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A Convergence Theorem for Immersions with <span class="mathjax-formula">$L^2$</span>-Bounded Second Fundamental Form

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A Convergence Theorem for Immersions with L

2

-Bounded Second Fundamental Form

CHEIKHBIRAHIMNDIAYE(*) - REINERSCHAÈTZLE(*)

ABSTRACT- In this short note, we prove a convergence theorem for sequences of immersions from some closed surfaceSinto some standard Euclidean spaceRn withL2-bounded second fundamental form, which is suitable for the variational analysis of the famous Willmore functional, wheren3. More precisely, under some assumptions which are automatically verified (up to subsequence and an appropriate MoÈbius transformation ofRn) by sequences of immersions from some closed surfaceSinto some standard Euclidean spaceRnarising from an appropriate stereographic projection ofSnintoRnof immersions fromSintoSn and minimizing theL2-norm of the second fundamental form withn3, we show that the varifolds limit of the image of the measures induced by the se- quence of immersions is also an immersion with some minimizing properties.

1. Introduction and statement of the results

The question of convergence in a suitable sense of sequences of im- mersions withL2-bounded second fundamental form and the regularity of the support of their varifolds limits are very important issues in the study of geometric variational problems arising from submanifold theory. A very well known example where this plays an important role is in the study of the celebrated Willmore functional who was (to the best of our knowledge) first considered in various works by Thomsen [11], and subsequently by Blaschke [1]. In 1965, Willmore [12] reintroduced, and studied it within the frame work of the conformal geometry of surfaces. The motivations of the

(*) Indirizzo degli Autori: Mathematisches Institut der Eberhard-Karls- UniversitaÈt TuÈbingen, Auf der Morgenstelle 10, D-72076 TuÈbingen, Germany.

E-mail addresses: ndiaye@everest.mathematik.uni-tuebingen.de schaetz@everest.mathematik.uni-tuebingen.de

Both authors were supported by the DFG Sonderforschungsbereich TR 71 Freiburg - TuÈbingen.

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study of Willmore functional stem from many areas of science. For ex- ample in molecular biology, it is known as the Helfrich Model [4], and is interpreted as surface energy for lipid bilayers. Another field where it plays an important role is solid mechanics. In fact, it arises as the limit energy for thin-plate theory, see [2]. Finally, in general relativity, the Willmore functional appears as the main term in the expression of the Hawking quasilocal mass, see [3].

In [7], J. Langer proved a convergence Theorem (after reparame- trization) in theC1-topology for sequences of immersions withLp-bounded second fundamental form withp>2. A crucial point in his argument is the fact that thanks to Sobolev embedding Theorem, such immersions areC1 sincep>2. This latter point is not true forpˆ2, as 2 is the borderline in the Sobolev embedding Theorem. Furthermore, as noticed by J. Langer [7] his result is no more true whenpˆ2. For these reasons, the issue of convergence theorem for sequence of immersions withL2-bounded second fundamental form which is suitable for the variational analysis of the Willmore functional is not a trivial matter.

In this note, we prove a convergence theorem for a sequence of im- mersions from some closed surfaceSinto some standard Euclidean space Rn(n3) withL2-bounded second fundamental form, which is suitable for applying the direct method of Calculus of Variations to study the Willmore functional. In fact our convergence theorem (see Theorem 1.1 especially Corollary 1.3) has been used in the study of the Willmore boundary pro- blem by R. SchaÈtzle [8], see section 4, last paragraph of page 290.

In order to state our result in a clear way, we first fix some notation. In the following, we use Rn, n3 to denote the standard n-dimensional Euclidean space, geuc for its standard metric, Rn‡1 the standard (n‡1)- dimensional Euclidean space andSn its unit sphere. We use the notation en‡1 to denote the north pole of Sn, namely en‡1:ˆ(0; ;1)2Rn‡1. Furthermore, we will use the following MoÈbius transformationF:Rn‡1[ f1g !Rn‡1[ f1gdefined by the following formula

F(x)ˆen‡1‡2 x en‡1

jx en‡1j2: …1†

The restriction ofFtoSnwill be called the stereographic projection ofSn toRn[ f1g. Forx2Rn,r>0,Br(x) will denotes the Euclidean ball of Rn with centerxand radiusr. Moreover forr>0,Drwill stand for the open disk ofR2of diameterr. We identifyRnwithR2Rn 2, and denote byPthe projection ofRnonto theR2factor. Forka nonnegative integer, Hk denotes the Hausdorff measure and Lk stands for the Lebesgue

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measure. GivenS a closed surface, andf :S !Rn an immersion we set gf :ˆf(geuc) the pull back of the standard metricgeuc byf. Furthermore, for an immersionf :S !Rn, we use the notationH~f to denote its mean curvature vector, Af to denote its second fundamental form, mgf the in- duced area measure onSbyf, namelydmgf is the volume form associated to gf, and mf ˆf(mgf) the varifold image of the immersion f, namely mf ˆf(mgf):ˆ(x ! H0(f 1(x))H2bf(S). We recall that mf ˆf(mgf) is an integral 2-varifold onRn, see [9] for more informations. Given an integer rectifiable 2-varifoldm onRn, we use the notation Amto denote its weak second fundamental form andsptmto denote its support, see [5] for more informations. For X a space,AX, andm a measure onX, we use the standard notationmbAto mean the restriction ofmonA. We also use the notationdiam(A) to denote the diameter ofA.

Now having fixed this notation, we are ready to state our main result which reads as follows:

THEOREM1.1. Let S be a closed surface, and fm:S !Rn, n3 be a family of immersions. Assuming that

1) R

SjAmj2dmgmE,mgm*mas varifolds, minteger rectifiable2-vari- fold, sptmcompact,jAmj2mgm*nas Radon measures with gm:ˆgfm, and Am:ˆAfm.

2) There exist e0>0, r1>0, such that if we denote by fxk;kˆ1; ;Kg sptm, the finite number of bad points of sptm verifyingn(fxkg)e20, then for every x2sptmn [Kkˆ1Br1(xk), we have there exists r(x)>0 such that mbBr(x)(x)ˆPNx

iˆ1mi(x)and this decomposition is unique up to relabelling and themi(x)'s are C1-graphs in Br(x)(x).

Then, (up to a subsequence) there exists m0depending only on E, n, S, sptm, n, r1, e0, and the bad points x1; ;xK, namely m0 ˆ m0(E;n;S;sptm;n;r1;e0;x1; ;xK)>0, such that for every mm0, we have the existence of a closed surface S1m, an immersion Fm:S1m !sptmRn such that

mbsptmn[K

kˆ1Br1(xk)(x ! H0(Fm1(x)))H2bFm(Sm;r1); and Z

Sm;r1

jAFmj2dmgFm Z

sptm

jAmj2dm;

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where Sm;r1:ˆfm1(sptmn [Kkˆ1Br1(xk)). Moreover, defining S1m;r1

S1mnFm1([Kkˆ1Br1(xk))for mm0, we have that the surfaceS1m verifies (still for mm0)

S1m;r1Sm;r1:

REMARK 1.2. As already said in the abstract, the assumptions of Theorem 1.1 are automatically satisfied for a sequence of immersions (up to a subsequence and an appropriate MoÈbius transformation ofRn) arising from an appropriate stereographic projection ofSn intoRnof immersions from S into Sn and minimizing theL2-norm of the second fundamental form. In fact up to a subsequence, apart ofsptmcompact, all the hypotheses in 1) follows from the minimizing property and the monotonicity formula as in (9) below. Moreover, up to an appropriate MoÈbius transformation ofRn, the compactness ofsptmfollows from a straightforward adaptation of the argument of Proposition 2.1 in [8] and the discussion right after until formula (2.7). On the other hand, the assumption 2) follows directly from the argument of Proposition 2.2 in [8].

Thus, we have the following corollary:

COROLLARY1.3. Let S be a closed surface, and fm:S !Rn, n3 be a family of immersions arising from the stereographic projection F:Sn !Rn[ f1g of immersions fromS into Sn where Fis as in(1).

Assuming that Z

S

jAfmj2dmgfm ! inf

f:S !Rn;fimmersion

Z

S

jAfj2dmf as m ! ‡ 1;

then up to a subsequence and an appropriate MoÈbius transformation of Rn, we have that the same conclusions of Theorem1.1hold.

REMARK1.4. We would like to point out that, if in addition the surface Sin Theorem 1.1 is orientable then the corresponding limit surfaceS1m is orientable as well. The same remark holds also for Corollary 1.3.

2. Proof of Theorem 1.1

In this section, we present the proof of Theorem 1.1 from which Cor- ollary 1.3 follows as already pointed out in Remark 1.2. We start by making two definitions following J. Langer [7].

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DEFINITION 2.1. Let S be a closed surface, q a point of S, f :S !Rn an immersion, n3, and A:Rn !Rn an Euclidean iso- metry which takes the origin to f(q) and whose differential takes enˆ(0;0; ;0;1)to the inward normal to f at q. We define Ur;qto be the q-component of(PA 1f) 1(Dr), namely the connected component of (PA 1f) 1(Dr)containing q.

DEFINITION2.2. LetSbe a closed surface, f :S !Rnan immersion, n3, and r;apositive real numbers. We say that f :S !Rnis an(r;a)- immersion if for every q2S,(A 1f)(Ur;q)is the graph of a C1-function h:Dr !Rn 2satisfyingkrhkL1 a, where A is the Euclidean isometry in the definition of Ur;q.

Now having fixed these definitions, we are ready to make the proof of Theorem 1.1. We divide the proof in 3 steps.

PROOF OFTHEOREM1.1

STEP 1. In this step, we show that, for everyx2sptmn [Kkˆ1Br1(xk), there exists43r(x)>rx>0 (wherer(x) is as in Theorem 1.1) such that (up to a subsequence), we have that there exists closed pairwise disjoint disks Dm;i(x) iˆ1; ;Ix and for every i, the sequence of measures mm;i(x):ˆ H2bfm(Dm;i(x))\B34rx(x) verifies the following properties

mm;i(x)*mi(x) weakly as varifolds in B34rx(x);

…2†

sptmm;i(x)!sptmi(x) locally in Hausdorff distance in B3

4rx(x);

…3†

and

mbB3

4rx(x)ˆXIx

1ˆ1

mi(x):

…4†

In order to achieve (2)-(4), we first recall that the Willmore energy offmis defined by W(fm):ˆ14R

S jH~mj2dmgm and is (thanks to the Gauss-Bonnet theorem) related toR

SjAmj2dmgmby the following formula W(fm)ˆ1

4 Z

S

jAmj2dmgm‡px(S);

…5†

whereH~mis the mean curvature vector offmandx(S) is the Euler-Poincare characteristic ofS. Thus, from the assumption ofL2-bounded second fun-

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damental form, we infer that W(fm

Z

S

jAmj2dmgmE;

…6†

whereEdepends only onEandS. Next, we take 05e05e0to be fixed later.

Thus, by assumption, we have that for every kˆ1; ;K, there holds n(fxkg)>e20. Hence, it is easy to see that, for everyx2sptmn [Kkˆ1Br1(xk), there exists 4r(x)>3rx>0 such that

n(Brx(x))5e20:

Now, since by assumption we havejAmj2mgm*nas Radon measures, then formlarge enough, we get

Z

Brx(x)

jAmj2dmgmn(Brx(x))5e20:

Hence, applying the graphical decomposition Lemma of L. Simon, see Lemma 2.1 in [10], we have that for everyx2sptmn [Kkˆ1Br1(xk), and form large enough, fm1(Brx(x))ˆPImx

iˆ1Dm;i(x) whereDm;i(x),iˆ1; ;Imx,IxmCE are closed pairwise disjoint sets. In addition, the following properties hold:

there exist affine 2-planes Lm;i(x)Rn, smooth functionsuxm;i:Vm;i(x) Lm;i(x) !Vm;i(x)?, where Vm;i(x)ˆV0m;i(x)n [Kkˆ1x;imdm;i;k(x), V0m;i(x) are simply connected and open,dm;i;k(x) are closed pairwise disjoint disks, with

r 1juxm;ij ‡ jruxm;ij C(E;n)e04n‡61 ;

and closed pairwise disjoint pimplesPm;i;1(x); ;Pm;i;Jmx;i(x)Dm;i(x) which are disks such that

fm(Dm;i(x)n [Jjˆ1mx;iPm;i;j(x))ˆgraph(uxm;i)\Brx 2(x);

…7†

and

XImx

iˆ1

XJx;im

jˆ1

diam(fm(Pm;i;j(x)))C(E;n)e012rx: …8†

On the other hand, using (A.16) in [6], we obtain r 2mm(Br(x))CW(fm) 8r>0;

…9†

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where mm:ˆmfmˆfm(mgm)ˆ(x ! H0(fm1(x))H2bfm(S) and gmˆgfmˆ fm(geuc). Thus, we derive

r 2mm(Br(x))C(E;n;S); 8r>0;

…10†

and

XIxm

iˆ1

XJmx;i

jˆ1

mm(fm(Pm;i;j(x)))C(E;n;S)e0r2x: …11†

Now, since

H2(graph(uxm;i)\Br2x(x)) 

1‡jruxm;ij2L1

q L2(PLm;i(x)(graph(uxm;i)\Br2x(x))



1‡C(E;n)e02n‡31 q

v2r2x; …12†

then, using (8) combined with the diameter bound of L. Simon [10] and (6) (which give the area control offm(Pm;i;j(x)) the image byfmof the pimples Pm;i;j(x)), we obtain

mgm(Dm;i(x))(1‡C(E;n)e02n‡31 )v2r2x 8m;i:

…13†

Next, using the monotonicity formula (A.6) in [6], we infer that s2mgm(Dm;i(x)\fm1(Bs(x))) 9

16(1‡d)r2xmgm(Dm;i(x)\fm1(B3

4rx(x)))‡

‡C(1‡d 1) Z

Brx(x)

jH~mj2dmm;

9

16(1‡d)rx2mgm(Dm;i(x))‡C(1‡d 1)e20; for ssuch that Bs(x)B3rx

4(x), and for every d>0. Now, we take t51 and choosedsuch that 9 2

16(1‡d)51‡t

2. Next, we come to the choice of e0 that we choose depending only onE,nandtsuch that 05e05e0 and (thanks to (13))

mgm(Dm;i(x)\fm1(Bs(x)))

v2s2 51‡t:

In particular

fm:Dm;i(x)\fm1(B3rx

4(x)) !B3rx

4(x) is an embedding:

…14†

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Now, setting,

mm;i(x):ˆ H2bfm(Dm;i(x))\B3rx

4(x)ˆmmbfm(Dm;i(x))\B3rx 4(x)ˆ

ˆfm(mgm)bfm(Dm;i(x))\B3rx 4(x);

and applying again the monotonicity formula as in (9), and using the fact thatImx CE, we can suppose that up to a subsequenceImx ˆIx, and for everyi

mm;i(x)*mi(x) weakly as varifolds in B3rx 4(x);

sptmm;i(x)!sptmi(x) locally in Hausdorff distance in B3rx 4(x);

and

mbB3rx

4(x)ˆXIx

1ˆ1

mi(x);

as desired.

REMARK2.3. Wewould like to point out thatrxdepends not only onxbut also onE,n,sptm,n,e0,r1andfx1; ;xKg. However, for the sake of clarity in the exposition, we have chosen to emphasize only the dependence tox.

STEP2. In this second step, we show the existence ofm~0 positive de- pending only onE,n,S,sptm,n,e0,r1and the bad pointsxk,kˆ1; ;K, namelym~0ˆm~0(E;n;S;sptm;n;e0;r1;x1; ;xK)>0 such that for every mm~0, we have the existence of an immersionFm:Sm;r1 !sptmRn with the following property

mFmˆGm(x)H2bFm(Sm;r1)m;

…15†

where

Gm(x)ˆ H0(Fm1(x)); x2Sm;r1:

For this end, we first use the fact thatsptmis compact to infer that there exists x1; ;xN2sptmn [Kkˆ1Br1(xk) such that sptmn [Kkˆ1Br1(xk) [Nkˆ1Brxk(xk) with N depending only on sptm; r1 and the bad points xk, kˆ1; ;K. Now, to continue, we recall that from the arguments of the proof of Step 1, we have that for every kˆ1; ;N, there exists m0k ˆm(xk;E;n;e0;rxk) depending only onxk,E,n,rxk, ande0such that for everymm0k, there holds:

fm1(Brxk(xk)) decomposes into finitely many closed pairwise disjoint setsDm;i(xk);

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and that for everymm0kandi,

the setDm;i(xk) contains closed pairwise disjoint disksPm;i;j(xk);

verifying

fm(Dm;i(xk)n [Jjˆ1xk;im Pm;i;j(xk)) is a graph of a smooth function;

and for everymm0k,iandj,

the image byfm of the pimples Pm;i;j(xk) (namelyfm(Pm;i;j(xk))) has very small diameter:

Now, settingm0:ˆmaxNkˆ1m0k, and recalling thatNdepends only onsptm;

r1 and the bad pointsxk,kˆ1; ;Kand for everykˆ1; ;N,rxk de- pends only onxk,E,n,n,sptm,e0,r1 and the bad pointsx1; ;xK (see Remark 2.3), we have thatm0 depends only onE,n,sptm,n,e0,r1, and x1; ;xK, namelym0ˆm0(E;n;sptm;n;r1;e0;x1; ;xK). Next, we have that by replacing the image of the pimplesPm;i;j(xk), namelyfm(Pm;i;j(xk)) (forkˆ1; ;Nandmm0) by extensions of the graphs, we obtain new immersions ^fm:Sm;r1 !Rn still for mm0, where we recall that Sm;r1:ˆfm1(sptmn [Kkˆ1Br1(xk)). We point out that^fm does not depend on the good pointsxk,kˆ1; ;N, since for eachxk, we have replaced the corresponding image (byfm) of the pimples and the result gives^fm. On the other hand, we have that the extensions can be done so that the ^fm's obtained are 3r

254;C(E;n)e04n‡61

-immersions with r5minNkˆ1rxk(for the definition of these type of immersions see Definition 2.2), and up to taking m0bigger, we have that formm0the following holds:

…16†Tfm(q)(fm(Sm;r1)) is very close toT^fm(q)(^fm(Sm;r1)); for everyq2Sm;r1; where for every q2Sm;r1, Tfm(q)(fm(Sm;r1)) and T^fm(q)(^fm(Sm;r1)) denote respectively the tangent space of fm(Sm;r1)) at fm(q) and of ^fm(Sm;r1) at

^fm(q). In addition, we have that thanks to (7), the extensions can be done so that after replacing the image by fm of the pimple Pm;i;j(xk), namely fm(Pm;i;j(xk)), we stay in the bigger balls B3rxk

4

(xk), and this for every kˆ1; ;N . Now, using (16) and the existence of nearest point projec- tion, we infer that there exists a projection

Pm:fm(Sm;r1) !^fm(Sm;r1);

still formm0. Next, given any pointz2^fm(Sm;r1), we have there existsi

(10)

andx2 fx1; xNgsuch thatz2fm(Dm;i(x))\B3rx

4(x). Now, using again (16) and the fact that sptmm;i(x)!sptmi(x) locally in Hausdorff distance in B3rx

4(x), see (3) in Step 1, we have that there existsm~0>m0depending only on E, n, S, n, sptm, e0, r1 and the bad points x; ;xK (namely m~0ˆm~0(E;n;S;n;sptm;r1;e0;x1; ;xK)) such that, formm~0, we have that by following a smooth transversal field to^fm, we can projectzto a point Qm(z) belonging tosptmi(x). We claim that this process gives a well-defined mapQm: ^fm(Sm;r1) !sptmformm~0. Indeed, let us suppose thatmm~0

andz2fm(Dm;i(y1)))\fm(Dm;j(y2))\B3ry1

4 (y1)\B3ry2

4 (y2) for two different pointsy1 andy22 fx1; xNg and show that by this process we obtain a unique value forQm(z). In order to do that, we argue as follows. First of all, using (14), we infer that z2fm(Dm;i(y1))\Dm;j(y2))\B3ry1

4 (y1)\B3ry2 4 (y2).

Thus, we have that there existsu53rz

8 andlsuch that Dm;i(z)\Dm;j(z)\Dm;l(z)6ˆ ; and Bu(z)B3ry1

4 (y1)\B3ry2 4 (y2):

…17†

In fact, sincez2fm(Dm;i(y1)\Dm;j(y2)), then there exists a pointqsuch that q2Dm;i(y1)\Dm;j(y2) and zˆfm(q):

…18†

On the other hand, we have also that z2B3ry1

16 (y1)\B3ry2

4 (y2). Now, from z2B3ry1

4 (y1)\B3ry2

4 (y2), we infer the existence of u53rz

8 such that Bu(z)B3ry1

8 (y1)\B3ry2

8 (y2). Next, since zˆfm(q) and z2B3rz

4(z), then q2fm1(B3rz

4(z)). Now, using Step 1, we have the following decomposition for fm1(Brz(z))

fm1(Brz(z))ˆXIz

iˆ1

Dm;i(z):

…19†

Thus, from (19), we infer that there existslsuch thatq2Dm;l(z). Hence, we have proved the existence oflanduwith the desired properties. To continue the proof of the claim we first define

mm;i;j;l(y1;y2;z):ˆ H2bfm(Dm;i(y1)\Dm;j(y2)\Dm;l(z))\B3ry1 4 (y1)\

\B3ry2

4 (y2)\B3rz 4(z):

Now, from Step 1 and the fact thatu53rz

8 , we get

mm;i;j;l(y1;y2;z)*ml(z) weakly as varifolds in Bu(z):

(11)

On the other hand, using again Step 1 and the fact that Bu(z) B3ry1

4 (y1)\B3ry2

4(y2), we infer that

mm;i;j;l(y1;y2;z)*mi(y1) weakly as varifolds in Bu(z);

and

mm;i;j;l(y1;y2;z)*mj(y2) weakly as varifolds in Bu(z):

So, from the uniqueness of varifolds limit, we have that ml(z)ˆmi(y1)ˆ mj(y2) inBu(z). Thus, by the uniqueness up to relabelling of the decom- position of m, and the fact that ryi54

3r(yi) for iˆ1;2, we have that mi(y1)ˆmj(y2) inB3ry1

4 (y1)\B3ry2

4 (y2). Hence, the projectionQm is well-de- fined formm~0. Thus, we have that there existsm~0depending only onE, n,S,sptm,n,e0,r1, andx1; ;xK such that for everymm~0, the fol- lowing composition is very well defined

FmˆQmPmfm:

Now, from the hypothesis that themi's areC1-graphs and the uniqueness of the decomposition, we infer thatFm:Sm;r1 !sptmRnis an immersion for everymm~0. Furthermore, setting

Gm(x)ˆ H0(Fm1(x)); x2Sm;r1; formm~0and using the definition ofFm, we have

mFmˆGm(x)H2bFm(Sm;r1)m;

formm~0as desired.

STEP 3. In this step, we finish the proof of Theorem 1.1. To do so, we first use Step 2 and fill the regions corresponding to the bad points, namely the set[Kkˆ1Br1(xk), to obtain (for mm~0) a closed surfaceS1m and im- mersionFm :S1m !sptmRnsuch that setting

S1m;r1:ˆS1mnFm1([Kkˆ1Br1(xk));

…20†

and recalling that

Sm;r1:ˆfm1(sptmn [Kkˆ1Br1(xk));

we obtain

FmˆFm in Sm;r1; and

S1m;r1 Sm;r1; …21†

(12)

still formm~0. On the other hand, from the definition ofFm(see Step 2) and (21), it is easy to see that

Z

Sm;r1

jAFmj2dmgFm Z

Sm;r1

jAfmj2dmgm‡em; …22†

withem !0 asm ! ‡ 1. Now, using assumption 1), we have also that Z

Sm;r1

jAfmj2dmgm ! Z

sptmn[Kkˆ1Br1(xk)

jAmj2dm as m ! ‡ 1:

…23†

Next, sincexk,kˆ1; ;Kare the bad point ofsptmverifyingn(fxkg)e20 forkˆ1; ;K, then we have

Z

sptmn[Kkˆ1Br1(xk)

jAmj2dm5 Z

sptm

jAmj2dm:

…24†

Thus, from (15), (21)-(24), we infer that there existsm0depending only on E, n, S, n, sptm, e0, r1 and the bad points x; ;xK, namely m0ˆm0(E;n;S;sptm;n;e0;r1;x1; ;xK), such that m0>m~0 and for everymm0there holds:

mbsptmn [Kkˆ1Br1(xk)(x ! H0(Fm1(x))H2bFm(Sm;r1) and Z

Sm;r1

jAFmj2dmgFm Z

sptm

jAmj2dm:

Hence, sincem0>m~0, then (21) completes the proof of the Theorem. p

REFERENCES

[1] W. BLASCHKE,Vorlesungen Ueber Differentialgeometrie III, Springer (1929).

[2] G. FRIESECKE- R. JAMES- S. MUÈLLER,A theorem on geometric rigidity and the derivation on nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math.,55, no. 11 (2002), pp. 1461-1506.

[3] S. W. HAWKING,Gravitational radiation in an expanding universe, J. Math.

Phys.,9(1968), pp. 598-604.

[4] W. HELFRICH, Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch.,C28(1973), pp. 693-703.

[5] J. E. HUTCHINSON, Second Fundamental Form for Varifolds and the Existence of Surfaces Minimizing Curvature. Indiana Univ. Math. J.,35, No 1 (1986), pp. 45-71.

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[6] E. KUWERT- R. SCHAÈTZLE,Removability of point singularity of Willmore surfaces, Ann. of Math.,160, no. 1, pp. 315-357.

[7] J. LANGER, A compactness theorem for surfaces with Lp-bounded second fundamental form, Math. Ann.,270(1985), pp. 223-234.

[8] R. SCHAÈTZLE, The Willmore boundary problem, Cal. Var. PDE,37(2010), pp. 275-302.

[9] L. SIMON,Lectures on Geometric Measure Theory, Proceedings of the Center for Mathematical Analysis Austrian National University, Volume 3.

[10] L. SIMON,Existence of surfaces minimizing the Willmore functional, Comm.

Anal. Geom., Vol1, no. 2 (1993), pp. 281-326.

[11] G. THOMSEN, UÈber konforme Geometrie I: Grundlagen der Konformen FlaÈchentheorie, Abh. Math. Sem. Hamburg (1923), pp. 31-56.

[12] T. J. WILLMORE,Note on embedded surfaces, Ann. Stiint. Univ. Al. I. Cuza Iasi, Sect. Ia Mat. (N. S) 11B (1965), pp. 493-496.

Manoscritto pervenuto in redazione il 30 agosto 2011.

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