk,m yield for 0 that
L1(Ck) 2048 .The selection principle in [14] yields that there
exists a set C
(0, 1024
)with
26
-
L1(C) 2048 and such that every s C lies in Ck for infinitely
many k .Now define the set
Dk =
s C
graph uk |B 34 +s
()L
|Ak|2 dH2 4096
kA()
|Ak|2 dH2 .
By a simple Fubini-type argument (as done before) it follows
that L1(Dk) 4096 ,and again by the selection principle there exists
a s
(0, 1024
)such that s Dk
for infinitely many k . It follows that uk is defined on the
circle B 34+s
() Land that graph uk |B 3
4 +s()L divides k into two connected topological discs k1,
k2,
one of them, w.l.o.g. k1, intersecting B 34().From the estimates
for the function uk and the choice of s we have
graph uk |B 34 +s
()L A() B( 34+ 512 )() \ B( 34 512 )().
From this inclusion and
k A() B( 34+ 512 )() \ B( 34 512 )() = k B( 34+ 512 )() \ B( 34
512 )()
we get thatk1 B( 34+ 512 )(),
and the Monotonicity formula yields
H2(k1) c2.
According to Lemma C.1 let wk C(B 3
4+s() L, L
)be an extension of uk
restricted to B 34+s
() L. In view of the estimates for uk and therefore for wk weget
that
graph wk B( 34+ 512 )().Now we can define the surface k by
k = k \ k1 graph wk.
By construction we have that k is an embedded and connected
C1,1-surface withgenus k = 0, which surrounds an open set k 3.The
problem is again that k might not be a comparison surface. But we
can do thesame correction as done before in Lemma 3.5 to get for
all 0
lim infk
kB( 34 512 )()
|Ak|2 dH2 c (3.67)
Thus by definition of the bad points could not have been a bad
point and thereforethe set of bad points is empty. Thus we have
shown that for every point thereexists a radius > 0, a 2-dim.
plane L and a function
u C1,(B() L
) W2,2
(B() L
)(3.68)
27
-
for some (0, 12
), with
B(x)L|D2 u|2 c2 (3.69)
for all x B() L and all < such that B(x) B() L, and such that
B() = graph u B(). (3.70)
By definition of and an approximation argument we have
W() infM
W( ) = infC1W2,2 ,I( )=
W( ).
On the other hand we have that the first variation of the
isoperimetric ratio of isnot equal to 0 as shown in Lemma 3.4.
Therefore there exists a Lagrange multiplier such that for all Cc
((, ) 3,3) with (0, ) = 0
ddt(W(t()) I(t())
)|t=0
= 0. (3.71)
Restricting to Cc ((, )B(),3) and using the graph representation
(3.70)this yields after some computation that u is a weak solution
of
kl(Ai jkl(D u) i ju
)+ i Bi(D u,D2 u) =
(i Ci(D u) + C0
)(3.72)
for some coefficients Ai jkl,Bi,Ci and C0 that perfectly fits
into the scheme ofLemma 3.2 in [14]. Since by (3.69) u fulfills the
assumptions of this lemma, weget by a bootstrap argument that u is
actually smooth.
Therefore we have finally shown that can locally be written as a
smooth graphand we get that H2() = (3) = 1 and = . As mentioned
before has theright volume and therefore has the right
isoperimetric ratio, especially M .Finally (3.5) yields that is a
minimizer of the Willmore energy in the set M andtherefore the
existence part of Theorem 1.1 is proved.
Last but not least we have to show that the function is
continuous and strictlydecreasing. For that let 0 < 0 < 1.
Choose according to the above 0 M0 suchthat W(0) = (0). As in
section 2 the Willmore flow t with initial data 0 ex-ists smoothly
for all times and converges to a round sphere. By a result of
Bryant in[3], which states that the only Willmore spheres with
Willmore energy smaller than8pi are round spheres, it follows that
W(t) is strictly decreasing in t. Thereforefor every (0, 1] there
exists a surface M with W()
-
4 Convergence to a double sphereIn this section we prove the
convergence to a double sphere stated in the introduc-tion in
Theorem 1.2. For that let k (0, 1) such that k 0. Choose
accordingto Theorem 1.1 surfaces k Mk such that W(k) = (k) 8pi.
After scalingand translation we may assume that 0 k and H2(k) = 1.
As in section 3 itfollows (after passing to a subsequence) that
k = H2xk in C0c (3),
where is an integral, rectifiable 2-varifold in3 with compact
support, (, ) 1-a.e. and weak mean curvature vector ~H L2(), such
that
W() lim infk
W(k) = lim infk (k) = 8pi.
The last equation follows from Theorem 1.1. Moreover we get as
in section 3 that
k spt in the Hausdorff distance sense.
Define again the bad points B with respect to a given > 0 as
in (3.10).As before there exist only finitely many bad points and
for every 0 spt \ Bthere exists a 0 = 0(0, ) > 0 such that
kB0 (0)|Ak|2 dH2 22 for infinitely many k .
Let 0 spt \ B and choose a sequence k k such that k 0. For
ksufficiently large we may apply the graphical decomposition lemma
to k, k and 2 , Lk, j is a 2-dim. plane and
the sets dk, j,m are pairwise disjoint closed discs in Lk, j,
and such that we have theestimates
m
diam dk, j,m +
n
diam Pkn c12 and 1
ukjL(k, j) + DukjL(k, j) c
16 .
We claim that for (0, 1) and all k sufficiently large (depending
on , )
graph ukj B 2 (k) , for at least two j {1, . . . , Jk}.
(4.1)
Suppose this is false. Notice that at least one graph has to
intersect with B 2 (k)since k k and because of the diameter
estimates for the Pkns. After passing to asubsequence we may assume
that
k B 2 (k) =graph uk
Nkn=1
Pkn
B 2 (k)29
-
and B 4 (0) B 2 (k) for all k. Let k = k , where k is the open
set surroundedby k. Since the isoperimetric ratio I(k) 0, it
follows that k 0 in L1. Letg C1c (B 4 (0),
3). We get thatk
g, k
dH2 = k div g 0, (4.2)where k is the outer normal to k = k. By
assumption we have
k
g, k
dH2 = graph ukB 4 (0)
g, k
dH2 +n
PknB 4 (0)
g, k
dH2 .The Monotonicity formula and the diameter estimates yield
that the second term isbounded by c2. Choose g = e3, where C1c (B 4
(0)) such that B 8 (0).We get (by choosing the right sign and after
rotation)
graph ukB 4 (0)
g, k
dH2 (Bk (piLk (k))Lk
)\m dk,m B 8 (0)(x + uk(x)).
It follows from the diameter estimates for the sets Pkn and the
bounds on uk thatdist(k, Lk) c 16 . Since k 0 we get for 0 and k
sufficiently large thatB
8
(0)(x+uk(x)) = 1 if x (B 16 (piLk (k)) Lk
)\m dk,m. The diameter estimates
for the discs dk,m finally yieldk
g, k
dH2 c2 c2. In view of (4.2) wearrive for 0 at a
contradiction.Now let < 02 such that (B 2 (0)) = 0 and therefore
k(B 2 (0)) (B 2 (0)).Let , (0, 12 ). For k sufficiently large we
may assume that B(1) 2 (k) B 2 (0)and by (4.1)
graph uk1 B(1) 2 (k) , and graph uk2 B(1) 2 (k) , .
Let xkj graph ukj B(1) 2 (k). In view of the diameter estimates
for the sets Pkn
we get
k(B 2 (0)) 2
j=1
(Bk, j (piLk, j (k))Lk, j
)\m dk, j,m B(1)(1)2 (xkj )(x + u
kj(x)) c2.
Since xkj graph ukj B(1) 2 (k) we have that xkj = z
kj + u
kj(zkj) with zkj Lk, j
such that |zkj piLk, j(k)| (1 )2 . Therefore B(1)(1) 2 (zkj) Bk,
j(piLk, j (k)).
Moreover it follows from the bounds for ukj that B(1)(1)2 (xkj
)(x + u
kj(x)) = 1 if
|x zkj | < (1)(1)1+c 162 . Therefore after all we get in view
of the diameter estimates
for the discs dk,m, j that
k(B 2 (0)) 2( (1 )(1 )
1 + c 16
)2pi(
2
)2 c2 3
2pi(
2
)2for 0 and , sufficiently small. Thus for all 0 spt \ B
(B 2(0)) 32pi
(
2
)2.
30
-
Now since the density exists everywhere by Theorem A.1, since is
integral andsince (B) = 0 (which follows from the Monotonicity
formula) we have shownthat 2(, ) 2 -a.e.. Since W() 8pi, the
Monotonicity formula in Theo-rem A.1 yields 2 2(, ) 14pi W() 2
-a.e. and therefore
2(, ) = 2 -a.e. and W() = 8pi.
Now define the new varifold =
12.
It follows that is a rectifiable 2-varifold in 3 with compact
support spt = spt and weak mean curvature vector ~H = ~H L2(), such
that 2(, ) = 1 -a.e.and W() = 4pi. The next lemma yields that is a
round sphere in the sense that = H2xBr(a) for some r > 0 and a
3. Therefore is a double sphere asclaimed and Theorem 1.2 is
proved.
Lemma 4.1 Let , 0 be a rectifiable 2-varifold in 3 with compact
support andweak mean curvature vector ~H L2() such that
(i) 2(, x) = 1 for -a.e. x 3,(ii) W() = 1
4
| ~H|2 d 4pi.
Then is a round sphere, namely = H2xBr(a) for some r > 0 and
a 3.
Proof: From Theorem A.1 it follows that the density exists
everywhere and that2(, x) 1 for all x spt . But then Theorem A.1
yields
W() = 4pi and 2(, x) = 1 for all x spt . (4.3)
Since , 0 it follows from Theorem A.1 that there exists a R >
0 such thatspt \ BR(x) , for all x 3. Let x0 spt . Since spt is
compact it followsfrom Theorem A.1 that | ~H(x)| = 4
(xx0)|xx0 |2 8R for -a.e. x spt \ B R2 (x0). On
the other hand by choosing x1 spt \ BR(x0) it follows that |
~H(x)| 8R for -a.e.x spt \ B R
2(x1). Since B R
2(x0) B R
2(x1) = it follows that | ~H(x)| 8R for -a.e.
x spt and therefore ~H L(). Using Allards regularity theorem
(Theorem24.2 in [15]) we see that spt can locally be written as a
C1,-graph u for some (0, 1). As in (3.49) it follows that u is a
weak solution of
2i, j=1
j(
det g gi jiF)=
det g ~H F
where F(x) = x + u(x), gi j = i j + iu ju. Since ~H Lp() for
every p 1,it follows from a standard difference quotient argument
(as for example in [7],Theorem 8.8) that u W2,p for every p 1 and
therefore
B|D2 u|2 c. (4.4)
From a classical result of Willmore [17] and an approximation
argument we get
W() = 4pi infsmooth
W() = infC1W2,2
W().
31
-
Therefore u solves the Euler-Lagrange equation (3.72) (but with
= 0 since we donot have any constraints) and again with the power
decay in (4.4) and Lemma 3.2in [14] it follows that u is smooth.
Thus spt is a smooth surface with Willmoreenergy 4pi and therefore
a round sphere due to Willmore [17].
Appendix
A Monotonicity formula
Following L. Simon [14] and Kuwert/Schtzle [10] we state here a
Monotonicityformula for rectifiable 2-variolds in 3 with square
integrable weak mean curva-ture vector ~H L2(). We use the
notation
2(,) = lim inf
(B(0))pi2
, W(, E) = 14
E| ~H|2 d for E 3 Borel.
Theorem A.1 Assume that ~H(x) Tx for -a.e. x 3. Then the
density
2(, x) = lim0
(B(x))pi2
exists for all x 3
and the function 2(, ) is upper semicontinuous. Moreover if 2(,)
= 0, thenwe have for all x0 3 and all 0 < <
(B(x0)) c2,
2(, x0) c((B(x0))
pi2+W(, B(x0))
),
B(x0)\B(x0)
14 ~H(x) + (x x0)
|x x0|22
d(x) 14pi
W() 2(, x0),
where denotes the projection onto Tx.
Remark A.2 Brakke proved in chapter 5 of [2] that ~H is
perpendicular for anyintegral varifold with locally bounded first
variation. Therefore the statements ofthis section apply to
integral varifolds with square integrable weak mean
curvaturevector.
B The graphical decomposition lemma of L. SimonHere we state the
graphical decomposition lemma of Simon proved in [14].
Theorem B.1 Let n be a smooth surface. For given and > 0
let
(i) B() = ,
(ii) H2( B()
) 2 for some > 0,
(iii)B()
|A|2 dH2 2.
32
-
Then there exists a 0 = 0(n, ) > 0 such that if 0 there exist
pairwise disjointclosed subsets P1, . . . , PN of such that
B 2() =
J
j=1graph u j
Nn=1
Pn
B 2 (),where the following holds:
1. The sets Pn are topological discs disjoint from graph u j.2.
u j C
( j, Lj
), where L j n is a 2-dim. plane and
j =(B j(piL j()) L j
)\
Mm=1
d j,m,
where j > 2 and the sets d j,m are pairwise disjoint closed
discs in L j.3. Let
(4 ,
2
)such that B() is transversal and B()
(Nn=1 Pn
)= .
Denote by {l}Ll=1 the components of B 2 () such that l B 8 () ,
. Itfollows (after renumeration) that
l B() = D,l =graph ul
Nn=1
Pn
B(),where D,l is a topological disc.
4. The following inequalities hold:M
m=1diam d j,m c(n)
B()
|A|2 dH2
14
c(n) 12 ,
Nn=1
diam Pn c(n, )
B()|A|2 dH2
14
c(n, ) 12 ,
1u jL( j) + Du jL( j) c(n)
12(2n3) .
C Useful resultsIn this section we state some useful results we
need for the proof of Theorem 1.1.Lemma C.1 is an extension result
adapted to the cut-and-paste procedure we useand is proved in
[13].Lemma C.1 Let L be a 2-dim. plane in n, x0 L and u C
(U, L
), where
U L is an open neighborhood of LB(x0). Moreover let |D u| c on
U. Thenthere exists a function w C(B(x0), L) such that
(i) w = u and w
=u
on B(x0),
(ii) 1||w||L(B(x0)) c(n)
(1||u||L(B(x0)) + ||D u||L(B(x0))
),
(iii) ||D w||L(B(x0)) c(n)||D u||L(B(x0)),
(iv)
B(x0)|D2 w|2 c(n)
graph u|B(x0)
|A|2 dH1 .
33
-
Proof: After translation and rotation we may assume that x0 = 0
and L = 2 {0}.Moreover we may assume that = 1, the general result
follows by scaling.
Let C(B1(0)) be a cutoff-function such that 0 1, = 1 on B 12(0),
= 0
on B1(0)\B 34(0) and |D | + |D2 | c(n), and define the function
w1 C(B1(0))
by
w1(x) = (1 (x)) u(
x
|x|
)+ (x)
?B1(0)
u.
It follows thatw1 = u,
w1
= 0 on B1(0),||w1||L(B1(0)) c(n)||u||L(B1(0)), ||D w1||L(B1(0))
c(n)||D u||L(B1(0)).
Using the Poincar-inequality we also getB1(0)
|D2 w1|2 c(n)||u||2W2,2 (B1(0)).
Next let w2 C(B1(0)) be the unique solution of the elliptic
boundary valueproblem given by
w2 = 0 in B1(0), w2 = u
on B1(0).
The solution w2 is explicitly given by
w2(x) = 12piB1(0)
1 |x|2|x y|2
u
(y) dy.
Using standard estimates it follows that
||w2||L(B1(0)) ||D u||L(B1(0)), |D w2(x)| 6
1 |x|2 ||D u||L(B1(0)),
||w2||2W1,2(B1(0)) c(n)(||D u||2L2(B1(0)) + ||D
2 u||2L2(B1(0))).
Next let w3 C(B1(0)) be given by
w3(x) = 12(|x|2 1
)w2(x).
It follows that
w3 = 0,w3
(x) = w2(x) = u
(x) on B1(0),
||w3||L(B1(0)) c||w2||L(B1(0)) c||D u||L(B1(0)),
||D w3||L(B1(0)) c||D u||L(B1(0)).Moreover
w3(x) = w2(x) + x D w2(x) in B1(0).Using again standard
estimates it follows that
B1(0)|D2 w3|2 c
(||D u||2L2(B1(0)) + ||D
2 u||2L2(B1(0))).
34
-
Finally define w C(B1(0)) by
w(x) = w1(x) + w3(x).
The properties of w1 and w3 yield
w = u,w
=u
on B1(0),
||w||L(B1(0)) c(||u||L(B1(0)) + ||D u||L(B1(0))
),
||D w||L(B1(0)) c||D u||L(B1(0)),B1(0)
|D2 w|2 c||u||2W2,2(B1(0)).
By subtracting an appropriate linear function from w, using
again the Poincar-inequality and the assumption |D u| c we can get
a better estimate for the L2-norm of D2 w, namely
B1(0)|D2 w|2 c
B1(0)
|D2 u|2 c
graph u|B1(0)|A|2,
and the lemma is proved.
The second lemma is a decay result we need to get a power decay
for the L2-normof the second fundamental form.
Lemma C.2 Let g : (0, b) [0,) be a bounded function such
that
g (x) g(2x) + cx for all x (0, b
2
),
where > 0, (0, 1) and c some positive constant. There exists
a (0, 1) anda constant c = c
(b, ||g||L(0,b)) such thatg(x) cx for all x (0, b) .
The last statement is a generalized Poincar inequality proved by
Simon in [14].
Lemma C.3 Let > 0, (0, 2
)and = B(0)\E, where E is measurable with
L1(p1(E)) 2 and L1(p2(E)) where p1 is the projection onto the
x-axis andp2 is the projection onto the y-axis. Then for any f C1()
there exists a point(x0, y0) such that
| f f (x0, y0)|2 C2
|D f |2 +C sup
| f |2,
where C is an absolute constant.
35
-
References
[1] Berndl, K., Lipowsky, R., Seifert, U., Shape transformations
of vesicles:Phase diagrams for spontaneous-curvature and
bilayer-coupling models,Phys. Rev. A 44, 1182-1202, 1991
[2] Brakke, K., The motion of a surface by its mean curvature,
Princeton Univ.Press, Princeton 1978
[3] Bryant, R. L., A duality theorem for Willmore surfaces, J.
Differential Geom.20 1984, no. 1, 23-53
[4] Castro-Villarreal, P., Guven, J., Inverted catenoid as a
fluid membrane withtwo points pulled together, Phys. Rev. E 76,
2007
[5] Deuling, H.J., Helfrich, W., Red blood cell shapes as
explained on the basisof curvature elasticity, Biophys. J. 16,
1976, 861-868
[6] Evans, L.C. and Gariepy, R.F., Measure Theory and Fine
Properties ofFunctions, Studies in Advanced Mathematics 1992
[7] Gilbarg, D. and Trudinger, N.S., Elliptic Partial
Differential Equations ofSecond Order, Springer 2001
[8] Helfrich, W., Elastic Properties of Lipid Bilayers: Theory
and Possible Ex-periments, Zeitschrift fr Naturforschung C - A
Journal of Biosciences, Vol.C 28, 693-703, 1973
[9] Kuwert, E., Li, Y., Schtzle, R., The large genus limit of
the infimum of theWillmore energy, Am. J. Math. 132, No. 1, 37-51
(2010)
[10] Kuwert, E., Schtzle, R., Removability of point
singularities of Willmoresurfaces, Ann. of Math. (2) 160 (2004),
No. 1, 315-357
[11] Li, P., Yau, S.T., A new conformal invariant and its
applications to the Will-more conjecture and the first eigenvalue
of compact surfaces, Invent. Math.69, 269-291 (1982)
[12] Nagasawa, T., Takagi, I., Bifurcating critical points of
bending energy underconstraints related to the shape of red blood
cells, Calc. Var. Partial Differ-ential Equations 16 (2003), no. 1,
63-111
[13] Schygulla, J., Flchen mit L2-beschrnkter zweiter
Fundamentalform nachLeon Simon, Diplomarbeit an der
Albert-Ludwigs-Universitt Freiburg(2008)
[14] Simon, L., Existence of Surfaces minimizing the Willmore
Functional, Com-munications in Analysis and Geometry (1993),
281-326
[15] Simon, L., Lectures on geometric measure theory,
Proceedings of the Centrefor Mathematical Analysis, Australian
National University, Vol. 3, 1983
[16] Thomsen, G., ber konforme Geometrie I: Grundlagen der
konformenFlchentheorie, Hamb. Math. Abh. 3, 31-56, 1923
[17] Willmore, T., Total curvature in Riemannian Geometry, Wiley
1982
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