LOCAL FOLIATION OF MANIFOLDS BY SURFACES OF WILLMORE TYPE

ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du

Tobias Lamm, Jan Metzger & Felix Schulze

Local foliation of manifolds by surfaces of Willmore type Tome 70, n^o4 (2020), p. 1639-1662.

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LOCAL FOLIATION OF MANIFOLDS BY SURFACES OF WILLMORE TYPE

by Tobias LAMM, Jan METZGER & Felix SCHULZE (*)

Abstract. —We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature.

This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

Résumé. —Nour prouvons l’existence d’un feuilletage local d’une variété riema- nienne de dimension trois autour des points critiques de la courbure scalaire par les points critiques non dégénérés de la fonctionnelle de Willmore sous la contrainte d’aire petite. On adapte une méthode développée par Rugang Ye pour construire un feuilletage par des surfaces à courbure moyenne constante.

1. Introduction

In this paper we consider the Willmore functional F(Σ) = 1

4 Z

Σ

H²dµ

for surfaces Σ immersed in a 3-dimensional Riemannian manifold (M, g).

HereH =λ₁+λ₂ denotes the sum of the principal curvatures of Σ.

More precisely, we consider the variational problem (1.1) inf{F(Σ)|Σ,→M with|Σ|=a}

Keywords:Willmore surfaces, local foliation.

2020Mathematics Subject Classification:53A30, 53C40.

(*) The first and second author were supported by the DFG with grants LA 3444/1-1 resp. ME 3816/1-2. The authors would like to thank the referee for the careful reading of the paper.

wherea∈(0,∞) is a (small) prescribed constant and|Σ|denotes the area of Σ with respect to the induced metric.

The Euler–Lagrange equation for this variational problem is (1.2) ∆H+H|A|^◦2+HRic(ν, ν) =λH.

Here ∆ denotes the Laplace–Beltrami operator on Σ,

◦

A is the trace free part of the second fundamental form A of Σ and Ric(ν, ν) is the Ricci- curvature of (M, g) in direction of the normal ν to Σ. Note that the left hand side is two times the first variation ofF and the right hand side of this expression is a Lagrange-parameter λ ∈ R multiplied with the first variation of the area functional.

In previous papers the first two authors have shown that if (M, g) is compact then there exists a small a₀ ∈ (0,∞) depending only on (M, g) such that the infimum in (1.1) is attained for all a ∈ (0, a0) on smooth surfaces Σ_a [6]. See [1] and [14] for alternative proofs and [11] for a recent parabolic approach. Existence and multiplicity results of Willmore surfaces in Riemannian manifolds have been studied previously in a perturbative setting in [12], [13] where the functionals F and the L²-norm of A^◦ are considered without a constraint.

Fora→0 the surfaces Σaconverge to critical points of the scalar curva- ture [5, 6, 9]. A similar result has been obtained previously for small isoperi- metric surfaces by Druet [2]. This was later generalized by Laurain [8] to surfaces with constant mean curvature.

It is natural to ask about the precise structure of this family when a tends to zero. A similar situation was considered by Ye [15] for surfaces of constant mean curvature, which are critical for the isoperimetric problem, that is to minimize area subject to prescribed enclosed volume. He proves that given a non-degenerate critical point p of the scalar curvature one can find a pointed neighborhood ˙U =U \ {p} which is foliated by hyper- surfaces of constant mean curvature. That is ˙U = S

H∈(H0,∞)Σ_H where ΣH has constant mean curvature H. For H → ∞ these surfaces become spherical and approach geodesic spheresSr(p) with radiusr≈ _H². Ye uses an implicit function argument to show that the ΣH can be constructed as graphs overSr(0). The main difficulty is that the operator linearizing the mean curvature has an approximate kernel corresponding to translations.

This approximate kernel can be dealt with by allowing a translation of the S_r(0) and using the non-degeneracy of the second derivative of the scalar curvature. Our result in this paper is to adapt the method of Ye to the case of the Willmore functional. More precisely, we get the following result:

Theorem 1.1. — Let(M, g)be a smooth Riemannian manifold and let p∈M be such that∇Sc(p) = 0and such that∇²Sc(p)is non-degenerate.

Then there exists a₀ ∈ (0,∞), a neighborhood U of p and for each a ∈ (0, a0) a spherical surface Σa which satisfies (1.2) for some λ ∈ R and

|Σa|=a. TheΣa are mutually disjoint andS

(0,a0)Σa=U\ {p}.

More detailed information on the structure of this foliation can be found in Section 3 where the implicit function argument is carried out. In particu- lar, we refer to Corollary 5.2 for some comments about the local uniqueness of the Σa.

The paper is organized as follows: In Section 2 we calculate the expan- sion of the Willmore functional on small geodesic spheres to set up the argument. In Section 3 we use the implicit function theorem to solve the equation in a very similar manner to Ye. First we solve the equation in the kernel of the linearized operator using the non-degeneracy condition on the scalar curvature and by a generic implicit function argument we solve perpendicular to the kernel. Proposition 4.4 in Section 4 establishes that the Σa indeed form a foliation as claimed. Finally in Section 5 we prove a local uniqueness reslut for the Σa as solutions to (1.2).

Remarks. — During the preparation of this manuscript, the authors learned that Norihisa Ikoma, Andrea Malchiodi and Andrea Mondino [4]

have an independent proof of Theorem 1.1.

2. The Willmore operator on geodesic spheres

In this section we compute the basic geometric quantities and the Will- more operator of small geodesic spheres. We consider a setup similar as in [15], i.e. we consider a pointp∈M³ and an orthonormal basis{ej}³_j=1 ofTpM which we use to identify TpM withR³. Furthermore, we consider the map

φ:R³⊃B_ρp(0)→M :x7→exp_p(xⁱe_i),

whereρ_p>0 is the injectivity radius ofp. Letegbe the pulled back metric ofM viaφ, withh ·,· idenoting the euclidean metric onR³. We consider the map Ψ_σ:R³→R³:x7→σxand denoteg:=σ⁻²Ψ^∗_σg.e

We now compute the second fundamental form of Sρ(0) ⊂ TpM for 0< ρ < σ⁻¹ρ_p. The normal to S_ρ w.r.t. g is given byx/|x|, and working w.l.o.g. at the north pole, i.e.ei for i= 1,2 are tangent vectors and e3 is

parallel to the normal, we obtain hij =g ei,∇e_jν

=g

ei,∇e_j

x^l

|x|el

=g

ei, δ^jl

ρ −x^lx^j ρ³

el

+x^l

ρg ei,Γ^k_jlek

= 1 ρ

g_ij−x^lx^j

ρ² g_il+x^lΓ^k_jlg_ik

.

This yields sincex^l= 0 forl= 1,2

(2.1) hⁱ_j= 1

ρ δⁱ_j+x^lΓⁱ_lj .

Furthermore, we have

0 =g ν, p^⊥_ν(e_i)

= x^k

ρ g_ki−xⁱ ρ,

wherep^⊥_ν(·) =e− hei, νiν is the orthogonal projection onto the subspace perpendicular toν. This gives

(2.2) δ_im= ∂

∂x^m x^kg_ki

=g_mi+x^k ∂

∂x^mg_ki, which we can use to compute

x^lΓⁱ_lj =1 2g^ik

x^l ∂

∂x^lgjk+x^l ∂

∂x^jglk−x^l ∂

∂x^kglj

=1 2g^ik

x^l ∂

∂x^lgjk+δjk−gjk−δkj+gkj

=1

2g^ikx^l ∂

∂x^lgjk. (2.3)

We denote partial derivatives with a semicolon, instead of a comma for covariant derivatives. From [10] and the definition ofgwe have the formula (2.4) g_ij(x) =g_ij(0) +σ²

3 R_ipqjx^px^q+σ³

6 R_ipqj,rx^px^qx^r +σ⁴

1

20Ripqj,rs+2

45RipqtRjrst

x^px^qx^rx^s+O(σ⁵|x|⁵), where the curvature terms are all corresponding tog and are evaluated at 0. Since

∂

∂x^mg^ij =−g^ivg^jw ∂

∂x^mg_vw

and differentiating further (using that the first derivatives ofg_ij vanish at 0) we obtain:

(2.5) g^ij(x) =g^ij(0)−σ²

3 Rⁱ_pq^j x^px^q−σ³

6 Rⁱ_pq,r^j x^px^qx^r+O(σ⁴|x|⁴).

Combining (2.4) and (2.5), we see 1

2g^ikx^l ∂

∂x^lg_jk= 1 2g^ik ∂

∂s(g_kj(sx)) _s=1

=g^ik σ²

3 R_kpqjx^px^q+σ³

4 R_kpqj,rx^px^qx^r +σ⁴

1

10Rkpqj,rs+4

45RkpqtRjrst

x^px^qx^rx^s

+O(σ⁵|x|⁵)

!

= 1

3σ²Rⁱ_pqjx^px^q+1

4σ³Rⁱ_pqj,rx^px^qx^r +σ⁴

1

10Rⁱ_pqj,rs−1

45Rⁱ_pqtRjrst

x^px^qx^rx^s +O(σ⁵|x|⁵).

(2.6)

Combining this with (2.1) and (2.3), we can thus write ρ·hⁱ_j =δⁱ_j+1

2g^ikx^l ∂

∂x^lgji= exp(aⁱ_j) where exp here is the exponential map on matrices and

aⁱ_j := 1

3σ²Rⁱ_pqjx^px^q+1

4σ³Rⁱ_pqj,rx^px^qx^r +σ⁴

1

10Rⁱ_pqj,rs−7

90Rⁱ_pqtRjrst

x^px^qx^rx^s+O(σ⁵|x|⁵).

This yields det3

δ_jⁱ+1

2g^ikx^l ∂

∂x^lg_jk

= exp(tr(aⁱ_j))

= 1−1

3σ²Rpqx^px^q−1

4σ³Rpq,rx^px^qx^r +σ⁴

−1

10Rpq,rs−7

90R^k_pqtRkrst+1

18RpqRrs

x^px^qx^rx^s+O(σ⁵|x|⁵).

Note that for the Gauss curvature we have from (2.1) K_Sρ = 1

ρ²det₂

δⁱ_j+1

2g^ikx^l ∂

∂x^lg_jk

= 1 ρ²det₃

δ_jⁱ+r

2g^ik ∂

∂rg_kj

,

since_∂r^∂g_k3= 0 ∀ k= 1,2,3. Combining this with the above computation, this yields

KS_ρ(x)

= 1 ρ²

1−1

3σ²Ricpqx^px^q−1

4σ³Ricpq,rx^px^qx^r

−σ⁴

Ricpq,rs

10 + 7

90R^k_pqtRkrst −RicpqRicrs

18

x^px^qx^rx^s

+ρ⁻²O(σ⁵|x|⁵).

(2.7)

Combining (2.1), (2.3) and (2.6) we get for the mean curvature, using Ricpq=−Rⁱ_pqi,

HS_ρ = 1 ρtr2

δⁱ_j+1

2g^ikx^l ∂

∂x^lgjk

= 1 ρ

2 + tr3

1

2g^ikx^l ∂

∂x^lgji

= 1 ρ

2−1

3σ²Ricpqx^px^q−1

4σ³Ricpq,rx^px^qx^r

−σ⁴ 1

10Ric_pq,rs+ 1

45Rⁱ_pqtR_irst

x^px^qx^rx^s

+ρ⁻¹O(σ⁵|x|⁵).

(2.8)

For the norm squared of the traceless second fundamental form, we have

|A|^◦2=¹₂(H²−4K), which yields

(2.9) |A|^◦2=ρ⁻²σ⁴ 1

9Rⁱ_pq^tRirst−1

18RicpqRicrs

x^px^qx^rx^s

+ρ⁻²O(σ⁵|x|⁵) We now aim to compute the laplacian of H on S₁. Using formula (3.2) in [3], we have

∆^S1H = ∆^gHe −Hess(He)(ν, ν) +g(∇H, ~e H)

= ∆^gHe −Hess(He)(x, x)−Hxⁱ∂He

∂xⁱ, (2.10)

whereH~ =−Hνis the mean curvature vector ofS₁, andHe is any extension ofH. We have

(2.11) ∆^gHe =g^ij ∂²He

∂xⁱ∂x^j −g^ijΓ^k_ij∂He

∂x^k.

From (2.4) and (2.5) we get g^ijΓ^k_ij =1

2g^ijg^kl ∂

∂xⁱg_lj+ ∂

∂x^jg_il− ∂

∂x^lg_ij

=g^klg^ij ∂

∂xⁱg_lj−1

2g^klg^ij ∂

∂x^lg_ij

= g_τ^kl+O(σ²)

g^ij_τ +O(σ²)

× σ²

3 (Rlipj+ Rlpij)x^p+O(σ³)

−1

2 g_τ^kl+O(σ²)

g^ij_τ +O(σ²)

× σ²

3 (Rilpj+ Riplj)x^p+O(σ³)

=2

3σ²Ric^k_px^p+O(σ³).

(2.12)

We choose an extension of the mean curvatureH onS1via He :=ρH_Sρ.

Combining this with (2.8) and (2.12), this yields (2.13) −g^ijΓ^k_ij∂He

∂x^k = 4

9σ⁴Ric_p^kRickqx^px^q+O(σ⁵).

Similarly, using (2.5) and (2.8) we obtain

g^ij ∂²He

∂xⁱ∂x^j =

g^ij−σ²

3 Rⁱ_pq^jx^px^q+O(σ³)

·

−2

3σ²Ric_ij−σ³

2 (Ric_ij,p+2 Ric_ip,j)x^p

−σ⁴ 1

5Ricij,pq+2

5Ricip,jq+2

5Ricip,qj+1

5Ricpq,ij

+ 4

45R^s_ij^tRspqt+ 8

45R^s_ip^tRsjqt

x^px^q+O(σ⁵)

=−2

3σ²Sc−σ³Sc,px^p

−σ⁴ 3

5Sc_,pq+1

5(∆ Ric)_pq+2

5Ric_p^sRic_sq + 4

45Ric^stR_spqt+8

45R^{si t}_p R_siqt

x^px^q+O(σ⁵), (2.14)

where we used that, due to the sign convention on the curvature tensor (Ricij = −R^t_ijt) and the second contracted Bianchi identity 2 Ric^t_i,t = Sc_,i, we have

Ric_ip,qj= Ric_ip,jq+ R_{qj i}^s Ric_sp+R_{qj p}^s Ric_si and thus

g^ijRicip,qj= 1

2Sc,pq+ Ric_q^sRicsp+ R_q^is_pRicsi. This yields, using (2.11) with (2.13) and (2.14) that

(2.15) ∆^gHe =−2

3σ²Sc−σ³Sc,px^p

−σ⁴ 3

5Sc_,pq+1

5(∆ Ric)_pq− 2

45Ric_p^sRic_sq + 4

45Ric^stRspqt+8

45R^{si t}_p Rsiqt

x^px^q+O(σ⁵).

Furthermore, using (2.8) we see (2.16) − ∂He

∂xⁱ∂x^jxⁱx^j =2

3σ²Ricpqx^px^q+3

2σ³Ricpq,rx^px^qx^r +σ⁴

6

5Ric_pq,rs+12

45Rⁱ_pqtR_irst

x^px^qx^rx^s+O(σ⁵) and combining (2.3) with (2.6) and (2.8)

xⁱx^jΓ^k_ij∂He

∂x^k = 1

2g^klx^sx^j ∂

∂x^sg_jl∂He

∂x^k

=−2

9σ⁴R^k_pqrRic_ksx^px^qx^sx^r+O(σ⁵) =O(σ⁵), (2.17)

since R^k_pqrRic_ksx^px^qx^rx^s = 0 by symmetry considerations. Combining (2.16) and (2.17), this yields

−Hess(He)(x, x) =− ∂He

∂xⁱ∂x^jxⁱx^j+xⁱx^jΓ^k_ij∂He

∂x^k

= 2

3σ²Ric_pqx^px^q+3

2σ³Ric_pq,rx^px^qx^r +σ⁴

6

5Ricpq,rs+12

45Rⁱ_pqtRirst

x^px^qx^rx^s +O(σ⁵).

(2.18)

By (2.8) we have

−Hxⁱ∂He

∂xⁱ = 4

3σ²Ricpqx^px^q+3

2σ³Ricpq,rx^px^qx^r (2.19)

+σ⁴ 4

5Ricpq,rs+8

45Rⁱ_pq^tRirst−2

9RicpqRicrs

x^px^qx^rx^s+O(σ⁵) Combining (2.10) with (2.15), (2.18) and (2.19) we arrive at

∆^S1H = ∆^gHe −Hess(He)(x, x)−Hxⁱ∂He

∂xⁱ

=−2

3σ²Sc +2σ²Ricpqx^px^q

−σ³(Sc,px^p−3 Ricpq,rx^px^qx^r)

−σ⁴ 3

5Sc_,pq+1

5∆ Ric_pq−2

45Ric_p^kRic_kq + 4

45Ric^klR_kpql+8

45R^{kl m}_p R_klqm

x^px^q

+σ⁴

2 Ricpq,rs+4

9R^k_pq^lRkrsl−2

9RicpqRicrs

x^px^qx^rx^s +O(σ⁵)

(2.20)

We now aim to compute the area constrained Willmore equation on S1, that is forλ∈Rthe quantity

(2.21) W_σ,λ:= ∆^S1H+H|A|^◦2+HRic(ν, ν) +σ²λH.

To deal with the Ricci term we do a Taylor expansion in normal coordinates on the original manifold aroundp. We get for the Ricci curvature ofegthat

Ric_pq(x) = Ric_pq(0) + Ric_pq;r(0)x^r+1

2Ric_pq;rs(0)x^rx^s+O(|x|³). Rescaling as before via the map Ψσ, we obtain for the Ricci curvature ofg (2.22) Ric_pq(x) =σ²Ric_pq+σ³Ric_pq;rx^r+σ⁴

2 Ric_pq;rsx^rx^s+O(σ⁵|x|³).

Recall that we denote partial derivatives with a semicolon, instead of a comma for covariant derivatives. Since the Christoffel symbols and deriva- tives thereof are of order at leastσ² we see that we have onS¹:

Ric(ν, ν) =σ²x^px^q

Ric_pq+σRic_pq,rx^r+σ²

2 Ric_pq,rsx^rx^s

+O(σ⁵)

and thus, combining this with (2.8)

(2.23) HRic(ν, ν) = 2σ²Ricpqx^px^q+ 2σ³Ricpq,rx^px^qx^r +σ⁴

Ricpq,rs−1

3RicpqRicrs

x^px^qx^rx^s+O(σ⁵).

Combining this with (2.20), (2.8) and (2.9) we arrive at the following propo- sition, where we replace the radiusρbyr.

Proposition 2.1. — Letp∈M³and{e_j}³_j=1 be an orthonormal basis ofTpM, via whichTpMcan be identified withR³. Furthermore, we consider the map

φ:R³⊃Bρ_p(0)→M :x7→exp_p(xⁱei),

whereρ_p>0is the injectivity radius ofp. Letegbe the pulled back metric of M viaφ, and consider the map Ψr:R³→R³:x7→rxand the rescaled metricg:=r⁻²Ψ^∗_rg. Then fore 0< r < ρ_p one has the following expansion of the area constrained Willmore equation(2.21)onS1:

Wr,λ(S₁) = 2r²

λ−Sc

3 + 2 Ric_pqx^px^q

−r³(Sc_,px^p+ 5 Ric_pq,sx^px^qx^s)

−r⁴ λ

3 Ricpq+3

5Sc,pq+1

5∆ Ricpq−2

45Ric_p^kRickq

+ 4

45Ric^klRkpql+ 8

45R^{kl m}_p Rklqm

x^px^q +r⁴

3 Ricpq,st+2

3R^k_pq^lRkstl−2

3RicpqRicst

x^px^qx^sx^t +O(r⁵).

3. The equation

In this section we prove Theorem 1.1 via the implicit function theorem.

We consider a setup similar to Ye [15]. Let (M, g) be given with injectivity radiusρ >0. Fix a base point p∈M and an orthonormal frame{ej}³_j=1 forTp(M). Consider the map:

c:R³⊃Bρ(0)→M :τ7→exp_p(τ),

where exp_p:T_pM →M denotes the exponential map ofM atp. Lete^τ_j be the parallel transports of theejtoc(τ) along the geodesict7→c(tτ)|_t∈[0,1]. Define the map

F_τ :R³⊃B_ρ(0)→M :x7→exp_c(τ)(xⁱe^τ_i).

Let Ω1 := {ϕ ∈ C^4,12(S₁) | kϕk

C^4,12(S₁) < 1} and for ϕ ∈ Ω1 let S_ϕ :=

{(1 +ϕ(x))x|x∈S₁}. For τ ∈B_ρ ⊂R³ and r∈(0, ρ/2) let S(r, τ, ϕ) = Fτ(Ψr(Sϕ)) where Ψrdenotes scaling byras in Section 2. Define

Φ : (0, ρ/2)e ×B_ρ(0)×Ω₁×R→C¹²(S₁) : (r, τ, ϕ, λ)7→Φ(r, τ, ϕ, λ)e whereΦ(r, τ, ϕ, λ) is the functione

(3.1) ∆H+H|A|^◦2+HRic(ν, ν) +λH

evaluated onS(r, τ, ϕ) with respect to the metricg and pulled back to S₁ via the parameterizationx7→Fτ(Ψr((1 +ϕ(x))x)).

Our goal is to findr0∈(0, ρ/2) and a map

(0, r0)→Bρ×Ω1×R :r7→(eτ(r),ϕ(r),e eλ(r)) so that

Φ(r,e eτ(r),ϕ(r),e λ(r)) = 0.e

Then for allr∈(0, r₀) the surfaces Σ_r:=S(r,τ(r),e ϕ(r)) solve the equatione

∆H+H|A|^◦2+HRic(ν, ν) +eλ(r)H = 0

as claimed. Up to reparameterization, the family (Σr)_r∈(0,r0)is the family of solutions as in Theorem 1.1, see Corollary 4.3 for details.

An equivalent way to defineΦ(r, τ, ϕ, λ) is to evaluate the operator (3.1)e onS_ϕwith respect to the metriceg^r,τ := (φ_τ◦Ψ_r)^∗g. To get a uniform scale inr, we consider instead the rescaled metricg^r,τ :=r⁻²eg^r,τ and define the rescaled function Φ(r, τ, ϕ, λ) to be the operator

(3.2) ∆r,τHr,τ+Hr,τ|A^◦r,τ|²+Hr,τRicr,τ(ν, ν) +r²λHr,τ

evaluated on Sϕ with respect to g^r,τ and pulled back to S1 via the pa- rameterizationx7→(1 +ϕ(x))xofS_ϕ. From the scaling of the geometric quantities, we get

Φ(r, τ, ϕ, λ) =r³Φ(r, τ, ϕ, λ).e By definition

(3.3) Φ(r, τ,0, λ) =W_r,λ(S₁),

whereWr,λ(S₁) is from Proposition 2.1 and the geometric quantities in the expression forWr,λ(S1) are evaluated atc(τ). Note that after shifting byτ, the metricgin Proposition 2.1 corresponds to the metricg^r,τ here.

The linearization of the Willmore operator Φ is denoted bye Wλ. It was calculated in [7, Section 3]. For a variation of an arbitrary surface Σ with normal speedf it is given by

(3.4) W_λf =LLf+1

2∇^?(H²∇f)−2∇^?(HA(∇f,^◦ ·)) +λLf+f Q,

where∇^?=−div,L=−∆− |A|²−Ric(ν, ν), and (3.5)

Q=|∇H|²+ 2ω(∇H) +H∆H+ 2h∇²H,Ai^◦ + 2H²|A|^◦2 + 2HhA, T^◦ i −H∇Ric(ν, ν, ν)−1

2H²|A|²+1

2H²Ric(ν, ν).

Hereω= Ric(ν,·)^T is the tangential projection of the 1-form Ric(ν,·) to Σ andT =R(·, ν, ν,·). All the geometric quantities inWλare evaluated on Σ with respect to the corresponding ambient geometry. For givenf ∈C⁴(S1) the family t 7→ S(r, τ, tf) is a normal variation of S(r, τ,0) with normal speedrf, so that

(3.6) Φeϕ(r, τ,0, λ)f =rWλf.

Here we evaluateWλ with respect to the metricg in M. Rescaling to the g^r,τ metric, we find that

(3.7) Φϕ(r, τ,0, λ)f =r⁴Wλf =Wr,τ,λf,

whereWr,τ,λ is the linearized Willmore operator with respect tog^r,τ: (3.8) Wr,τ,λf =Lr,τLr,τf+1

2∇^?_r,τ(H_r,τ² ∇r,τf)

−2∇^?_r,τ(Hr,τ

◦

Ar,τ(∇r,τf,·) +r²λLr,τf +Qr,τf.

Here we use the subscriptr,τ to denote quantities evaluated with respect to the metricg^r,τ.

In the limit r→0 the metric g^r,τ converges to the Euclidean metric so that in the limit we have

W_0,τ,λf =L₀(L₀+ 2)f = (−∆)(−∆−2)f.

The kernel of this operator is given by

K:= kerW0,τ,λ= Span{1, x¹, x², x³},

where the xⁱ are the standard coordinate functions on S₁. We split this kernel into two parts:

(3.9) K₀:= Span{1} and K₁:= Span{x₁, x₂, x₃}.

As in [15], the function spaceC^4,12(S1) splits as a direct sum into K and its L²-orthogonal complement K^⊥. It is standard to verify that we have the direct sum decomposition of the target with respect to theL²-scalar product:

C^0,12 =K+W0,τ,λ(K^⊥).

Define theL²-orthogonal projection maps

P0:C^0,12(S1)→K0 and P1:C^0,12(S1)→K1.

The mapsT₀:K₀→RandT₁:K₁→R³ identifyK₀andK₁ withRand R³ according to the basis given in equation (3.9). Moreover, for i∈ {0,1}

letPei=Ti◦Pi. Denote by{e1, e2, e3}the standard basis ofR³. Lemma 3.1. — We have

Pe0(Φ(r, τ, ϕ, λ)) = 8πr²

λ+1

3Sc(c(τ))

+O(r⁴) +Pe0

Z 1

0

Φϕ(r, τ, tϕ, λ)ϕdt

and

Pe₁(Φ(r, τ, ϕ, λ)) = 4π

3 r³∇eiSc(c(τ))e_i+O(r⁵) +Pe₁

Z 1

0

Φϕ(r, τ, tϕ, λ)ϕdt

.

Proof. — Start by writing

Φ(r, τ, ϕ, λ) = Φ(r, τ,0, λ) + Z 1

0

Φϕ(r, τ, tϕ, λ)ϕdt.

By (3.3), fori∈ {0,1}

Pei(Φ(r, τ,0, λ)) =Pei(Wr,λ(S1)),

WhereW_r,λ(S₁) is evaluated at the base point c(τ). The right hand side can be calculated term by term from the expansion of Wr,λ(S1) given in Proposition 2.1:

Pe0(Wr,λ(S1)) = 8πr²

λ+1

3Sc(c(τ))

+O(r⁴) and Pe₁(Wr,λ(S₁)) = 4π

3 r³∇e_iSc(c(τ))ei+O(r⁵).

Note that all terms that contain an odd number of xⁱ-factors integrate to zero. For the other terms we used that R

S₁xⁱx^p = ^4π₃δip and a similar expression for integrals involving four factors of components ofx.

Lemma 3.2. — For every τ∈R³ and everyλ∈Rwe have that Φ_ϕr(0, τ,0, λ) = ∂

∂r _r=0

W_r,τ,λ= 0.

Proof. — For the proof, we have to calculate _∂r^∂

_r=0Wr,τ,λ from its ex- pression (3.8) taking into account its definition (3.4) and (3.5). Since we compute _∂r^∂ Wr,τ,λ at r = 0 we see that all terms that are product of at

least two quantities that vanish at (r, ϕ) = (0,0) do not contribute to the derivative. In particular

(3.10) ∂

∂r _r=0

|∇Hr,τ|²+ 2ωr,τ(∇Hr,τ) + 2h∇²Hr,τ,A^◦r,τi +2H_r,τ² |A^◦r,τ|²+ 2Hr,τhA^◦r,τ, Tr,τi+r²λLr,τ

= 0.

From the proof of [15, Lemma 1.3], we quote equation (1.15) _∂r^∂

_r=0g^r,τ = 0, its consequence _∂r^∂

_r=0∆r,τ = 0, equation (1.17) _∂r^∂

_r=0Ar,τ = 0,

∂

∂r

_r=0Ricr,τ = 0, and assertion (1), that is _∂r^∂

_r=0Lr,τ = 0. These iden- tities also imply that _∂r^∂

_r=0H_r,τ = 0 and _∂r^∂ _r=0

◦

A_r,τ = 0. From these formulas we find that

∂

∂r _r=0

Lr,τLr,τ + 2Hr,τ∆r,τHr,τ −1

2H_r,τ² |Ar,τ|²

= 0.

Since Ric_r,τ = O(r²) as in (2.22) and ∇r,τRic_r,τ(ν_r,τ, ν_r,τ, ν_r,τ) =O(r³) by a similar argument, also

∂

∂r _r=0

−Hr,τ∇r,τRicr,τ(νr,τ, νr,τ, νr,τ) +1

2H_r,τ² Ricr,τ(ν, ν)

= 0.

To treat the final remaining terms inW_r,τ,λ rewrite it to 1

2∇^∗_r,τ(H_r,τ² ∇r,τf)−2∇^∗_r,τ(H_r,τA^◦_r,τ(∇r,τf,·))

=−Hr,τh∇Hr,τ,∇r,τfi −1

2H_r,τ² ∆r,τf+ 2

◦

A_r,τ(∇Hr,τ,∇r,τf) + 2Hr,τh∇^∗_r,τA^◦r,τ,∇r,τfi+ 2Hr,τhA^◦r,τ,∇²_r,τfi

=−1

2H_r,τ² ∆r,τf+ 2A^◦r,τ(∇Hr,τ,∇r,τf) + 2Hr,τhA^◦r,τ,∇²_r,τfi + 2Hr,τωr,τ(∇r,τf).

(3.11)

In the last equality we used the Codazzi equation in the form −∇^∗A^◦ =

1

2∇H +ω. By inspection we see that each term in this expression has vanishing derivative inr-direction. Hence

∂

∂r _r=0

Wr,τ,λf = 0

as claimed.

Letϕ₀ be the unique solution of the PDE (3.12) W_0,τ,λϕ₀=

−4

3Sc +4 Ric_pqx^px^q

_r=0

.

Note that the right hand side of this equation is an element of K^⊥ and henceϕ0∈K^⊥ is indeed uniquely defined.

Lemma 3.3. — Let(M, g)be a three dimensional manifold and p∈M such that∇Sc(p) = 0and∇²Sc(p)is non-degenerate. For this base point there existsr₀ ∈(0,∞), an open neighborhood U ⊂C^4,12(S₁) ofϕ₀, and functions

λ: [0, r0)×U →R: (r, ϕ)7→λ(r, ϕ) and

τ: [0, r0)×U →R³: (r, ϕ)7→τ(r, ϕ) so that

Pei Φ(r, τ(r, ϕ), r²ϕ, λ(r, ϕ))

= 0 for i∈ {0,1}, (3.13)

τ(0, ϕ₀) = 0, and λ(0, ϕ₀) =−1 3Sc(p).

(3.14)

Proof. — Calculate:

Φ_ϕ(r, τ, tr²ϕ, λ) = Φ_ϕ(0, τ,0, λ) +r Z 1

0

Φ_ϕr(sr, τ, str²ϕ, λ) ds +tr²

Z 1

0

Φϕϕ(sr, τ, str²ϕ, λ)ϕds.

Moreover, we have r

Z 1

0

Φ_ϕr(sr, τ, str²ϕ, λ) ds=r² Z 1

0

Z 1

0

sΦ_ϕrr(usr, τ, ustr²ϕ, λ) duds +r³

Z 1

0

Z 1

0

stΦϕϕr(usr, τ, ustr²ϕ, λ)ϕduds +rΦϕr(0, τ,0, λ).

Hence

r²Φ_ϕ(r, τ, tr²ϕ, λ) =r²Φ_ϕ(0, τ,0, λ) +r³Φ_ϕr(0, τ,0, λ) +r⁴

Z 1

0

tΦϕϕ(sr, τ, str²ϕ, λ)ϕds +r⁴

Z 1

0

Z 1

0

sΦϕrr(usr, τ, ustr²ϕ, λ) duds+O(r⁵).

By equation (3.7) we have Φϕ(0, τ,0, λ) =W_0,τ,λ and from Lemma 3.2 we get Φϕr(0, τ,0, λ) = 0. Therefore

Pei r²Φϕ(r, τ, tr²ϕ, λ)

=O(r⁴) for i∈ {0,1}.

It follows from Lemma 3.1 that the system (3.13) is equivalent to 8π

λ+Sc

3

=−r⁻²Pe₀ Z 1

0

Φ_ϕ(r, τ, tr²ϕ, λ)ϕdt

+O(r²) =O(r²) and

4π

3 ∇e_iScei =rPe1

Z 1

0

Z 1

0

tΦϕϕ(sr, τ, str²ϕ, λ)ϕϕdsdt

+rPe1

Z 1

0

Z 1

0

Z 1

0

sΦϕrr(usr, τ, ustr²ϕ, λ)ϕdudsdt

+O(r²)

=O(r).

By assumption∇Sc(p) = 0. Hence, at r= 0, this system is satisfied for an arbitraryϕ₀∈K^⊥, ifλ|_r=0=−¹₃Sc(p) andτ|_r=0= 0.

The derivative with respect toλand τ at r= 0 of the left hand side of this system is given by the matrix

8π ^8π₃ ∇Sc|r=0

0 ^4π₃∇²Sc|r=0

=

8π 0 0 ^4π₃ ∇²Sc|r=0

.

By assumption ∇²Sc|_r=0 is non-degenerate. Hence, it follows from the implicit function theorem that there exist functions λ=λ(r, ϕ) andτ = τ(r, ϕ) as claimed at least for (r, ϕ) in a neighborhood of (0, ϕ0) ∈ R×

C¹²(S₁).

Lemma 3.4. — Assume that (M, g), p, φ0, r0, U, λ and τ are as in Lemma 3.3. Then there existsr1∈(0, r0]and a function

ϕ: [0, r1)→U :r7→ϕ(r) such that

Φ(r, τ(r, ϕ(r)), r²ϕ(r), λ(r, ϕ(r))) = 0 and ϕ(0) =ϕ₀.

In particular, for small enoughr, we have constructed a surface of Will- more type with Lagrange multiplierλ(r, ϕ(r)).