Willmore minimizers withprescribed isoperimetric ratio

Johannes Schygulla*Mathematisches Institut der Albert-Ludwigs-Universitt Freiburg

Eckerstrae 1, D-79104 Freiburg, Germanyemail: johannes.schygulla@math.uni-freiburg.de

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

(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

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

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

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.

5

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

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.

7

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

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.

9

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

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

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

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

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