Minimizers of the Willmore functional with a small area constraint

Ann. I. H. Poincaré – AN 30 (2013) 497–518

www.elsevier.com/locate/anihpc

Minimizers of the Willmore functional with a small area constraint

Tobias Lamm

, Jan Metzger

aGoethe-Universität Frankfurt, Institut für Mathematik, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany bInstitute for Analysis, Karlsruhe Institute of Technology (KIT), Kaiserstr. 89, 76128 Karlsruhe, Germany

cUniversität Potsdam, Institut für Mathematik, Am Neuen Palais 10, 14469 Potsdam, Germany Received 10 January 2012; received in revised form 9 October 2012; accepted 22 October 2012

Available online 12 November 2012

Abstract

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

©2012 Elsevier Masson SAS. All rights reserved.

Keywords:Willmore functional; Minimizers; Direct method

1. Introduction

For a three-dimensional complete Riemannian manifold(M, g)and an immersionf:Σ→Mthe Willmore func- tional is defined by

W(f )=1 2

Σ

H²dμ,

whereH is the induced mean curvature andμthe induced area measure. In the following we letAresp. ˚Abe the second fundamental form resp. the trace-free second fundamental form of the immersionf. Critical points ofWare called Willmore surfaces and they are solutions of the Euler–Lagrange equation

H+H|A˚|²+HRic(ν, ν)=0,

where Ric denotes the Ricci curvature of(M, g)andνis the normal vector toΣinM.

In the literature other possible definitions of the Willmore functional for immersions in a Riemannian manifold were considered, for example:

Σ

|A|²dμ,

Σ

|A˚|²dμ, or

Σ

H²+κ^M dμ.

* Corresponding author at: Institute for Analysis, Karlsruhe Institute of Technology (KIT), Kaiserstr. 89, 76128 Karlsruhe, Germany.

E-mail addresses:[email protected](T. Lamm),[email protected](J. Metzger).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.10.003

Here κ^M denotes the sectional curvature ofM. In a curved ambient manifold the Gauss equation and the Gauss–

Bonnet theorem yield W(Σ )=1

2

Σ

|A|²dμ+

Σ

G(ν, ν)dμ+2π

1−q(Σ )

whereq(Σ )is the genus ofΣ,G=Ric−¹₂Scgis the Einstein tensor, Ric denotes the Ricci curvature and Sc the scalar curvature of(M, g). Hence, the functionals above differ by lower order terms involving the curvature of(M, g).

In particular, if(M, g)=(R³, δ)the only difference is a multiple ofq(Σ ).

In this paper we are interested in surfaces of Willmore type, i.e. critical points ofWsubject to an area constraint.

These surfaces are solutions of the Euler–Lagrange equation H+H|A˚|²+HRic(ν, ν)+λH=0,

with the Lagrange parameterλ. In[9]we studied spherical surfaces of Willmore type with positive mean curvature in small geodesic balls. Assuming a certain lower bound on the Lagrange parameter, we showed that such surfaces can only concentrate at critical points of the scalar curvature ofM. This paper establishes the existence of minimizers of Wwith fixed small area and verifies that the minimizers satisfy the assumptions of[9]. Our main result is as follows:

Theorem 1.1.Let(M, g)be a compact, closed Riemannian manifold. Then there exists a constanta0>0such that for all a∈(0, a0)there is a smooth spherical surface Σa of positive mean curvature that minimizes the Willmore functional among all immersed surfaces with areaa.

For any sequenceai→0there is a subsequencea_i such thatΣa_i is asymptotic to a geodesic sphere centered at a pointp∈MwhereScattains its maximum.

The existence of aW^2,2∩W^1,∞conformal immersion minimizingWwith prescribed small area was recently and independently obtained by Chen and Li[2].

Kuwert and Schätzle [8]constructed smooth minimizers in a conformal class of the Willmore functional inR³. Existence was recently generalized to arbitrary co-dimension by Kuwert and Li[6]in the classW^2,2∩W^1,∞and by Rivière[14].

The existence of a smooth minimizer of the Willmore functional inRⁿwith prescribed genus was first proved by Simon[16]under a Douglas-type condition, which was established by Bauer and Kuwert[1](see also[14]for a new proof of this result). Recently, Schygulla[15]suitably modified the arguments of Simon in order to prove the existence of a minimizing Willmore sphere inR³with prescribed isoperimetric ratio.

Under suitable curvature assumptions, Kuwert, Mondino, and Schygulla [7] recently showed the existence of smooth spherical minimizers of the functionals

Σ|A|²dμand

Σ(1+ |H|²)dμin Riemannian manifolds. The role of the curvature assumptions is to ensure a uniform bound on the area of surfaces in a minimizing sequence.

Some other existence and non-existence results of critical points ofWby Mondino can be found in[11,12].

Previously, in a joint work with F. Schulze[10], we proved the existence of a foliation of the end of an asymp- totically flat manifold with positive mass by spherical surfaces of Willmore type. For the existence result we studied perturbed geodesic spheres and we used the implicit function theorem to find suitable deformations of these spheres.

In this paper we use the direct method of the calculus of variations in order to construct the minimizer of W. A major difficulty is the invariance ofWunder diffeomorphisms. We overcome this problem by showing that spherical surfaces with small enough area have small diameter since the Willmore energy is a priori close to 8π. Hence the surface is contained in a small geodesic ball around a pointp∈Mwhere the metricgis a small perturbation of the Euclidean metric. Thus we can apply a result of De Lellis and Müller[3,4]which gives the existence of aW^2,2∩ W^1,∞conformal parametrizationF :S²→Σ⊂R³of Σ. Therefore, instead of studying minimizing sequences of immersionsfk:Σk→MofW, we consider minimizing sequences of parametrizationsFk∈W^2,2∩W^1,∞(S²,R³).

Within this class we are able to show the existence of a minimizer ofW.

In order to show the higher regularity of the minimizer inW^2,2∩W^1,∞we suitably modify the arguments of[7]

and[15]. In our situation their arguments heavily simplify since the smallness of the area rules out bad points for the minimizing sequence and we can use the fact that there exists a limiting parametrization.

As the last step we show that the minimizers we construct satisfy the assumptions of the main result in[9]and hence we conclude that the minimizing surfaces have to concentrate around a maximal point of the scalar curvature ofMas the area tends to zero.

In the following we give a brief outline of the paper. In Section2we review manifolds with bounded geometry and show that the diameter estimates of[16]extend to these ambient manifolds. Moreover we show that by a scaling argument one can adjust the area of a surface without changing the Willmore energy too much. This fact will be crucial in the proof of the smoothness of the minimizer ofWwith prescribed area.

In Section3we study the Willmore functional for immersionsF∈W^2,2∩W^1,∞(S²,R³)and we show the lower semi-continuity and the differentiability of the functional.

In Section4 we construct the smooth minimizer and hence prove the existence part of Theorem1.1using the methods described above.

In Section5we prove an integral estimate and estimate for the Lagrange parameter of the minimizer. These esti- mates then allow us to apply the results of[9].

In Appendix A we derive a variant of the stability inequality of minimal surfaces for surfaces of Willmore type with positive mean curvature. Using the methods of[5]we are then able to partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian manifolds. As a corollary we obtain that the only complete surfaces of this type inR³are round spheres.

2. Preliminaries

2.1. Manifolds with bounded geometry

In this section we recall some basic properties of manifolds with bounded geometry. The main point is that such manifolds have uniformly controlled normal coordinates.

Definition 2.1.Let(M, g)be a complete Riemannian manifold. We say that(M, g)has bounded geometry, if there exists a constant 0< C_B<∞such that for eachp∈Mwe have inj(M, g, p)C_B⁻¹and if the Riemann tensor and its first derivative are bounded|Rm| + |∇Rm|C_B.

In the following we useB_r to denote a Euclidean ball centered at the origin of radiusrandBr(p)⊂Mto denote a geodesic ball of radiusrcentered atp∈M.

Remark 2.2. If(M, g) has bounded geometry with constant C_B, then there exist constants h₀<∞ and ρ₀>0, depending only onC_B, such that for everyp∈M, we can introduce normal coordinatesφ:B_ρ0→Bρ0(p)for the metricgsuch that in these coordinates the metricgsatisfies

g=g^E+h, with

sup

B_ρ0

|x|⁻²|h| + |x|⁻¹|∂h| +∂²hh₀.

Hereg^Edenotes the Euclidean metric induced by the normal coordinates,|x|denotes the Euclidean distance topand

∂the connection ofg^E.

If(M, g)is compact or asymptotically flat, then it is of bounded geometry for some constantCB. 2.2. Area, diameter and the Willmore energy

To proceed, we need a generalization of Lemma 1.1 from[16] to general ambient manifolds. Although this is straightforward, we present the proof here to show where the non-flat ambient geometry has to be taken into account.

For ease of presentation, we split this into three separate statements.

Lemma 2.3.(Cf.[9, Lemma 2.2].) Letg=g^E+honB_ρ0 be given, such that sup

Bρ0

|x|⁻²|h| + |x|⁻¹|∂h| h0.

Then there exists a purely numerical constantc >0such that ifρ1:=min{ρ0, ¹

c√

h₀}, then for all surfacesΣ⊂B_r withr∈(0, ρ1)we have that

|Σ|r²W(Σ ).

The following proposition is very similar to the calculations in[16, Section 1]. We add only minor modifications in order to deal with the non-flat background.

Proposition 2.4.Letg=g^E+honBρ₀ be given, such that sup

Bρ0

|x|⁻²|h| + |x|⁻¹|∂h| h₀.

Then there exists a purely numerical constantc, such that for every smooth surfaceΣ⊂B_ρ0 with∂Σ⊂∂B_ρ0 and 0∈Σwe have that

πc

1+h₀r²

r⁻²|Σ_r| +W(Σ_r) for allrρ. HereΣ_r:=Σ∩B_r.

Proof. InB_ρ0 consider the position vector fieldx. We denote byca constant that is purely numerical, but which may change from line to line. For all surfacesΣ as in the statement we have

|divΣx−2|ch0|x|²

in view of the assumption onh. Furthermore, we calculate that inB_ρ0 d|x| = x

|x|.

In particular, away from the origin, we have divΣ

|x|⁻²x

= |x|⁻²divΣx−2|x|⁻³d|x| x^T

. ThusdivΣ

|x|⁻²x

−2|x|⁻⁴x^⊥2ch₀, (1)

wherex^⊥denotes the projection ofxonto the normal bundle ofΣ.

Choose 0< s < r < ρ0such thatΣ intersects∂Br and∂Bs transversely (note that the set of radii satisfying this condition is dense in(0, ρ0)). Let

Σs,r:=Σ∩(Br\ ¯Bs)

and integrate Eq. (1) onΣs,rto obtain

Σs,r

divΣ

|x|⁻²x dμ−

Σs,r

2|x|⁻⁴|x^⊥|²dμ

ch₀|Σ_s,r|. (2) Using Stokes, we infer that

Σs,r

divΣ

|x|⁻²x dμ=

Σs,r

H|x|⁻²x, νdμ−

Σ∩∂B_s

|x|⁻²x, ηdσ+

Σ∩∂B_r

|x|⁻²x, ηdσ. (3)

Hereνis the normal vector ofΣ andηdenotes the co-normal of∂B_s∩Σand∂B_r∩ΣinΣrespectively. We chose the orientation so thatηpoints in direction of∇r. To proceed, we note that

Σ∩∂B_σ

|x|⁻²x, ηdσ=σ⁻²

Σ∩∂B_σ

x, ηdσ=σ⁻²

Σ_σ

divΣx^Tdσ,

so that

Σ∩∂B_σ

|x|⁻²x, ηdσ−2σ⁻²|Σ_σ| +σ⁻²

Σ_σ

H x^⊥, ν

dμ

ch0|Σ_σ|. (4) Furthermore

Σs,r

|x|⁻⁴x^⊥2−1

2H|x|⁻²x, νdμ=

Σs,r

|x|⁻²x^⊥−1 4H ν

²− 1

16H²dμ. (5)

Inserting (3)–(5) into (2), we infer that

Σ_s,r

|x|⁻²x^⊥−1 4H ν

²dμ−1 2s⁻²

Σ_s

Hx, νdμ+s⁻²|Σ_s|

r⁻²|Σr| +1

8W(Σs,r)−1 2r⁻²

Σr

Hx, νdμ+ch0|Σr|.

SinceΣis smooth at the origin, we can lets→0 and drop the square term on the left to obtain πr⁻²|Σ_r| +1

8W(Σ_r)−1 2r⁻²

Σr

Hx, νdμ+ch₀|Σ_r|.

Using Cauchy–Schwarz on the third term and recalling thatr < ρ₀we infer the estimate πc

1+h0r²

r⁻²|Σr| +W(Σr)

(6) for a purely numerical constantc. Since the values ofrfor which (6) holds are dense in(0, ρ0]we arrive at the claimed estimate by approximation. 2

The next lemma shows that the diameter of a surfaceΣcontained in a Riemannian manifold(M, g)is bounded in terms of its area and its Willmore energy. We define

diam(Σ ):=max d(M,g)(p, q): p, q∈Σ

to be the extrinsic diameter ofΣ. Hered_(M,g)(p, q)denotes the geodesic distance ofpandq in the ambient mani- foldM.

Lemma 2.5.Let(M, g)be a manifold withC_B-bounded geometry. Then there exists a constantCdepending only on C_B such that for all smooth connected surfacesΣwe have

diam(Σ )C

|Σ|^1/2W(Σ )^1/2+ |Σ| .

Proof. Leth₀andρ₀be as in Remark2.2. Choosep, q∈Σsuch thatd:=d_(M,g)(p, q)=diam(Σ ). Assume for now thatr∈(0,^d₂)is chosen such thatr < ρ₀.

LetN be the largest integer smaller thand/rand letp₀=p. Forj=1, . . . , N−1 we choosep_j∈Σat distance (j+¹₂)r top₀, which is possible sinceΣ is connected. Then the geodesic ballsBr/2(p_j)are pairwise disjoint for j=0, . . . , N−1. Using Proposition2.4withp_jas center and summing overj yields

N πc

W(Σ )+

1+h0r² r⁻²|Σ|

. (7)

With this in mind, we let r:=min

ρ₁ 2 ,1

4 |Σ|

W(Σ )

,

whereρ₁=min{ρ₀, ¹

c√

h0}is such that Lemma2.3applies withρ₀andh₀as above.

We have to check thatr < d/2. Assume for the contrary thatd/2r. Then in particulardρ₁and Lemma2.3 implies that

|Σ|

W(Σ )d2r1 2

|Σ| W(Σ ),

a contradiction. Hencer < d/2 and thusN_2r^d. Revisiting (7) thus yields dc

rW(Σ )+

1+h₀r² r⁻¹|Σ| and since

r⁻¹max 2

ρ₁,4

W(Σ )

|Σ|

we find that dc

1+r²h₀

|Σ|W(Σ )+c/ρ₁|Σ|

c

1+r²h0

|Σ|W(Σ )+c

ρ₀⁻¹+ h0

|Σ|.

This yields the claimed estimate. 2 2.3. Area adjustment by scaling

Lemma 2.6.Let(M, g)be a manifold withC_B-bounded geometry. Then there existsρ₁>0with the following prop- erty. Letr∈(0, ρ1),p∈M, and let x be the position vector field with respect to geodesic normal coordinates in Br(p). Denote byΦ:Br/4(p)×(−∞,2)→Br(p)the flow associated tox. Then for everya∈RandΣ⊂Br/4(p) with|Σ| ∈(^a₂,^3a₂)there existst0∈Rwith|Φt₀(Σ )| =aand|t0|2^||Σ_a^|−a|.

Proof. Letρ0andh0as in Remark2.2. We chooseρ1∈(0, ρ0)such that

|∇x−Id|1 2

onBρ₁(p)for allp∈M. Note that thenρ₁depends only onC_B.

Letr∈(0, ρ1)andΣ⊂Br(p)be an arbitrary surface with|Σ|a/2. Then we calculate that d

dt

t=0

Σ=

Σ

Hx, νdμ=

Σ

divΣxdμΣa 2.

If we considerΣ as in the statement of the lemma, we can apply this estimate toΣ_t=Φ_t(Σ )as long asΣ_t∈B_r(p) and|Σ_t| ∈(^a₂,^3a₂). In particular the area of Σ_t is a continuous and strictly increasing function oft. In addition we have that

Σ_t+a fort⁺:=max

0,2a− |Σ| a

, and Σ_t−a fort⁻:=min

0,2|Σ| −a a

. This yields the claim. 2

Lemma 2.7.Let (M, g)be a manifold withC_B-bounded geometry and letρ₁, be as in Lemma2.6. There exists a constantCdepending only onC_Bwith the following property. Letr∈(0, ρ1)and letΣ⊂Br/2(p). Then

d dt

t=0

Σt

|A_t|²dμtC|Σ|^1/2

Σ

H²dμ 1/2

+Cr

Σ

|A|²dμ+C(1+r)|Σ|.

Here we use the notation of Lemma2.6, so thatΣ_t=Φ_t(Σ ),A_t denotes the second fundamental form ofΣ_t anddμt

its induced measure.

Proof. We use the Gauss equation to write

Σ

|A|²dμ=

Σ

H²dμ−2

Σ

G(ν, ν)dμ−4π

1−g(Σ )

whereg(Σ )is the genus ofΣandG(ν, ν)=Ric−¹₂Scg. Therefore we have d

dt

t=0

Σt

|A_t|²dμt=2δx,νW(Σ )−δ_x,νV(Σ )

where

V(Σ )=2

Σ

G(ν, ν)dμ.

We estimate the variations ofWandVseparately. We have that

|∇x−Id|C|x|² and |∇²x|C (8)

whereC is a constant depending only onC_B.

Consider the functionx, νonΣ. A calculation shows that with respect to an adapted ON-Frame{e1, e2, e3=ν}, we have

x, ν = ∇ei,eix, ν −H∇νx, ν +2∇eix, e_kA_ik− x, ν|A|²+ x, e_k∇iA_ik

=H− x, ν|A|²+ x, e_k∇ekH+O(1)+O(r)∗A.

In the last equality we used the Codazzi equation to rewrite divA= ∇H+Ric(ν,·)together with the fact thatx= O(r). In addition we used (8) and use the notationO(1)for terms which are bounded by a constantCandO(r)∗A for terms bounded byCr|A|. Here as usualCdepends only onC_B.

We calculate the variation of the Willmore functional with respect to scaling:

δ_x,νW=

Σ

x, ν

H+H|A˚|²+HRic(ν, ν) dμ

=

Σ

H x, ν + x, νH|A˚|²+ x, νHRic(ν, ν)dμ

=

Σ

H²−1

2H³x, ν +H∇kHx, ek +H

O(1)+O(r)∗A

dμ. (9)

Integrate by parts the third term on the right to calculate further

Σ

H∇kHx, e_kdμ=

Σ

1 2∇k

H²

x, e_kdμ= −1 2

Σ

H²∇kx, e_kdμ

= −1 2

Σ

H²divΣx−H³x, νdμ.

Inserting this into Eq. (9) and taking into account (8) yields that δx,νW=

Σ

H

O(1)+O(r)∗A dμ.

This can be estimated as follows:

|δx,νW|C|Σ|^1/2

Σ

H²dμ 1/2

+Cr

Σ

|A|²dμ.

We proceed with the variation ofV(Σ ). From[10, (75)]we obtain that 1

2δx,νV(Σ )=

Σ

x, ν_M

∇νG(ν, ν)+H G(ν, ν)

−2G ν,∇

x, ν dμ.

Straightforward estimates show that

δ_x,νV(Σ )C(1+r)|Σ| +Cr|Σ|^1/2

Σ

|A|²dμ 1/2

.

This implies the claim. 2

Lemma 2.8.Let(M, g)be a manifold withC_B-bounded geometry and letρ₁, be as in Lemma2.6. For every constant C₀ there exists a constant C₁ with the following properties. If r∈(0, ρ1), a ∈(0, C0r²)and Σ⊂B_r/4(p) with

|Σ| ∈(^a₂,^3a₂)then there exists a surfaceΣ⊂Br(p)with|Σ| =aand

Σ

|A|²dμ−

Σ

|A|²dμ

C₁r||Σ| −a| a

1+

Σ

|A|²dμ

.

Proof. LetΣ be as in the statement. Using Lemma2.6we can findΣ⊂Br(p)with|Σ| =a in the formΣ= Φt₀(Σ ), whereΦt is as in Lemma2.6. In addition, we have that|t0|2^||Σ_a^|−a|2.

To analyze the amount the second fundamental form has changed, we assume for definiteness thatt0>0. From Lemma2.7we find that for allΣt=Φt(Σ )witht∈ [0, t0]we have that

d dt

1+

Σt

|At|²dμt

Cr

1+

Σt

|At|²dμt

,

where the constantC only depends onCB,C0andρ1. In particular, we used that the area of allΣt is bounded by

3C₀

2 r^23C₂⁰ρ1r. Integrating this ordinary differential inequality on[0, t0]and using the fact that|t0|2 is a priori bounded, we arrive at the claimed estimate. 2

3. Analytical aspects of the Willmore functional

In this section we consider the Willmore functional in the space of parametrizations which are in a subset of W^2,2(S²,R³)∩W^1,∞(S²,R³). We assume thatR³is equipped with a smooth metricgof which we assume that with respect to standard coordinates all components and derivatives up to second order thereof are bounded. When we refer to coordinates onR³we use Greek indices, and when referring to coordinates onS²we use Latin indices.

We define the space B:= F∈W^2,2

S²,R³

∩W^1,∞

S²,R³ g,¯ g¯^#∈L^∞ S²

∩W^1,2 S²

where we denote byg¯the pull-back metricF^∗gonS²and byg¯^#its inverse. The function spaces and all tensor norms are defined with respect to the standard metric on the sphere and the standard metricg^EonR³.

3.1. Definition of the Willmore functional First we establish that the Willmore functional

W(F )=1 2

F (S²)

H²dμ

is well defined forF ∈B. Denote byhij the second fundamental form ofF (S²), byΓ¯ the Christoffel symbols ofg¯ and byΓ the Christoffel symbols of the ambient metricg. The Weingarten equation gives

F_ij^α= −h_ijν^α+ ¯Γ_ij^mF_m^α−Γ_βγ^α F_i^βF_j^γ. (10) Here we use the shorthand notation

F_i^α=∂F^α

∂xⁱ and F_ij^α= ∂²F^α

∂xⁱ∂x^j.

In fact we can take Eq. (10) as the definition of the second fundamental form. Note that sinceg,¯ g¯^#∈W^1,2(S²)we have thatΓ¯∈L²(S²)so that the second fundamental form is inL²(S²). Taking the trace of (10) gives

H= −

¯

g^ijF_ij^α+

Γ_βγ^α ◦F F_i^βF_j^γ

(g_αδ◦F )ν^δ. (11)

In particular, we can write the Willmore functional as W(F )=1

2

S²

g¯^ijF_ij^α+

Γ_βγ^α ◦F F_i^βF_j^γ

(g_αδ◦F )ν^δ2

| ¯g|dx (12)

where| ¯g| =det(g)¯ and dxdenotes the standard volume element onS². ClearlyWis continuous onBwhere we equip Bwith the topology induced by convergence ofF inW^2,2(S²,R³)∩W^1,∞(S²,R³)andg¯andg¯^#inW^1,2∩L^∞. 3.2. Lower semi-continuity

In this section we show lower semi-continuity ofWinBwith respect to weak convergence.

Proposition 3.1.Assume thatF_k, F ∈Bare parametrizations of surfacesΣ_kandΣand thatgis a smooth metric on R³such that all coefficients ofgwith respect to standard coordinates onR³and all their derivatives are bounded. If

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

F_k F weakly inW^2,2

S²,R³, g^E , F_k F^∗ weakly-* inW^1,∞

S²,R³, g^E ,

¯

g_k^∗ g¯ weakly-* inL^∞ S²

and

¯

g^#_k^∗ g¯^# weakly-* inL^∞ S²

,

(13)

where(g¯_k)_ij:=g(^∂Fk

∂xⁱ,^∂F_∂x^k

j)are the coefficients of the induced metric onΣ_k, pulled back toS², then W(F )lim inf

k→∞ W(F_k).

Proof. Let| ¯g_k| =det(gk)_ij, and let ν_k be the normal ofΣ_k with respect tog. The above convergence (13) implies that for any 1< q <∞we have

(g¯_k)_ij→ ¯g_ij inL^q S²

, (g¯_k)^ij→ ¯g^ij inL^q S²

, νk→ν inL^q

S²

, νk ∗

ν inL^∞ S²

,

| ¯g_k| → | ¯g| inL^q S²

, | ¯g_k|^∗ | ¯g| inL^∞ S²

. (14)

From Eq. (12) we find that W(Fk)=1

2

S²

g¯_k^ij

(Fk)^α_ij+Γ_βγ^α ◦Fk(Fk)^β_i(Fk)^γ_j

(g◦Fk)αβν_k^β2

|gk|dx.

We split this expression into two parts W(Fk)=W1(Fk)+W2(Fk) with

W1(Fk)=1 2

S²

¯ g_k^ij ∂²F_k^α

∂xⁱ∂x^j(g◦Fk)αβν_k^β 2

|gk|dx

and

W2(Fk)=1 2

S²

2g¯^ij_k(Fk)^α_ij(gαβ◦Fk)ν_k^βg¯_k^ab

Γ_εμ^δ ◦Fk

(Fk)^ε_a(Fk)^μ_b(gδρ◦Fk)ν_k^ρ

+

¯

g^ij_kΓ_βγ^α ◦F_k(F_k)^β_i(F_k)^γ_j(g_αβ◦F_k)ν_k^β2

|g_k|dx.

By the Sobolev embedding and smoothness ofgit follows thatg_αβ◦F_k converges inL^∞(S²)tog_αβ◦F and that Γ_βγ^α ◦F_kconverges inL^∞(S²)toΓ_βγ^α ◦F. In view of the convergence (14) it thus follows thatW2is continuous with respect to the convergence in (13):

W2(Fk)→W2(F ) fork→ ∞.

To analyzeW1(Fk)we letφbe the integrand in the definition ofW1(Fk):

W1(F_k)=1 2

S²

φ

¯

g^ij_k, (g_αβ◦F_k), ν_k,

|g_k|, (F_k)^α_ij dx.

Thenφis smooth with respect to all variables,φ0, andφis convex with respect to the last set of variables(F_k)^α_ij as it is the concatenation of the following three maps: The linear (in(F_k)^α_ij) map

g¯_k^ij, (gαβ◦Fk), νk,

|gk|, (Fk)^α_ij

→ ¯g_k^ij(Fk)^α_ij(gαβ◦Fk)ν_k^β, the convex mapξ →ξ²forξ ∈Rand the linear multiplication by√

|gk|. Lower semi-continuity then follows as in the proof of[17, Theorem 1.6]. 2

3.3. Differentiability

Given a mapF ∈B and a smooth vector fieldX∈X(R³)we have thatX◦F is inW^2,2∩W^1,∞(S²,R³, g^E).

Furthermore, for small enoughε >0 the mapF_t:=F+t (X◦F )is inBfor allt∈(−ε, ε)since the inverse of the metricg¯_t induced byF_t is a smooth function ofg¯_t.

We can thus calculate the variation ofWin direction ofX δ_XW(F )= ∂

∂t

t=0

W(F_t).

A fairly long but standard calculation shows thatW is indeed differentiable and that its variation in directionX is given by

δ_XW(F )=

S²

−Hg¯^ijg

∇_F²_i,FjX, ν

−2Hg¯^ikg¯^{j l}h_ijg(∇FiX, F_j)

+H²g(∇νX, ν)−HRic(X, ν)+1

2H²div^TXdμ

where Ric denotes the Ricci curvature of g, ∇ the connection of g and ∇² the second covariant derivative of vector fields with respect to g. There is also a formulation of the Euler–Lagrange equation for variations X∈ W^2,2∩W^1,∞(S²,R³)which are not induced by a smooth ambient variation. To consider such vector fields, deriva- tives ofXhave to be calculated with the pull-back∇^∗=F^∗∇ of the Levi-Civita connection ofg. This affects only the way in which the second derivatives are calculated. We obtain the following expression:

δ_XW(F )=

S²

−Hg¯^ijg

∇^∗2

F_i,F_jX, ν

−2Hg¯^ikg¯^{j l}h_ijg

∇_F^∗_iX, F_j

−HRic(X, ν)+1

2H²div^TXdμ.

Note that ifF isC⁴(S²,R³), we can integrate by parts the terms involving derivatives ofXand write the variation in a more familiar form:

δ_XW(F )=

S²

g(X, ν)

H+H|A˚|²+HRic(ν, ν) dμ.

4. Direct minimization

In this section we construct minimizers for the Willmore functional subject to a small area constraint by direct minimization. We assume that(M, g)is compact without boundary.

Fix a pointp∈M. Forr <inj(M, g, p)we consider the geodesic spheresS_r(p). By[11]these surfaces satisfy W

S_r(p)

=8π−4π r²

3 Sc(p)+O r³

. (15)

In particular, for a givenε >0 there exists a constant 0< a₀=a₀(ε)such that for|S_r(p)|a₀we haveW(S_r) 8π+ε.

Fix somea∈(0, a0). We consider a minimizing sequence forW of surfacesΣ_k with|Σ_k| =a. By comparison with geodesic spheres, we can assume thatW(Σ_k)8π+ε. Thus, in view of Lemma2.5there exists a constantC such that

diam(Σk)C

a₀^1/2+a₀

for allk. By choosinga₀small enough, we can ensure that 4 diam(Σk) <inj(M, g)uniformly. By compactness ofM, by choosing a₀ small enough, and by passing to a subsequence if necessary, we can assume that all the Σ_k are contained inBρ0/16(p)for some suitablep∈M, whereρ₀is as in Remark2.2. We decorate all geometric quantities onΣkby the subscriptk, i.e.Hk, νk, . . ..

By the Gauss equation, we have W(Σ_k)=8π+

Σk

|A˚_k|²dμk+

Σk

G(ν_k, ν_k)dμk.

By assumption we haveW(Σ_k)8π+ε. Since the curvature of(M, g)is bounded, we can estimate the last term by Ca₀. Thus we obtain the estimate

A˚_k²_L2(Σk)ε+Ca₀.

From|A_k|²= |A˚_k|²+¹₂H_k², we also get Ak²_L2(Σ_k)8π+2ε+Ca0.

In the following proposition we show that we can pass this sequence to a (weak) limit, and thatW is lower semi- continuous under this limit.

Proposition 4.1.Let(M, g)be compact without boundary. Then there existsε >0, depending only on the geometry ofM, such that the following holds. IfΣkis a sequence of immersed surfaces with

|Σ_k| =a < ε and W(Σ_k) <8π+ε (16)

then there exists a family of parametrizations F_k:S²→Σ_k

such that theF_kconverge weakly inW^2,2and weakly-* inW^1,∞to a limiting parametrization F :S²→Σ⊂(M, g)

such that|Σ| =aand W(Σ )lim inf

k→∞ W(Σ_k).

Proof. Let(ρ0, h0)be as in Remark2.2. Using the reasoning prior to the statement of this proposition, we can assume without loss of generality, thatΣ_k⊂Bρ₀(p)for somep∈M. Introducing normal coordinatesx:B_ρ0 →Bρ₀(p)we can pull-back theΣ_k for allkand the metricgtoB_ρ0 andghas the form

g=g^E+h

withhas in Remark2.2. Assuming thata₀is small enough, it is easy to see that Eq. (16) implies thatΣ_ksatisfies W^E(Σ_k) <8π+2ε

whereW^E denotes the Willmore functional computed with respect to the Euclidean background metricg^E. Via the Gauss equation with respect to the Euclidean background, this implies that

A˚^E_k

L²(Σk,g^e)<2ε

onΣ. The estimates of De Lellis and Müller[3,4]imply that there exist conformal (with respect to the Euclidean back- ground) parametrizationsF_k:S²→Σ_k which are uniformly bounded inW^2,2(S²,R³)∩W^1,∞(S²,R³). Moreover, the metricg¯_k is conformal to the standard metric onS²with conformal factoru_k which satisfiesu_kC⁰(S²)Cε.

Thus there is a subsequence of theFkwhich we relabel toFk, such that for any given 1p <∞we have

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

F_k F weakly inW^2,2

S²,R³, g^E , F_k F^∗ weakly-* inW^1,∞

S²,R³, g^E , F_k→F inW^1,p

S²,R³, g^E ,

¯

g_k^∗ g¯ weakly-* inL^∞ S²

and

¯

g_k^#∗ g¯^# weakly-* inL^∞ S²

,

(17)

for a functionF ∈W^1,∞∩W^2,2(S²,R³)and a metricg¯∈L^∞(S²). DenoteΣ=F (S²), then this convergence implies that|Σ| =limk→∞|Σ_k| =a. Proposition3.1implies the lower semi-continuity ofWwith respect to the convergence in Eq. (17). 2

In the following we show that the limiting parametrizationF ∈W^2,2∩W^1,∞(S², M) of Σ of the minimizing sequence F_k for the Willmore functional is smooth. This follows from suitable modifications of a recent result of Kuwert, Mondino and Schygulla[7]on the existence of smooth spheres minimizing

Σ|A|²dμresp.

Σ(|H|²+1)dμ in Riemannian manifolds satisfying suitable curvature conditions and a result of Schygulla[15]on the existence of a minimizing Willmore sphere with prescribed isoperimetric ratio inR³. Both of these results rely on the fundamental existence result (and especially the approximate graphical decomposition lemma) of Simon[16].

In the following we indicate how to modify the arguments of Section 3 in[7]in order to handle the present situation.

First of all we note that because of the above proposition we get that the Radon measures onM μ_k(E)=

F_k⁻¹(E)

dμgk(y) and α_k(E)=

F_k⁻¹(E)

|A_k|²dμgk,

whereg_k is the metric on S²induced by F_k, converge weakly to limiting Radon measuresμandα. Note that for W<8π+εthe monotonicity formula implies that the density ofμis one on its support. We have that