Foliations of small tubes in Riemannian manifolds by capillary minimal discs

Nonlinear Analysis 70 (2009) 4422–4440

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Foliations of small tubes in Riemannian manifolds by capillary minimal discs Mouhamed Moustapha Fall ∗ , Carlo Mercuri Sissa, Via Beirut 2-4, Trieste, 34014, Italy

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Article history: Received 8 October 2008 Accepted 8 October 2008 MSC: 53A10 53C21 35R35

a b s t r a c t Letting Γ be an embedded curve in a Riemannian manifold M , we prove the existence of minimal disc-type surfaces centered at Γ inside the surface of revolution of M around Γ , having small radius, and intersecting it with constant angles. In particular we obtain that small tubular neighborhoods can be foliated by minimal discs. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Minimal surfaces Exponential map Variational perturbative methods for PDE’s

1. Introduction Minimal surfaces are surfaces with mean curvature vanishing everywhere. These include, but are not limited to, surfaces of minimal area subjected to various constraints. In this paper we are interested in minimal surfaces which intersect a given hyper-surface with a constant angle. Such surfaces, called capillary minimal surfaces, are critical points of an energy functional under some constraints, see (3). Capillary surfaces correspond to the physical problem of describing an incompressible liquid in a container in the absence of gravity. A great deal of work has been devoted to capillarity phenomena from the point of view of existence, uniqueness and topological properties of solutions mainly in the non-parametric case and in the more general situation of presence of gravity (see the book of Finn, [7], for an account of the subject). Some answers to these questions have been obtained by many authors. We refer for example to the papers [9,10,4,12,16,18,17,20,21] and the references therein. Here we prove existence results of capillary surfaces with prescribed topology in Riemannian manifolds. Roughly speaking, we first show the existence of a class of capillary (minimal) disc-type surfaces embedded in a Riemannian surface of revolution (see below). In particular, shrinking enough the thickness of the surface of revolution, this class constitutes a foliation. Secondly we have existence of minimal disc-type surfaces embedded in a geodesic tube of a curve which intersect perpendicularly the boundary of the tube. The method we use is perturbative in nature and the main idea goes back to Ye, [23], subsequently employed with success by many authors to obtain existence of (large) constant mean curvature hyper-surfaces. For example, one can see [5,6,13– 15]. Before stating our results, some preliminaries are required. A surface of revolution is a surface created by rotating a parametric curve [a, b] 3 s → (κ(s), φ(s)) ∈ R2 lying on some plane around a straight line (the axis of rotation) in the same plane.

∗

Corresponding address: SISSA/ISAS, Functional Analysis, Via Beirut 2/4, 34014 Trieste, Italy. Tel.: +39 40 3787 479; fax: +39 40 3787 528. E-mail addresses: [email protected] (M.M. Fall), [email protected] (C. Mercuri).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.024

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

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The resulting surface C 1 therefore always has azimuthal symmetry. Examples of surfaces of revolution include cylinder (excluding the ends), hyperboloid, paraboloid, sphere, torus, etc. In more generality one can obtain surfaces of revolution in Rn+1 , n ≥ 2 using the standard parametrization S (s, z ) = (κ(s), φ(s) Θ (z )) , where z 7→ Θ (z ) ∈ S n−1 , φ(s) 6= 0 ∀s ∈ [a, b]. Assuming that the rotating curve is parametrized by arc length namely

(φ 0 (s))2 + (κ 0 (s))2 = 1, clearly the disc Ds,1 centered at (κ(s), 0) (on the axis of rotation) with radius φ(s) parametrized by Bn1 3 x 7→ (κ(s), φ(s) x) , has zero mean curvature and intersects the above surface of revolution with a constant angle equal to arccos φ 0 (s), where Bn1 stands for the unit ball of Rn centered at the origin, namely Ds,1 is a capillary surface. Motivated by capillarity problems, for questions of stability, see [7], it is not restrictive to assume that the angle of contact is in (0, π ), namely φ 0 (s) ∈ (−1, 1) or equivalently

κ 0 (s) 6= 0. We shall extend these definitions of surface of revolution in a Riemannian setting. Let (M n+1 , g ) be the Riemannian manifold, and Γ an embedded curve parametrized by a map γ : [0, 1] → M . We consider a local parallel orthogonal frame E1 , . . . , En of N Γ along Γ . This determines a coordinate system by

[0, 1] × Rn 3 (x0 , y) 7→ f (x0 , y) := expγ (x0 ) (yi Ei ) ∈ M. For a small parameter ρ > 0, consider the Riemannian surface of revolution C ρ around Γ in M parametrized by (s, z ) −→ f (ρ S (s, z )) = f (ρ κ(s), ρ φ(s)Θ (z )) = expγ (ρ κ(s)) (ρ φ(s)Θ i (z ) Ei ), where z 7→ Θ (z ) ∈ S n−1 , and call its interior Ωρ := int C ρ which is nothing but a tubular neighborhood for Γ if ρ is small enough. Here we are assuming always that φ(s) 6= 0 and that (φ 0 (s))2 + (κ 0 (s))2 = 1. For any s ∈ [a, b], we consider the following set Ds,ρ := f (ρ κ(s), ρ φ(s) Bn1 ), it is clear that ∂ Ds,ρ ⊂ C ρ and we have that the mean curvature HDs,ρ of Ds,ρ , see Section 4.1, satisfies HDs,ρ = O (ρ)

in Ds,ρ

(1)

while the angle between the unit outer normals (see also Section 4.2) can be expanded as

hNDs,ρ , NC ρ i = φ 0 (s) + O (ρ) on ∂ Ds,ρ .

(2)

Our aim is to perturb Ds,ρ to a capillary minimal submanifold, Ds,ρ , of Ωρ centered on Γ with contact angle arccos φ (s) along ∂ Ds,ρ ⊂ C ρ , as happens in Rn+1 . 0

Theorem 1.1. Suppose we are in the situation described above. Let [a0 , b0 ] ⊂ [a, b] be such that φ(s)φ 00 (s) > 0 for every s ∈ [a0 , b0 ]. Then there exists ρ0 > 0 such that for any s ∈ [a0 , b0 ] and ρ ∈ (0, ρ0 ), there exists an embedded minimal disc Ds,ρ ⊂ Ωρ , intersecting C ρ by an angle equal to φ 0 (s) along its boundary. Moreover Ds,ρ is a normal graph over the set Ds,ρ for which the norm (in the C 2,α -topology) of this function defining the graph tends to zero uniformly as ρ tends to zero. Furthermore there exists a tubular neighborhood Oρ of γ ([a0 , b0 ]) foliated by such minimal discs for which each leaf intersects ∂ Oρ transversally along its boundary. Remark 1.2. • When we parametrize in particular C ρ with κ(s) = s, and if we require the capillary discs to be perpendicular to C ρ , we obtain the conditions φ 0 = 0 and φ 00 6= 0. This means that non-degenerate extrema of the width φ determine the location of such surfaces. • An example is the hyperboloid, φ(s) = cosh s and κ(s) = sinh s. Here one may see M as a Lorentzian manifold modeled on the Minkowski space Rn1 . Letting q ∈ M and E0 a unit time-like vector of Tq M and γ (x0 ) = expq (x0 E0 ), one can see D0,ρ as a space-like minimal disc in the geodesic sphere of radius ρ . An interesting particular case which is not covered by Theorem 1.1 is when φ ≡ 1 and κ = Id, namely when we deal with geodesic tubes. In this situation (recall that in this case the angle of contact is π2 ) it is the geometry of the manifold to determine the position of the discs. More precisely, we have that C ρ is the geodesic tube of radius ρ > 0 around Γ , C ρ = {q ∈ M : distg (q, Γ ) = ρ},

and its interior is nothing but

Ωρ := {q ∈ M : distg (q, Γ ) < ρ}. In this case due to invariance by translations along the axis of rotation, we reduced our problem of finding minimal surfaces to a finite-dimensional one. Namely we have obtained the following.

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Theorem 1.3. There exists a smooth function ψρ : [a, b] → R such that, for ρ small, if s0 is a critical point of ψρ the set Ds0 ,ρ can be smoothly perturbed to an embedded minimal hyper-surface Ds0 ,ρ ⊂ Ωρ intersecting C ρ perpendicularly along its boundary. Furthermore, for any integer k, there exists a constant ck (independent of ρ ) such that

n

X

hRp (Ej , Ei )Ej , Ei i

ψρ −

i ,j

≤ ck ρ 2 ,

C k [a,b]

where Rp is the Riemann tensor of M at p = γ (ρ s). Some remarks are due: let Γ 3 p → Ψ (p) = i,j hRp (Ej , Ei )Ej , Ei i, any strict maxima or minima of Ψ imply the existence of minimal surfaces. In particular suppose at some point p0 = γ (ρ s0 ) interior to Γ , there hold

Pn

dΨ (p0 )[γ˙ (ρ s0 )] = 0

and

2 d Ψ (p0 )[γ˙ (ρ s0 ), γ˙ (ρ s0 )] > c ,

for some constant c independent of ρ . By the implicit function theorem, there exists a curve (0, ρ0 ) 3 ρ 7→ sρ with sρ → s0 such that sρ is a critical point of ψρ . Hence for every ρ ∈ (0, ρ0 ), there exists an embedded minimal disc Dsρ ,ρ , centered at γ (ρ sρ ), contained in Ωρ that intersects ∂ Ωρ perpendicularly along its boundary. Remark 1.4. • We have that

Ψ (p) =

n X hRp (Ej , Ei )Ej , Ei i = S (p) + 2 Ricp (γ˙ (ρ s), γ˙ (ρ s)), i ,j

where S (p) =

n X

hRp (Eα , Eβ )Eα , Eβ i

α,β=0

is the scalar curvature of M at p = γ (ρ s), E0 = γ˙ (ρ s) and Ricp is the Ricci tensor of M at p. From Theorem 1.3, we have that if s 7→ Ricp (γ˙ (ρ s), γ˙ (ρ s)) is locally constant along Γ then stable critical points of the scalar curvature yields existence of minimal discs. m m • Recall that if (M1 1 , g1 ) and (M2 2 , g2 ) are two manifolds, the Riemann tensor R of the (Riemannian) Cartesian product m1 +m2 M := (M1 × M2 , g1 ⊕ g2 ) decomposes as R = R1 ⊕ R2 since the connection ∇ is given by ∇X1 +X2 (Y1 + Y2 ) = 1 ∇X1 Y1 + ∇X22 Y2 for any X1 , Y1 (resp. X2 , Y2 ) vector fields of M1 (resp. M2 ), where ∇ i is the connection of Mi . Clearly for any p2 ∈ M2 , the set (M1 )(p2 ) := {(p1 , p2 ) ∈ M : p1 ∈ M1 } is a submanifold of M , diffeomorphic to M1 . In particular if m1 = 1, R1 = 0, by Theorem 1.3 we obtain that stable critical points of the mapping S |(M1 )(p2 ) yield existence of minimal discs inside (small) geodesic tubes around the curve (M1 )(p2 ), where as before S is the scalar curvature of M . • As a simple byproduct of our analysis, we find that if Γ is a closed curve, we have at least 2 (equal to the Lusternik–Schnierelman category of Γ , see [3]) solutions (without any assumptions on the curvature of M ). • We believe that this result might be generalized to higher codimensions namely if N ` , 1 < ` < n, is an `-dimensional submanifold of M n+1 and considering the following surface of revolution with axis of rotation R` S (s, z ) = (κ 1 (s), . . . , κ ` (s), φ(s) Θ (z )), where z 7→ Θ (z ) ∈ S n−` , one could obtain (n − ` + 1)-dimensional minimal disc-type submanifolds of M centered on N `. Let us describe the proof of the theorems above. We first recall, see [17], that Capillary hyper-surfaces with constant contact angle arccos φ 0 (s) are stationary for the energy functional

E (D) = Area(D ∩ Ωρ ) − φ 0 (s) Area(Ωρ0 ),

(3)

among (orientable smooth) surfaces D ⊂ Ωρ with ∂ D ⊂ ∂ Ωρ and Ωρ0 ⊂ ∂ Ωρ is the part (on one side of D) for which the angle is measured. Moreover the Euler–Lagrange equations are nothing but HD = 0 in D, hND , N∂ Ωρ i = φ 0 (s) on ∂ D.

(4)

Here HD is the mean curvature of D while ND and N∂ Ωρ are outer unit normals of D and ∂ Ωρ respectively. Since we look for stationary surfaces with a given profile for this energy functional, clearly by (1)–(2) a manifold of approximate solutions is given by Zρ := {Ds,ρ : s ∈ [a, b]}. For any given hyper-surface Ds,ρ ∈ Zρ , we parametrize (locally) a neighborhood of Ds,ρ (in the manifold in M ) by a mapping F s : R × Bn1 → M for which F s (t , ∂ Bn1 ) ⊂ ∂ Ωρ , for every t, while the direction F∗s (∂t ) is nearly normal to Ds,ρ , and moreover Ds,ρ = F s (0, Bn1 ), see (8). This allows us to parametrize any set D nearby Ds,ρ satisfying ∂ D ⊂ ∂ Ωρ by a function w : Bn1 → R such that D (w) = F s (w, Bn1 ). We call H (s, ρ, w) the mean curvature of D (w) and B (s, ρ, w) the angle between the normals N∂ D (w) and NC ρ of ∂ D (w) and ∂ Ωρ respectively.

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One of the main features in this work in (technical) Sections 4.1 and 4.2 is to calculate H (s, ρ, w) as a nonlinear elliptic partial differential operator, depending on ρ and s acting on w coupled with the mixed boundary operator which we denote by B (s, ρ, w). In these calculations it is important to gather various different types of error terms, some of which depend linearly and some nonlinearly on w , and some of which are inhomogeneous terms vanishing to some order in ρ . It turns out to be helpful to rescale the local coordinates y by ε(s) = ρφ(s) which is the radius of the discs. The final expression, Proposition 4.5, for the mean curvature of D (w) then is

φ H (s, ρ, w) = −Lρ,s (w) + O (ρ 2 ) + ρ Q (w) in D (w), κ0 where Lρ,s is the linearized mean curvature operator about D (0) = Ds,ρ : −

Lρ,s (w) = −∆ w + ρ Ls (w) in D (w); also the angle between the normals satisfies (see Proposition 4.6)


ρ −1 B (s, ρ, w) − φ 0 (s) = Bρ,s (w) + O (1) + ρ Q¯ (w), where

Bρ,s (w) =



(κ 0 (s))2

 ∂w + φφ 00 w + ρ L¯ s (w) on ∂ D (w). ∂η

Here Ls (resp. L¯ s ) is a second order (resp. first order) differential operator and Q (w), Q¯ (w) are quadratic in w , see also the end of Section 2 for more precise definitions. It turns out that the problem of finding w such that D (w) solves (4) namely

H (s, ρ, w) = 0 in D (w), B (s, ρ, w) = φ 0 (s) on ∂ D (w), can be transformed to a fixed point problem for which the solvability is based on the invertibility of Lρ,s on a suitable space of functions w such that Bρ,s (w) = 0. If φφ 00 > 0, the operator Lρ,s (resp. −Lρ,s ) is invertible by means of usual Sobolev inequalities. Hence after suitable adjustment of the disc D (w), we readily prove the first theorem. This program is carried out in Section 5.1. Now in the situation where φ ≡ 1 and κ = Id, it is clear that the linearized mean curvature Lρ,s about any D ∈ Zρ may have small (possibly zero) eigenvalues on the space of functions for which Bρ,s (w) = ∂w + ρ L¯ s (w) = 0. This is ∂η related to the invariance by translations along the axis of rotation in the ‘‘flat’’ case. Hence Lρ,s may not be invertible on such space. However restricting again ourselves on space of function orthogonal to the constant function 1, we can perturb Zρ to a manifold Zρ (constituted by sets having constant and small mean curvatures, see Section 5.3.1) which turns out to be a natural constraint for E , namely the critical point of E |Zρ is also stationary for E . For that we use an argument from Kapouleas in [11] which was successfully employed in [15]. We will follow the argument of the latter. We refer to Section 5.3.1. It is worth noticing that this method is also closely related to variational-perturbative methods introduced by Ambrosetti and Badiale in [1] and subsequently used with success to get existence and multiplicity results for a wide class of variational problems in some perturbative setting. We refer the reader to the book by Ambrosetti Malchiodi [2] for more details and related applications. Remark 1.5. • If φ 00 ≡ 0, namely when φ(s) = cs + d, S parametrizes the cone we cannot conclude. However we notice that the proof of Theorem 1.3 highlights that near a point γ (ρ s0 ) for which φ 00 (s0 ) = 0, there will be capillary surfaces with constant and small mean curvatures, see Remark 5.1. • An interesting question is also to perturb the set expγ (ρ κ(s)) (ρ φ(s)S n−1 ) to a closed minimal submanifold of C ρ . One can see also the work by Secchi [19]. • Another problem can be set as follows. Let U be a smooth bounded domain of M, and Γ ,→ ∂ U be a smooth curve. We let y˜ = (y1 , . . . , yn ) and N∂ U be a unit interior normal field along ∂ U. Choosing an oriented orthogonal frame (E1 , . . . , En−1 ) along Γ in ∂ U, one obtains a coordinate system by letting, for any y = (˜y, yn ) = (y1 , . . . , yn−1 , yn ), F (x0 , y˜ , yn ) := expM

i exp∂γ U (x0 ) (y Ei )

(yn N∂ U ).

Now consider the set F (ρ κ(s), ρ φ(s)Bn+ ), where Bn+ = {x = (x1 , . . . , xn−1 , xn ) ∈ Rn : |x| = 1, xn > 0}. One may be tempted to perturb the set above into capillary minimal surfaces that meet the ‘‘half ’’-surface of revolution n −1 F (ρ κ(s), ρ φ(s)S+ )

by an angle equal to arccos φ 0 . In this case, as we believe, a result like Theorem 1.1 would carry over. On the other hand one would need maybe to impose some conditions on the principal curvature of ∂ U along Γ in order to obtain a variant of Theorem 1.3.

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2. Preliminaries and notations We consider (φ, κ) : [a, b] → R2 smooth with κ 0 (s), φ(s) 6= 0 for every s ∈ [a, b] moreover we assume that s is the arc length of the rotating curve s 7→ (φ(s), κ(s)) precisely

(φ 0 (s))2 + (κ 0 (s))2 = 1 ∀s ∈ [a, b]. We also assume that x0 is the arc length of γ , and we will let E0 := γ 0 . We choose a parallel (local) orthonormal frame E1 , . . . , En of N Γ along Γ . This determines a coordinate system by defining f (x0 , y) := expγ (x0 ) (yi Ei ) for y = (y1 , . . . , yn ) which therefore defines coordinate vector fields: Yi := f∗ (∂yi ).

Y0 := f∗ (∂x0 ),

We will adopt the convention that the indices i, j, k, . . . ∈ {1, . . . , n} while α, β, . . . ∈ {0, . . . , n} with Yα = Y0 when α = 0. By construction, ∇Xi Y0 |Γ ∈ T Γ so that we can define

h∇Xi Y0 , Y0 i|Γ = −Γ00 (Ei ). There also holds

∇Yi Yj (y) = O(|y|)γ Yγ .

(5)

If q = f (x0 , y) ∈ M near the point p = f (x0 , 0) ∈ Γ , we can expand the metric gαβ (q) = hYα , Yβ i in y, (we refer to [13] Proposition 2.1 for the proof). Lemma 2.1. In the above coordinates (x0 , y), for any i, j = 1, . . . , n, we have gij (q) = δij + g0j (q) =

2 3

1 3

hRp (Y , Ei )Y , Ej i +

1 6

h∇Y Rp (Y , Ei )Y , Ej i + Op (|y|4 );

hRp (Y , E0 )Y , Ej i + Op (|y|3 );

g00 (q) = 1 − 2Γ00 (Y ) + hRp (Y , E0 )Y , E0 i + Op (|y|3 ), where Y := yi Ei . Notation for error terms: Any expression of the form L(ω) (resp. L¯ (ω)) denotes a linear combination of the function ω together with its derivatives with respect to the vector fields Yi up to order 2 (resp. order 1). The coefficients of L or L¯ might depend on ρ and s but, for all k ∈ N, there exists a constant c > 0 independent of ρ ∈ (0, 1) and s ∈ [a, b] such that

kL(ω)kC k,α (Bn ) ≤ c kωkC k+2,α (Bn ) , 1

1

kL¯ s (ω)kC k,α (Bn ) ≤ c kωkC k+1,α (Bn ) . +

1

Similarly, any expression of the form Q (ω) (resp Q¯ (ω)) denotes a nonlinear operator in the function ω together with its derivatives with respect to the vector fields Yi up to order 2 (resp. 1). The coefficients of the Taylor expansion of Q a (ω) in powers of ω and its partial derivatives might depend on ρ and s and, given k ∈ N, there exists a constant c > 0 independent of ρ ∈ (0, 1) and s ∈ [a, b] such that

  kQ (ω1 ) − Q (ω2 )kC k,α (Bn ) ≤ c kω1 kC k+2,α (Bn ) + kω2 kC k+2,α (Bn ) kω1 − ω2 kC k+2,α (Bn ) , 1

1

1

1

provided kωi kC k+2,α (Bn ) ≤ 1, i = 1, 2. Also 1

  kQ¯ (ω1 ) − Q¯ (ω2 )kC k,α (Bn ) ≤ c kω1 kC k+1,α (Bn ) + kω2 kC k+1,α (Bn ) kω1 − ω2 kC k+1,α (Bn ) , 1

1

1

1

provided kωi kC k+2,α (Bn ) ≤ 1. We also agree that any term denoted by O (r d ) (with r ∈ R may depend on s) is a smooth 1

function on Bn1 that might depend on s but satisfies



O (r d )

|r |d k,α n ≤ c C (B ) 1

for a constant c independent of s.

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

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3. On the surface of revolution around Γ We start by fixing the following notations which will be useful later. Notations: Throughout the following of this paper,

ε(s) = ρ φ(s) and ε1 (s) = ρ κ(s)

for every s ∈ [a, b].

In terms of cylindrical coordinates, letting Θ (z ) : Rn−1 → S n−1 , the surface of revolution C ε around Γ can be parametrized by

C ε (s, z ) := f (ε1 (s), ε(s)Θ (z )) = expγ (ε1 (s)) (ε(s)Θ i (z ) Ei ). The tangent plane is spanned by the vector fields Z0c = C ρ (∂x0 ) = ε10 Y0 + ε 0 Υ , Zjc = C ρ (∂z j ) = ε Υj ,

j = 1, . . . , n,

where

Υ = Θ i Yi . We recall also from [13]. Lemma 3.1. Let q = C ρ (s, z ) ∈ C ρ , there hold

hΥ , Υ iq = 1, hΥ , Y0 iq = 0, hΥ , Υj iq = 0. Lemma 3.2. In the notations above, the first fundamental form of C ρ has the following expansions

ε2 − 2ε|ε10 |2 Γ00 (Θ ) + ε 2 |ε10 |2 hR(Θ , E0 )Θ , E0 i + O (ε5 ), φ2 hZlc , Zkc i = ε 2 hΘl , Θk i + O (ε4 ), hZ0c , Z0c i =

hZ0c , Zkc i = O (ε4 ). Proof. Recalling that

|ε10 |2 + |ε 0 |2 =

ε2 φ2

hence we obtain, using also Lemmas 2.1 and 3.1, that


hZ0c , Z0c i = |ε10 |2 1 − 2ε Γ00 (Θ ) + ε 2 hR(Θ , E0 )Θ , E0 i + O (ε3 ) + |ε 0 |2 =

ε2 − 2ε|ε10 |2 Γ00 (Θ ) + ε 2 |ε10 |2 hR(Θ , E0 )Θ , E0 i + O (ε 5 ). φ2

The other expansions are easy consequences of Lemmas 3.1 and 2.1.



3.1. The unit normal field to the surface of revolution Call M := ε 0 X0 − ε10 Υ and set N˜c (s, z ) = M + α0 Z0c + αk Zkc . Note that this vector field is normal (not necessary unitary) to the surface whenever we can determine αk so that hN˜c , Zkc i =

hN˜c , Z0c i = 0 for all k = 1, . . . , n. This therefore leads to solving a linear system. Observe that

hM , Z0c i = ε10 ε 0 hY0 , Y0 i − ε10 ε 0 hΥ , Υ i
= ε10 ε 0 1 − 2εΓ00 (Θ ) + ε 2 hR(Θ , E0 )Θ , E0 i − 1 + O (ε3 ) ,

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

hence

hM , Z0c i = −2ε10 ε 0 εΓ00 (Θ ) + ε10 ε 0 ε 2 hR(Θ , E0 )Θ , E0 i + O (ε 5 ). Also we have

hM , Zkc i = εε 0 hX0 , Yk i − ε10 εhΥ , Yk i = O (ε 4 ). If we use Lemma 3.2, we have

α0 hZ0c , Z0c i = −αk hZkc , Z0c i − hM , Z0c i = αk O (ε4 ) + 2εε10 ε 0 Γ00 (Θ ) − ε2 ε10 ε0 hR(Θ , E0 )Θ , E0 i + O (ε 5 ) so

α0 =

0 3
ε 0 ε10 2 0 0 2 2 |ε1 | 2 ε Γ ( Θ ) − |ε | h R ( Θ , E ) Θ , E i + 4 φ φ ε 0 Γ00 (Θ )Γ00 (Θ ) + O (ε3 ) + αk O (ε4 ). 0 0 0 1 ε2 ε2

Since

αk hZkc , Zlc i + α0 hZ0c , Zlc i = −hM , Zlc i and using (6)

αk hZkc , Zlc i + αk O (ε6 ) + O (ε5 ) = O (ε 4 ) we get


αk ε2 hΘl , Θk i + O (ε4 ) = O (ε 4 ), therefore

αk = O (ε2 ). Recalling that ε = ρ φ while ε1 = ρ κ we define α¯ 0 by the relation

α0 = φ 0 α¯ 0 + O (ε3 ). Namely

α¯ 0 = 2ε10 φ Γ00 (Θ ) − ε0 ε10 φhR(Θ , E0 )Θ , E0 i + 4ε10 κ 0 ε Γ00 (Θ )Γ00 (Θ ). Now let us compute the norm of this normal vector field. Since N˜c (s, z ) := M + α0 Z0c + αk Zkc we have by construction

hN˜c , N˜c i = hM , M i + a20 hZ0c , Z0c i + αk αl hZkc , Zlc i + 2α0 hM , Z0c i + 2αk hM , Zkc i + 2αk α0 hZkc , Z0c i. Notice that

α0 hZ0c , Z0c i = −hM , Z0c i − αk hZkc , Z0c i = −hM , Z0c i + O (ε 6 ) and

α02 hZ0c , Z0c i = −α0 hM , Z0c i + O (ε6 ), hence

hN˜c , N˜c i = hM , M i + α0 hM , Z0c i + O (ε6 ). Now observe that

hM , M i = |ε 0 |2 hX0 , X0 i + |ε10 |2 hΥ , Υ i =


ε2 + |ε 0 |2 −2ε Γ00 (Θ ) + ε 2 hR(Θ , E0 )Θ , E0 i + O (ε3 ) φ2

and

α0 hM , Z0c i = −2εε10 ε 0 Γ00 (Θ )α0 + O (ε4 )α0 = −4

ε2 0 2 0 2 0 |ε | |ε1 | Γ0 (Θ )Γ00 (Θ ) + O (ε5 ). ρ2

So we have


φ2 ˜ ˜ |ε 0 |2 2 φ4 0 2 0 2 0 2 0 h N , N i = 1 + φ − 2 ε Γ ( Θ ) + ε h R ( Θ , E ) Θ , E i − 4 |ε | |ε1 | Γ0 (Θ )Γ00 (Θ ) + O (ε3 ). c c 0 0 0 ε2 ε2 ε2

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4429

Finally we conclude that

! 0 4 0 2 ε 4 2 ε1 0 2 4 ε ˜ −1 |ε0 |2 2 0 |ε 0 |2 |Nc | = 1 + φ Γ0 (Θ ) + 3 φ ε + 2 |ε | φ Γ00 (Θ )Γ00 (Θ ) − hR(Θ , E0 )Θ , E0 i + O (ε 3 ) φ ε ε ε 2 and, setting Hc (Θ , Θ ) :=

 3

|ε 0 |2 |ε 0 |2 + 2 12 2 ε ε



1

φ 2 Γ00 (Θ )Γ00 (Θ ) − hR(Θ , E0 )Θ , E0 i, 2

we can simply write

  ε ˜ −1 1 0 |Nc | = 1 + |ε 0 |2 φ 2 Γ0 (Θ ) + Hc (Θ , Θ ) + O (ε 3 ). φ ε We collect all these in the following. Proposition 3.3. There exists an interior (non unit) normal vector field of C ρ which has the following expansions N˜ C ρ (s, z ) = −ε10 Υ + ε 0 Y0 + (φ 0 α¯ 0 + O (ε 3 ))Z0c + αk Zkc , where

α¯ 0 = 2ε10 φ Γ00 (Θ ) − ε 0 ε10 φhR(Θ , E0 )Θ , E0 i + 4ε10 κ 0 εΓ00 (Θ )Γ00 (Θ ); αk = O (ε 2 ). Moreover

−1
ρ N˜ C ρ = 1 + |φ 0 |2 ε Γ00 (Θ ) + ε 2 Hc (Θ , Θ ) + O (ε3 ), where Hc (Θ , Θ ) = |φ 0 |2 + 2φ 4 Γ00 (Θ )Γ00 (Θ ) −




1 2

hR(Θ , E0 )Θ , E0 i + O (ε3 ).

4. Discs centered on Γ with boundary on C ρ For δ > 0, Bnδ will denote the ball of Rn with radius δ centered at the origin. For any s, we consider the disc D s,ε of radius ε centered at γ (ε(s)) given by D s,ε := f (ε1 (s), ε(s) Bn1 ),

parametrized by Bn1 3 x 7→ D s,ε (x) = f (ε1 , ε x). Notations

ε¯ (s, t ) := ε(s + ε(s)t ) ε¯ 1 (s, t ) := ε1 (s + ε(s)t ); 0 ε¯ (s, t ) := ∂t ε(s + ε(s)t ) = ε(s)ε0 (s + ε(s)t ) ε¯ 10 (s, t ) := ∂t ε1 (s + ε(s)t ) = ε(s)ε10 (s + ε(s)t ). Notice that

ε¯ 0 (s, t ) = (ε 0 + εε 00 t + ε 3 O (t 2 ))ε,

ε¯ 10 (s, t ) = (ε10 + εε100 t + ε 3 O (t 2 ))ε.

(7)

A parametrization of the neighborhood of the disc (in M ) centered at p = γ (ε1 (s)) ∈ Γ with radius ε (ρ small) can be defined by F s (t , x) := f (¯ε1 (s, t ), ε¯ (s, t ) x) ∀x ∈ Bn1 , |t |  1. Note that by construction, F s (t , ∂ Bn1 ) ⊂ C ρ ,

∀|t |  1

and more precisely, for every |t |  1 F s (t , x) = C ρ (s + ε t , ε(s + ε t ) x) ∀x ∈ ∂ Bn1 = S n−1 . We consider the following vector fields induced by F s

ε T0 := F∗ (∂t ) = ε¯ 10 Y0 (t ) + ε¯ 0 X (t ), ε¯ Tj := F∗ (∂yj ) = ε¯ Yj (t ). Here X = xi Ei .

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Lemma 4.1. At q = F s (t , x), we have

hYi (t ), Yj (t )iq = δij + 2ε 2

hY0 (t ), Yj (t )iq =

3

ε2 3

hRp (X , Ei )X , Ej i +

ε3 6

h∇X Rp (X , Ei )X , Ej i + Op (ε4 ) + ε 3 O (t ) + ε 4 Op (t 2 );

hRp (X , E0 )X , Ej i + Op (ε3 ) + ε 3 Op (t ) + ε 4 Op (t 2 );

hY0 (t ), Y0 (t )iq = 1 − 2ε¯ Γ00 (X ) + 2ε2 U00 (X )t + ε 2 hRp (X , E0 )X , E0 i + Op (ε3 ) + ε 3 Op (t ) + ε 4 Op (t 2 ), where p = γ (ε1 (s)) ∈ Γ and 0 ε U00 (X ) = ε10 Γ00 − ε 0 Γ00 (X ).

Proof. There holds d dt

hY0 (t ), Y0 (t )iq

t =0

= 2h∇εT0 Y0 , Y0 i t =0 = 2ε¯ 10 h∇Y0 Y0 , Y0 i t =0 + 2ε¯ 0 h∇X Y0 , Y0 i t =0 0 = 2ε(ε10 Γ00 − ε 0 Γ00 (X )) + Op (ε3 ).

We have used (7). Hence the last expansion follows. On the other hand one has d dt

hY0 (t ), Yi (t )iq

t =0

= h∇εT0 Y0 , Yi i|t =0 + hY0 , ∇εT0 Yi i|t =0 = ε¯ 10 h∇Y0 Y0 , Yi i t =0 + 2ε¯ 0 h∇X Y0 , Yi i t =0 + ε¯ 10 hY0 , ∇Y0 Yi i t =0 + 2ε¯ 0 hY0 , ∇X Yi i t =0 .

Since by construction h∇E0 E0 , Ei i + hE0 , ∇E0 Ei i = 0 and also h∇X E0 , Ei i + hE0 , ∇X Ei i = 0 on Γ , we infer that

hY0 (t ), Yi (t )iq dt d

= Op (ε3 ). t =0

In the same vein, the first expansion follows similarly.



Using the above lemma, and (7) we get: Lemma 4.2. The following expansions hold

hTi , Tj i = δij +

ε2 3

hRp (X , Ei )X , Ej i +

ε3 6

h∇X Rp (X , Ei )X , Ej i + Op (ε 4 ) + ε 3 O (t ) + ε 4 Op (t 2 );

hT0 , Tj i = ε 0 xi + Op (ε3 ) + ε 4 Op (t ) + ε 5 Op (t 2 ); hT0 , T0 i =


ε2 + |ε10 |2 −2ε¯ Γ00 (X ) + 2ε 2 U00 (X )t + Op (ε5 ) + ε 5 Op (t ) + ε 6 Op (t 2 ), 2 φ

where p = γ (ε1 (s)) ∈ Γ . Observe that all disc-type surfaces nearby Ds,ρ with boundary contained in C ρ can be parametrized by Gs (x) := F s (w(x), x), for some smooth function w :

(9) Bn1

→ R. We will denote D

s,ε(s)

(w) = G ( ). s

Bn1

4.1. Mean curvature of perturbed disc D (w) It is not difficult to see that the tangent plane of D (w) = D s,ε(s) (w) is spanned by the vector fields Zj = Gs∗ (∂xj ) = ε wxj T0 + ε¯ (s, w) Tj . From Lemma 4.2, it is clear that at the point q = Gs (x) = F s (w(x), x) there hold

hTi , Tj iq = δij +

ε2

ε3

h∇X Rp (X , Ei )X , Ej i + O (ε 4 ) + ε 3 L(w) + ε 4 Q (w); 6 hT0 , Tj iq = ε0 xj + O (ε3 ) + ε 4 L(w) + ε 5 Q (w); 3

hRp (X , Ei )X , Ej i +


ε2 hT0 , T0 iq = 2 + |ε10 |2 −2ε¯ Γ00 (X ) + 2ε 2 U00 (X )w + Op (ε 5 ) + ε 5 L(w) + ε 6 Q (w). φ

(10)

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4431

Observing that ε¯ (s, w) = ε + εε 0 w + ε 3 Q (w) and using (10), we get the first fundamental form hij := hZi , Zj i,

ε −2 hij = (1 + 2ε 0 w) δij + ε 0 (wxi xj + wxj xi ) +

ε2 3

hRp (X , Ei )X , Ej i +

ε3 6

h∇X Rp (X , Ei )X , Ej i

+ O (ε 4 ) + ε 3 L(w) + ε2 Q (w).

(11)

4.1.1. The normal vector field Consider the vector field N˜ D = Y0 + ak Zk . Observe that it is normal (not necessary unitary) to the disc whenever we can find ak such that hN˜ D , Zk i = 0 for any k = 1, . . . , n. Namely ak satisfies D s,ε(s) (w)ak hik = −hY0 , Zi i.

(12)

Since

hY0 , Zi i = ε¯ 10 wxi hY0 , Y0 i + ε¯ 0 wxi hY0 , X i + ε¯ hY0 , Yi i then from (7) and (11), we get the formula

ε 2 ak = εε10 wxk hY0 , Y0 i + εhY0 , Yk i + O (ε4 ) + ε 4 L(w) + ε 3 Q (w).

(13)

Also since ε¯ = ε + ε 2 L(w), ε¯ 1 = ε1 + ε 2 L(w), we get

ε 2 ak = −εε10 (1 − 2ε Γ00 (X ))wxk −

2ε 3 3

hR(X , E0 )X , Ek i + O (ε4 ) + ε 4 L(w) + ε 3 Q (w)

and thus ak = −

ε10 2ε (1 − 2ε Γ00 (X ))wxi − hR(X , E0 )X , Ei i + O (ε 2 ) + ε 2 L(w) + ε Q (w). ε 3

Moreover using also (12) we have

hN˜ D , N˜ D i = hY0 , Y0 i − ak al hkl = hY0 , Y0 i − ak (O (ε 3 ) + ε 2 L(w) + ε 4 Q (w))

= hY0 , Y0 i − O (ε) + L(w) + ε 2 Q (w) O (ε3 ) + ε 2 L(w) + ε 2 Q (w) = hY0 , Y0 i + O (ε4 ) + ε 3 L(w) + ε 2 Q (w). Hence

−1 ˜ hND , N˜ D i = |hY0 , Y0 i|−1 + O (ε4 ) + ε 3 L(w) + ε 2 Q (w).

(14)

Therefore

−1 ˜ hND , N˜ D i = 1 + ε¯ Γ00 (X ) + O (ε 2 ) + ε 2 L(w) + ε 2 Q (w).

(15)

We then conclude that the unit normal has the following expansions: ND = 1 + ε Γ00 (X ) + O (ε 2 ) + ε 2 L(w) + ε 2 Q (w) Y0




+

 0  ε 2ε − 1 (1 + ε Γ00 (X ))wxk − hR(X , E0 )X , Ek i + O (ε 2 ) + ε 2 L(w) + ε Q (w) Zk . ε 3

Sometimes we will simply need to write ND in the more compact form ND = Y0 + O (ε 2 ) + ε L(w) + ε 2 Q (w) α Yα .




(16)

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4.1.2. The second fundamental form Observe that in the scaled variables y = ε x, since the functions O (ε m ), L(w) and Q (w) are depending on x whereas the vector fields Yα depend on y = ε x, we have for any Ei (O (ε m )) = O (ε m−1 ),

Ei (ε m L(w)) = ε m−1 L(w),

Ei (ε m Q (w)) = ε m−1 Q (w).

Having this in mind, we state the following Lemma 4.3. There holds

hT0 , ∇Zi ND i = O (ε3 ) + ε L(w) + ε 2 Q (w). Proof. Using (16) and recalling that T0 = ε10 Y0 + ε 0 X + ε L(w)α Yα , we have

hT0 , ∇Zi ND i w=0 = hT0 , ∇εYi (1 + ε Γ00 (X ))Y0 i + O (ε3 ) = ε Γ00 (Ei )hT0 , Y0 i + ε(1 + ε Γ00 (X ))hT0 , ∇Yi Y0 i + O (ε 3 ) = εε10 Γ00 (Ei )hY0 , Y0 i + εε10 (1 + ε Γ00 (X ))hY0 , ∇Yi Y0 i + O (ε3 ) = εε10 Γ00 (Ei ) − εε10 Γ00 (Ei ) + O (ε3 ). Hence we get the result.



Let us now estimate the second fundamental form of D (w). Lemma 4.4.



h∇Zi Zj , ND i = ε 1 − ε Γ00 (El )xl wxi xj + ε 2 h∇Yi Yj , Y0 i + O (ε4 ) − ε 2 wxj Γ00 (Ei ) + wxi Γ00 (Ej ) + ε3 L(w) + ε 3 Q (w). Proof. We have

h∇Zi Zj , ND i = εh∇Zi (wxj T0 ), N i + h∇Zi (¯ε Tj ), ND i. We first estimate h∇Zi (wxj T0 ), ND i. Observe that

∂ hw j T0 , ND i = h∇Zi (wxj T0 ), ND i + hwxj T0 , ∇Zi ND i, ∂ xi x which implies that

h∇Zi (wxj T0 ), ND i =

∂ hw j T0 , ND i − wxj hT0 , ∇Zi ND i. ∂ xi x

Formula (12) shows that

2 hY0 , N˜ D i = hY0 , Y0 i + ak hZk , Y0 i = hY0 , Y0 i − ak al hZk , Zl i = N˜ D and then

hY0 , ND i = N˜ D . From the fact that hY0 , X i = 0 when w = 0 and that

hZk , X i = ε xk + O (ε4 ) + ε 2 L(w) + ε 5 Q (w) we obtain ak hZk , X i = O (ε 2 ) + ε L(w) + ε 2 Q (w), from which the following hold

hε T0 , ND i = ε¯ 10 hY0 , ND i + ε¯ 0 hX , ND i = ε¯ 10 N˜ D + ε¯ 0 (ε 2 + ε L(w) + ε 2 Q (w)) = εε10 (1 − ε Γ00 (X )) + O (ε4 ) + ε 3 L(w) + ε 4 Q (w). From this, we deduce that


∂ hwxj T0 , ND i = ε10 1 − ε Γ00 (X ) wxi xj − εε10 Γ00 (Ei )wxj + ε 3 L(w) + ε 2 Q (w). i ∂x We conclude using also Lemma 4.3 that


h∇Zi (wxj T0 ), ND i = ε10 1 − ε Γ00 (X ) wxi xj − εε10 Γ00 (Ei )wxj + ε 3 L(w) + ε 2 Q (w).

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4433

It remains the term h∇Zi (¯ε Tj ), ND i = εwxi h∇T0 (¯ε Tj ), ND i + ε¯ h∇Ti (¯ε Tj ), ND i. Since

ε¯ (s, w) = ε(s) + ε L(w) + ε 3 Q (w),

(18)

we can write

h∇Ti (¯ε Tj ), ND i = εh∇Ti Tj , N i + h∇Ti ((ε2 L + ε 3 Q )Tj ), ND i. Recalling that ∇Yi Yj = (O (ε) + ε L + ε 2 Q )α Yα also hTi , ND i = ε 2 + ε L + ε 2 Q , thus

h∇Ti (¯ε Tj ), ND i = εh∇Yi Yj , Y0 i|w=0 + O (ε 3 ) + ε 3 L(w) + ε3 Q (w). Moreover (7) and (18) yield

h∇T0 (¯ε Tj ), ND i|w=0 = εεε10 h∇Y0 Yj , ND i + εεε 0 h∇X Yj , ND i = −εεε10 Γ00 + O (ε4 ). This implies that

h∇T0 (¯ε Tj ), ND i = −εεε10 Γ00 + O (ε4 ) + ε 2 L(w) + ε 3 Q (w). Finally, collecting these and using (18) it turns out that

h∇Zi (¯ε Tj ), ND i = εεh∇Yi Yj , Y0 i|w=0 + O (ε4 ) − εεε10 Γ00 (Ej )wxi + ε 4 L(w) + ε 3 Q (w). The result follows from (17) and (19).

(19)



We also need to expand more precisely h∇Yi Yj , Y0 i|w=0 . By construction it vanishes on Γ and Yl h∇Yi Yj , Y0 i = h∇Yl ∇Yi Yj , Y0 i + h∇Yi Yj , ∇Yl Y0 i. Furthermore by (5) and since (see for instance [8] Lemma 9.20)

∇Yl ∇Yi Yj |γ (x0 ) = −

1 3

R(El , Ei )Ej + R(El , Ej )Ei ,




it follows that

ε h∇Yi Yj , Y0 i = − (hR(X , Ei )Ej , E0 i + hR(X , Ej )Ei , E0 i) + O (ε2 ). 3

We conclude from Lemma 4.4 that the Second fundamental form qij = h∇Zi Zj , N D i of the perturbed disc D s,ε(s) (w) centered at the point γ (ε(s)) with radius ε(s) is given by



ε3 hR(X , Ei )Ej , E0 i + hR(X , Ej )Ei , E0 i qij = εε10 1 − ε Γ00 (El )xl wxj xi − 3
+ O (ε4 ) − ε 2 ε10 wxj Γ00 (Ei ) + wxi Γ00 (Ej ) + ε 4 L(w) + ε 3 Q (w). We recall that if Eα is an orthogonal basis of Tp M , then Ricp (X , Y ) = −hRp (X , Eα )Y , Eα i

∀X , Y ∈ Tp M.

Finally we obtain

qij hij =


ε10 2ε 1 − ε Γ00 (X ) ∆w − Ricp (X , E0 ) + O (ε 2 ) − 2ε10 Γ00 (∇w) + ε 2 L(w) + ε Q (w), ε 3

where X = xl El . Proposition 4.5. In the above notations, the mean curvature H (s, ρ, w) of Ds,ρ (w) has the following expansions

φ 2ρ φ 2 H (s, ρ, w) = ∆w − Ricp (X , E0 ) + O (ρ 2 ) + ρ L(w) + ρ Q (w). 0 κ 3 κ0 In particular if Γ is a geodesic, Γ00 = 0 then

φ 2ρ φ 2 H ( s , ρ, w) = ∆ w − Ricp (X , E0 ) + O (ρ 2 ) + ρ 2 L(w) + ρ Q (w). κ0 3 κ0

(20)

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4.2. Angle between the normals By construction, at q = Gs (x) we have F s (w(x), x) = C ε (s + εw(x), ε(s + εw(x))), for every x ∈ ∂ Bn1 . Recall from Sections 3.1 and 4.1.1 that N˜ C ρ (s, z ) = −ε10 (s)X + ε 0 (s)Y0 + α0 Z0c + αk Zkc , where α0 = φ 0 α¯ 0 + O (ε 3 ) and αk = O (ε 2 ) also N˜ D = Y0 + ak Zk . One easily verifies that

αk α0 c hZ0 , Y0 iq + hZkc , Y0 iq ρ ρ αk ak c 0 0 − κ ak hX , Zk iq + φ ak hZk , Y0 iq + α0 hZ0 , Zk iq + al hZk , Zlc iq . ρ ρ

hN˜ D , ρ −1 N˜ C ε (s + εw)iq = −κ 0 hX , Y0 iq + φ 0 hY0 , Y0 iq +

We have to expand

κ 0 (s + εw) = κ 0 (s) + εκ 00 (s)w + ε 2 Q (w)

φ 0 (s + εw) = φ 0 (s) + εφ 00 (s)w + ε 2 Q (w).

We will also need the following result which uses just the expansions of the metric Lemma 4.1

hZ0c , Zk i = O (ε2 ) + ε 2 L(w) + ε 3 Q (w), hZlc , Zk i = O (ε2 ) + ε 2 L(w) + ε 3 Q (w), hX , Zk iq = (ε + ε 0 εw)xk + ε 3 L(w) + ε 5 Q (w). We use the fact that ak hZl , Zk i = −hY0 , Zk i to obtain

αk ρ −1 hN˜ D , N˜ C ε (s + εw)iq = (φ 0 + εwφ 00 )hY0 , Y0 iq + (κ 0 + ε κ 00 w)α0 hY0 , Y0 iq + hZkc , Y0 iq ρ ak αk 0 00 0 c c − (κ + εκ w)ak hX , Zk iq − φ ak al hZk , Zl iq + α0 hZ0 , Zk iq + al hZk , Zl iq + ε 3 L(w) + ε 2 Q (w). ρ ρ

Now from (21) we get

−φ 0 ak al hZk , Zl iq +

α0 αk ak hZ0c , Zk iq + al hZk , Zlc iq = O (ε 3 ) + ε 3 L(w) + ε 2 Q (w) ρ ρ

and also since

hZkc , Y0 iq = O (ε3 ) + ε 3 L(w) + ε 4 Q (w), one has

ρ −1 hN˜ D , N˜ C ε (s + εw)iq = (φ 0 + εwφ 00 )hY0 , Y0 iq + (κ 0 + εwκ 00 )α¯ 0 hY0 , Y0 iq − (κ 0 + εκ 00 w)ak hX , Zk iq + O (ε 3 ) + ε 3 L(w) + ε2 Q (w). From (21) and recalling the formula for ak in (13) we get

∂w hY0 , Y0 iq − (1 + ε0 w)hY0 , X iq + O (ε3 ) + ε 3 L(w) + ε 2 Q (w) ∂η ∂w = −ε10 hY0 , Y0 iq + O (ε3 ) + ε 3 L(w) + ε 2 Q (w) ∂η

ak hX , Zk iq = −ε10

and then we deduce that


∂w hY0 , Y0 iq + 2ε 2 κ 00 κ 0 w Γ00 (X ) ρ −1 hN˜ D , N˜ C ε (s + εw)iq = φ 0 1 + κ 0 α¯ 0 hY0 , Y0 iq + (εwφ 00 + κ 0 ε10 ) ∂η + O (ε3 ) + ε 3 L(w) + ε 2 Q (w). Using (14), we have that


∂w |Y0 |q + 2ε 2 κ 00 κ 0 w Γ00 (X ) ρ −1 hND , N˜ C ε (s + εw)iq = φ 0 1 + κ 0 α¯ 0 |Y0 |q + (εwφ 00 + κ 0 ε10 ) ∂η + O (ε3 ) + ε 3 L(w) + ε 2 Q (w). Since α0 = φ 0 (s + εw)α¯ 0 + O (ε 3 ), one has

α0 = (φ 0 (s) + φ 00 (s)ε(s)w)α¯ 0 + ε 2 Q (w)α¯ 0

(21)

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4435

so that

α0 = φ 0 (s)α¯ 0 − 2φ 00 (s)ε10 εwΓ00 (X ) + ε3 L(w) + ε 2 Q (w). Moreover notice that

Γ00 (X )|q = Γ00 (X ) + ε 2 L(w) + ε 3 Q (w) and also

|φ 0 (s + εw)|2 = (φ 0 + 2εwφ 00 )φ 0 + ε 2 Q (w), we have that

−1 ρ N˜ C ρ (s + εw) = 1 + (φ 0 + 2εφ 00 w + ε 0 w)εφ 0 Γ00 (X ) + |φ 0 (s)|2 ε 2 Hc (X , X ) + O (ε 3 ) + ε 3 L(w) + ε3 Q (w) from which we deduce that

−1
ρ N˜ C ρ (s + εw) |Y0 |q = 1 − (κ 0 )2 ε Γ00 (X ) + ε 2 3 − (φ 0 )2 + (φ 0 )4 + 2(φ 0 )2 φ 4 Γ00 (X )Γ00 (X ) q

+

2 − (φ 0 )2


hRp (X , E0 )X , E0 i + εw −ε 0 + 2εφφ 00 + ε 0 φ 0 Γ00 (X ) + 2ε2 U00 (X )w + O (ε3 ) + ε 3 L(w) + ε 2 Q (w).

2 Consequently we may expand the angle as

−1
hND , NC ε (s + εw)iq = ρφ 0 1 + κ 0 α¯ 0 |Y0 |q N˜ C ρ (s + εw) + 2ε 2 κ 00 κ 0 w Γ00 (X ) q  
0 2 0 00 0 0 ∂w + O (ε3 ) + ε 3 L(w) + ε 2 Q (w). + 1 − (κ ) ε Γ0 (X ) εwφ + κ ε1 ∂η  

∂w hND (w) , NC ε (s + εw)iq = φ 0 (s) 1 + (κ 0 )2 ε Γ00 (X ) + 1 − (κ 0 )2 ε Γ00 (X ) εwφ 00 + κ 0 ε10 ∂η
+ φ 0 ε 2 3 − 2(κ 0 )4 − (φ 0 )2 + (φ 0 )4 + 2(φ 0 )2 φ 4 Γ00 (X )Γ00 (X )
2 − (φ 0 )2 0 2 φ ε hRp (X , E0 )X , E0 i + εφ 0 w −ε0 + 2εκ 00 κ 0 + 2εφφ 00 + ε 0 φ 0 Γ00 (X ) + 2

+ 2ε2 U00 (X )w + O (ε3 ) + ε 3 L(w) + ε 2 Q (w). We define

B (s, ρ, w) := hND (w) , NC ε (s + εw)iq . Now we conclude this section by collecting all these in the following. Proposition 4.6. In the above notations, there holds

B (s, ρ, w) = φ (s) 1 + (κ ) ρφ 0

0 2

Γ00


(X ) + ρ

  00 0 2 ∂w + φφ w + O (ρ 2 ) + ρ 2 L¯ (w) + ρ 2 Q¯ (w), (κ ) ∂η

while if φ 0 (s) = 0, one has

B (s, ρ, w) = ρ(κ 0 )2

∂w + O (ρ 3 ) + ρ 3 L¯ (w) + ρ 2 Q¯ (w). ∂η

In particular if Γ is a geodesic, we get precisely

   ∂w ρ 2 φ 2 hRp (X , E0 )X , E0 i + ρ (κ 0 )2 + φφ 00 w 2 ∂η 3 3¯ 2 ¯ +O (ρ ) + ρ L(w) + ρ Q (w).

B (s, ρ, w) = φ 0 (s)



1+

1 + (κ 0 )2

5. Existence of capillary minimal submanifolds 5.1. Case where φ(s0 )φ 00 (s0 ) > 0 We may assume that φ(s)φ 00 (s) > 0 for all s ∈ Is0 (δ) := [s0 − δ, s0 + δ] for some δ > 0 small. We define the following operator by

φφ 00 (Ls w, v) := ∇w∇v dx + 0 2 (κ ) Bn1 Z

I ∂ Bn1

wv dσ .

4436

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

It is clear that from the inequality (see [22], Theorem A.9)

Z

Z

w dx ≤ C (n) 2

Bn1

!

I

2

w dσ , 2

|∇w| dx + ∂ Bn1

Bn1

∀w ∈ H 1 ,

(22)

s,ρ

the operator Ls is coercive if ρ is small. We call w1 the unique solution to the equation

(Ls w1 , v) := −φ

I ∂ Bn1

Γ00 (X )v dσ .

Namely it solves the problem

−∆w1 = 0 in Bn1 , ∂w1 φφ 00 + 0 2 w1 = −φ Γ00 (X ) ∂η (κ )

on ∂ Bn1 .

By elliptic regularity theory, there exists a constant c > 0 (independent of ρ and s) such that

s,ρ

w 1

C 2,α

≤ c ∀s ∈ Is0 (δ).

Moreover we have that for all k ≥ 0

k s,ρ

∂ w1

∂ sk

≤ ck ∀s ∈ Is0 (δ),

C 2,α

for some constant ck which neither depends on s nor on ρ small. Clearly by construction there holds



s,ρ

H (s, ρ, w1 ) = O (ρ) s,ρ B (s, ρ, w1 ) = φ 0 (s) + O (ρ 2 )

s,ρ

in , Ds,ρ (w1 ), s,ρ on ∂ Ds,ρ (w1 ).

We define the space

Cs2,ρ,α

 ∂ s,ρ B (s, ρ, w1 + v) [w] = 0 := w ∈ C ( ) : ∂v v=0   00 φφ ∂w 2,α + 0 2 w + ρ L¯ s (w) = 0 . = w ∈ C (Bn1 ) : ∂η (κ ) 

2,α

Bn1

s,ρ

We consider the linearized mean curvature operator about D s,ρ (w1 ) (see Proposition 4.5), Lρ,s (w) : C 2,α (Bn1 ) → C 0,α (Bn1 ) defined by

Lρ,s (w) := −

φ ∂ s,ρ [w] = −∆ w + ρ Ls (w). H ( s , ρ, w + v) 1 0 κ ∂v v=0

We define also Φ (s, ρ, x), Qs,ρ (w) ∈ C 0,α (Bn1 ) by duality as


φ Φ (s, ρ, x), w 0 := − 0 κ

Z Bn1

s,ρ

H (s, ρ, w1 )w 0 dx + ρ −1

I

s,ρ

∂ Bn1

B (s, ρ, w1 ) − φ 0 (s) w 0 ds




and for every w ∈ C 2,α

Qs,ρ (w), w 0 :=




Z Bn1

Q (w)w 0 dx +

I ∂ Bn1

Q¯ (w)w 0 ds,

∀w0 ∈ L2 .

Clearly the solvability of the system



s,ρ

H (s, ρ, w1 + w) = 0 s,ρ B (s, ρ, w1 + w) = φ 0 (s)

s,ρ

inDs,ρ (w1 + w), s,ρ on ∂ Ds,ρ (w1 + w)

(23)

is equivalent to the fixed point problem

 −1  w = − Lρ,s |C 2,α Φ (s, ρ, x) + ρ Qs,ρ (w) . s,ρ

Furthermore one has

kQs,ρ (w)kC 0,α = O(kwkC 2,α ) kwk2C 2,α ; kQs,ρ (w1 ) − Qs,ρ (w2 )kC 0,α = O(kw1 kC 2,α , kw2 kC 2,α )kw1 − w2 kC 2,α ,

(24)

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4437

also by construction, there exists a constant c > 0 (independent of ρ and s) such that

kΦ (s, ρ, ·)kC 0,α ≤ c ρ ∀s ∈ Is0 (δ). ,α By (22) the operator Lρ,s is coercive on Cs2,ρ if ρ is small enough and also by elliptic regularity theory, Lρ,s is an isomorphism ,α ,α from Cs2,ρ into C 0,α (Bn1 ) therefore we can solve the fixed point problem (24) in a ball of Cs2,ρ with radius C ρ for some C > 0 which does not depend neither on ρ small nor s. ,α , with kw s,ρ kC 2,α ≤ C ρ such that Thus for ρ small and s ∈ Is0 (δ) there exists a function w s,ρ ∈ Cs2,ρ



s,ρ

H (s, ρ, w1 + w s,ρ ) = 0 s,ρ B (s, ρ, w1 + w s,ρ ) = φ 0 (s)

s,ρ

in Ds,ρ (w1 + w s,ρ ), s,ρ on ∂ Ds,ρ (w1 + w s,ρ ).

s,ρ

Namely Ds,ρ (w1 + w s,ρ ) is a capillary submanifold of Ωρ with constant contact angle arccos φ 0 (s) if ρ is small enough by s,ρ C 2,α bound up to the boundary of w ˜ s,ρ = w1 + w s,ρ . Furthermore it follows from the construction that, for all k ≥ 0

k s,ρ

∂ w

˜

∂ sk

C 2,α

≤ ck ρ ∀s ∈ Is0 (δ),

(25)

for some constant ck which does not depend on s nor on ρ small.

5.2. Foliation by minimal discs s,ρ

Denote w ˜ s,ρ = w1 + w s,ρ . From (8), Lemma 4.1 and (25) the mapping Ψρ

Is0 (δ) × Bn1 3 (s, x) −→ F s (w ˜ s,ρ (x), x) = f (¯ε1 (s, w ˜ s,ρ (x)), ε¯ (s, w ˜ s,ρ (x))x) has a Jacobian determinant which expands as

ρ 2n+2

1 − |x|2 (φ 0 )2 φ 2n + Os (ρ)







and hence since (φ 0 )2 ∈ (0, 1) (see Section 1), Ψρ is a local homeomorphism if ρ is small enough. In particular it is a homeomorphism of a neighborhood of (s0 , 0) which implies that there exist 0 < δ 0 < δ and % > 0 such that

Ψρ (s, Bn% ) ∩ Ψρ (s0 , Bn% ) = ∅ ∀s 6= s0 ∈ Is0 (δ 0 ), for every ρ sufficiently small. In this way the family of discs Ds,ρ% (w ˜ s,ρ ), s ∈ Is0 (δ 0 ) with radius ρ% φ(s) centered at γ (ρ κ(s)) constitutes a foliation of a neighborhood of γ (ρ κ(s0 )) for which each leaf Ds,ρ% (w ˜ s,ρ ) is a minimal disc intersecting C ρ% transversely along its boundary (the angle of contact may not be equal to arccos φ 0 (s)). 5.3. φ ≡ 1 and κ = Id, C ρ is the geodesic tube around Γ In this situation, C ρ = {q ∈ M : distg (q, Γ ) = ρ}

and its interior is

Ωρ = {q ∈ M : distg (q, Γ ) < ρ}. By [17], it is well known that (smooth) minimal surfaces D ⊂ Ωρ with ∂ D ⊂ C ρ are stationary for the area functional relative to C ρ which is D 7→ Area(D ∩ Ωρ ) under variations Ψt : D → M such that ∂ Ψt (D) ⊂ C ρ . Moreover the Euler–Lagrange equations are given by in D, hND , NC ρ i = 0 on ∂ D.

HD = 0

(26)

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M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

5.3.1. A finite-dimensional reduction s,ρ For every s ∈ [a, b] and X = xi Ei , we let w1 be the solution of the following problem: 2

−∆w1 = − Ricp (X , E0 ) in Bn1 , 3 ∂w1 = 0 on ∂ Bn1 , ∂η where p = γ (ρ κ(s)). By elliptic regularity theory, there exists a constant c > 0 (independent of ρ and s) such that

s,ρ

w 1

C 2,α

≤ c ∀s ∈ [a, b].

(27)

As in Section 5.1, we let

Cs2,ρ,α

 ∂ s,ρ B (s, ρ, ρw1 + v) [w] = 0 := w ∈ C ( ) : ∂v v=0   ∂w = w ∈ C 2,α (Bn1 ) : + ρ L¯ (w) = 0 . ∂η 

2,α

Bn1

s,ρ

,α As explained in the first section, the linearized mean curvature operator about D s,ρ (ρ w1 ) restricted on Cs2,ρ defined by

Lρ,s (w) := −

∂ s,ρ H (s, ρ, ρ w1 + v) [w] = −∆ w + ρ Ls (w) ∂v v=0

,α may have small (possibly zero) eigenvalues. Hence it may not be invertible on Cs2,ρ . However instead of solving (26), we will ,α prove that there exist a constant λs,ρ ∈ R and a function w s,ρ ∈ Cs2,ρ such that



H (s, ρ, w s,ρ ) = λs,ρ B (s, ρ, w s,ρ ) = 0

in Ds,ρ (w s,ρ ), on ∂ Ds,ρ (w s,ρ ).

(28)

ToRachieve this we let P be the L2 projection on the space of functions w ∈ L2 which are orthogonal to the constant function 1, Bn w dx = 0. Now if ρ is small enough, the Poincare inequality implies together with elliptic regularity theory that the 1

,α operator P ◦ Ls,ρ is an isomorphism from P Cs2,ρ into P C 0,α (Bn1 ). Here letting


Φ (s, ρ, x), w 0 := −

Z

s,ρ

Bn1

H (s, ρ, ρw1 )w 0 dx +

I

s,ρ

∂ Bn1

B (s, ρ, ρw1 )w 0 ds

one has

kΦ (s, ρ, ·)kC 0,α ≤ c ρ 2 ∀s ∈ [a, b]. Consequently for ρ small, our fixed point problem

 −1  w = P ◦ Lρ,s |C 2,α P ◦ Φ (s, ρ, x) + ρ P ◦ Qs,ρ (w) s,ρ

,α ,α admits a unique solution w s,ρ ∈ P Cs2,ρ , in a ball of radius c ρ 2 of P Cs2,ρ . More precisely

Z Bn1

ws,ρ dx = 0 and kws,ρ kC 2,α (Bn ) ≤ c ρ 2 ∀s ∈ [a, b].

(29)

1

Furthermore it follows from the construction that, for all k ≥ 0

k s,ρ

∂ w

∂ sk

C 2,α (Bn1 )

≤ ck ρ 2 ∀s ∈ [a, b], s,ρ

for some constant ck which neither depends on s nor on ρ small. We then conclude that P ◦ H (s, ρ, ρ w1 + w s,ρ ) = 0 and hence the existence of a real number λs,ρ ∼ ρ 2 such that (28) is satisfied. s,ρ We have to mention that by (29), provided ρ is small, the corresponding disc Ds,ρ := Ds,ρ (w ˜ s,ρ ) with w ˜ s,ρ = ρ w1 +ws,ρ is embedded into Ωρ . This defines a one-dimensional manifold of sets satisfying (28):

Zρ := {Ds,ρ ⊂ Ωρ , ∂ Ds,ρ ⊂ ∂ Ωρ : s ∈ [a, b]}. Remark 5.1. We notice that, in Section 5.1, the same argument as above implies that whenever φ 00 (s0 ) = 0, there will be a capillary disc centered at γ (ρ s0 ) with a constant and small mean curvature.

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

4439

Variational argument We will show that in fact problem (26) can be reduced to a finite-dimensional one. We now define the reduced functional ϕρ : [a, b] → R by

ϕρ (s) := Area(Ds,ρ ) for any Ds,ρ ∈ Zρ . We have to show the following. Lemma 5.2. There exists ρ0 small such that for any ρ ∈ (0, ρ0 ) if s is a critical point of ϕρ then λs,ρ = 0. Proof. Let λ ∈ R and let q = γ (ρ(s + λ t )). Then provided t is small, it is clear that the hyper-surface Dq,ρ can be written as a normal graph over Dp,ρ , p = γ (ρ s) by a smooth function gp,ρ,t ,λ . This defines the variation vector field

ζp,ρ,λ

∂ gp,ρ,t ,λ = ND . ∂ t t =0 p,ρ

Let Z be the parallel transport of λE0 along geodesics issued from p = γ (ρ s). Then, we can easily get the estimates:

kζ − Z k ≤ c ρ|λ|. Assume that s is a critical point of ϕρ then from the first variation of area, see [17], 0 =

dϕρ (ρ(s + λt ))



= λρϕρ0 (s) t =0 I HDs,ρ hζ , NDs,ρ ids +

dt

Z =n Ds,ρ

D s,ρ

∂ Ds,ρ

hζ , N∂ Ds,ρ i,

D s,ρ

where N∂ Ds,ρ ∈ T Ds,ρ stands for the normal of ∂ Ds,ρ in Ds,ρ . Therefore by construction one has 0 = λs,ρ

Z Ds,ρ

hζ , NDs,ρ ids.

(30)

Notice that

hζ , NDs,ρ i − hZ , Y0 i = hζ − Z , NDs,ρ i + hZ , NDs,ρ − Y0 i, so using the fact that NDs,ρ = Y0 + O (ρ), see Section 4.1.1, we have

hζ , ND i − λ ≤ c ρ|λ|. s,ρ Inserting this in (30), we get

−λ λs,ρ Area(Ds,ρ ) ≤ c ρ λs,ρ |λ| Area(Ds,ρ ) but since Area(Ds,ρ ) = Area(ρ Bn1 ) + Os (ρ 2+n ) by (11); (27) and (29), it follows that

−λ λs,ρ ≤ c ρ λs,ρ |λ| . 2

2

Therefore taking λ = −λs,ρ , we see that λs,ρ ≤ c ρ λs,ρ and this implies that λs,ρ=0 .







We shall end the proof of the Theorem 1.3 by giving the expansion of ϕρ . From (29) the first fundamental form hij of a disc Ds,ρ expands as

ρ −2 hij = δij +

ρ2

3 where p = γ (ρ s) ∈ Γ . From the formula

p

hRp (X , Ei )X , Ej i +

det(I + A) = 1 +

1

2 we obtain the volume form:

ρ3 6

h∇X Rp (X , Ei )X , Ej i + Os (ρ 4 ),

tr(A) + O(|A|2 ),

p ρ2 ρ3 ρ −n det(h) = 1 − hRp (X , Ei )X , Ei i + h∇X Rp (X , Ei )X , Ei i + Os (ρ 4 ) 6 12 R and since by oddness Bn h∇X Rp (X , Ei )X , Ej idx = 0 we deduce that 1 ! n ρ2 X n 4 ϕρ (s) = Area(Ds,ρ ) = Area(Bρ ) 1 − hRp (Ej , Ei )Ej , Ei i + Os (ρ ) . 6n i,j=1

4440

M.M. Fall, C. Mercuri / Nonlinear Analysis 70 (2009) 4422–4440

Thus setting

ψρ (s) :=

6n

ρ

2

ϕρ (s) 1− Area(Bnρ )

! =

n X

hRp (Ej , Ei )Ej , Ei i + Os (ρ 4 )

i,j=1

we get the result. Acknowledgments The authors would like to thank Prof. Antonio Ambrosetti for directing their attention to these questions. They are grateful to Prof. Antonio Ambrosetti and Prof. Andrea Malchiodi at SISSA/ISAS for the helpful discussions. They have been supported by M.U.R.S.T within the PRIN 2008 Variational Methods and Nonlinear Differential Equations. References [1] A. Ambrosetti, M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1131–1161. [2] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on Rn , in: Progress in Mathematics, Birkhäuser Verlag, Basel, Boston, Berlin, 2005. [3] A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, in: Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge Univ. Press, 2007. [4] W. Bürger, E. Kuwert, Area-minimizing disks with free boundary and prescribed enclosed volume, J. Reine Angew. Math. 621 (2008) 1–27. [5] M.M. Fall, Embedded disc-type surfaces with large constant mean curvature and free boundaries, preprint SISSA 2007. [6] M.M. Fall, F. Mahmoudi, Hypersurfaces with free boundary and large constant mean curvature: Concentration along submanifolds, Ann. Sc. Norm. Sup. Pisa, preprint SISSA 2007 (in press). [7] R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, New York, 1986. [8] Gray A. Tubes, Advanced Book Program, Addison Wesley, Redwood City, CA, 1990. [9] M. Grüter, J. Jost, On embedded minimal discs in convex bodies, Annales de l’institut Henri Poincaré (C) Analyse non linéaire 3 (5) (1986) 345–390. [10] J. Jost, Existence results for embedded minimal surfaces of controlled topological type I, Ann. Sci. Sup. Pisa. Classe di Scienze Sr. 4 13 (1) (1986) 15–50. [11] N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differ. Geom. 33 (3) (1991) 683–715. [12] H.B. Lawson, Lectures on Minimal Submanifolds, Vol. I, Second edition., in: Mathematics Lecture Series, vol. 9, Publish or Perish, Wimington, Del, 1980. [13] F. Mahmoudi, R. Mazzeo, F. Pacard, Constant mean curvature hypersurfaces condensing along a submanifold, Geom. Funct. Anal. 16 (2006) 924–958. [14] R. Mazzeo, F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom. 13 (4) (2005) 633–670. [15] F. Pacard, X. Xu, Constant mean curvature spheres in Riemannian manifolds, preprint 2005. [16] A. Ros, The isoperimetric problem, in: Lecture Series Given During the Caley Mathematics Institute Summer School on the Global Theory of Minimal Surfaces at the MSRI, Berkley, California, 2001. [17] A. Ros, R. Souam, On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997) 345–361. [18] A. Ros, E. Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1) (1995) 19–33. [19] S. Secchi, A note on closed geodesics for a class of non-compact Riemannian manifolds, Adv. Nonlinear Stud. 1 (1) (2001) 132–142. [20] M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984) 547–560. [21] M. Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1–2) (1988) 19–64. [22] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, third edition, Springer, Berlin, 2000. [23] R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (2) (1991) 381–396.