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.
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 forp2, as 2 is the borderline in the Sobolev embedding Theorem. Furthermore, as noticed by J. Langer [7] his result is no more true whenp2. 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, Rn1 the standard (n1)- dimensional Euclidean space andSn its unit sphere. We use the notation en1 to denote the north pole of Sn, namely en1:(0; ;1)2Rn1. Furthermore, we will use the following MoÈbius transformationF:Rn1[ f1g !Rn1[ f1gdefined by the following formula
F(x)en12 x en1
jx en1j2: 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
measure. GivenS a closed surface, andf :S !Rn an immersion we set gf :f(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 f(mgf) the varifold image of the immersion f, namely mf f(mgf):(x ! H0(f 1(x))H2bf(S). We recall that mf f(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:gfm, and Am:Afm.
2) There exist e0>0, r1>0, such that if we denote by fxk;k1; ;Kg sptm, the finite number of bad points of sptm verifyingn(fxkg)e20, then for every x2sptmn [Kk1Br1(xk), we have there exists r(x)>0 such that mbBr(x)(x)PNx
i1mi(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 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
k1Br1(xk)(x ! H0(Fm1(x)))H2bFm(Sm;r1); and Z
Sm;r1
jAFmj2dmgFm Z
sptm
jAmj2dm;
where Sm;r1:fm1(sptmn [Kk1Br1(xk)). Moreover, defining S1m;r1:
S1mnFm1([Kk1Br1(xk))for mm0, we have that the surfaceS1m verifies (still for mm0)
S1m;r1Sm;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].
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 en(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 [Kk1Br1(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) i1; ;Ix and for every i, the sequence of measures mm;i(x): 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)XIx
11
mi(x):
4
In order to achieve (2)-(4), we first recall that the Willmore energy offmis defined by W(fm):14R
S jH~mj2dmgm and is (thanks to the Gauss-Bonnet theorem) related toR
SjAmj2dmgmby the following formula W(fm)1
4 Z
S
jAmj2dmgmpx(S);
5
whereH~mis the mean curvature vector offmandx(S) is the Euler-Poincare characteristic ofS. Thus, from the assumption ofL2-bounded second fun-
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 k1; ;K, there holds n(fxkg)>e20. Hence, it is easy to see that, for everyx2sptmn [Kk1Br1(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 [Kk1Br1(xk), and form large enough, fm1(Brx(x))PImx
i1Dm;i(x) whereDm;i(x),i1; ;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)V0m;i(x)n [Kk1x;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)e04n61 ;
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 [Jj1mx;iPm;i;j(x))graph(uxm;i)\Brx 2(x);
7
and
XImx
i1
XJx;im
j1
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
where mm:mfmfm(mgm)(x ! H0(fm1(x))H2bfm(S) and gmgfm fm(geuc). Thus, we derive
r 2mm(Br(x))C(E;n;S); 8r>0;
10
and
XIxm
i1
XJmx;i
j1
mm(fm(Pm;i;j(x)))C(E;n;S)e0r2x: 11
Now, since
H2(graph(uxm;i)\Br2x(x))
1jruxm;ij2L1
q L2(PLm;i(x)(graph(uxm;i)\Br2x(x))
1C(E;n)e02n31 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))(1C(E;n)e02n31 )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(1d)r2xmgm(Dm;i(x)\fm1(B3
4rx(x)))
C(1d 1) Z
Brx(x)
jH~mj2dmm;
9
16(1d)rx2mgm(Dm;i(x))C(1d 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(1d)51t
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 51t:
In particular
fm:Dm;i(x)\fm1(B3rx
4(x)) !B3rx
4(x) is an embedding:
14
Now, setting,
mm;i(x): H2bfm(Dm;i(x))\B3rx
4(x)mmbfm(Dm;i(x))\B3rx 4(x)
fm(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 Ix, 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)XIx
11
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,k1; ;K, namelym~0m~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
mFmGm(x)H2bFm(Sm;r1)m;
15
where
Gm(x) H0(Fm1(x)); x2Sm;r1:
For this end, we first use the fact thatsptmis compact to infer that there exists x1; ;xN2sptmn [Kk1Br1(xk) such that sptmn [Kk1Br1(xk) [Nk1Brxk(xk) with N depending only on sptm; r1 and the bad points xk, k1; ;K. Now, to continue, we recall that from the arguments of the proof of Step 1, we have that for every k1; ;N, there exists m0k m(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);
and that for everymm0kandi,
the setDm;i(xk) contains closed pairwise disjoint disksPm;i;j(xk);
verifying
fm(Dm;i(xk)n [Jj1xk;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:maxNk1m0k, and recalling thatNdepends only onsptm;
r1 and the bad pointsxk,k1; ;Kand for everyk1; ;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, namelym0m0(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)) (fork1; ;Nandmm0) by extensions of the graphs, we obtain new immersions ^fm:Sm;r1 !Rn still for mm0, where we recall that Sm;r1:fm1(sptmn [Kk1Br1(xk)). We point out that^fm does not depend on the good pointsxk,k1; ;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)e04n61
-immersions with r5minNk1rxk(for the definition of these type of immersions see Definition 2.2), and up to taking m0bigger, we have that formm0the following holds:
16Tfm(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 k1; ;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
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~0m~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)6 ; 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 zfm(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 zfm(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))XIz
i1
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): 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):
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)mi(y1) 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 i1;2, we have that mi(y1)mj(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
FmQmPmfm:
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) H0(Fm1(x)); x2Sm;r1; formm~0and using the definition ofFm, we have
mFmGm(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[Kk1Br1(xk), to obtain (for mm~0) a closed surfaceS1m and im- mersionFm :S1m !sptmRnsuch that setting
S1m;r1:S1mnFm1([Kk1Br1(xk));
20
and recalling that
Sm;r1:fm1(sptmn [Kk1Br1(xk));
we obtain
FmFm in Sm;r1; and
S1m;r1 Sm;r1; 21
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
jAfmj2dmgmem; 22
withem !0 asm ! 1. Now, using assumption 1), we have also that Z
Sm;r1
jAfmj2dmgm ! Z
sptmn[Kk1Br1(xk)
jAmj2dm as m ! 1:
23
Next, sincexk,k1; ;Kare the bad point ofsptmverifyingn(fxkg)e20 fork1; ;K, then we have
Z
sptmn[Kk1Br1(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 m0m0(E;n;S;sptm;n;e0;r1;x1; ;xK), such that m0>m~0 and for everymm0there holds:
mbsptmn [Kk1Br1(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
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Manoscritto pervenuto in redazione il 30 agosto 2011.