Double bubble with small volume in compact manifolds

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Introduction

1.1 The Isoperimetric Problem

One of the most studied problem in mathematics is the isoperimetric problem: it concerns the problem of enclosing a given volume with least-possible area in the euclidean space Rn, that is of solving the minimization problem:

inf

V ol(Ω)=V Area(∂Ω), (1.1)

where V is the fixed volume and Ω ⊂ Rn_{. The function that associates at each volume V ,}

the above infimum value is called isoperimetric profile.

This problem is well understood, and its solution is known to be given by a round sphere (e.g. [26] and [41]). However, if one attempts to pose this problem in more generality, for instance by considering an ambient manifold instead of the euclidean space, one can say very little about the solution (if any) itself. For example in case of non compact ambient manifold, one might not gain the existence of a solution. In the case of a compact ambient manifold, one gets the existence of a minimizer, but in dimension n ≥ 7 the solution might not be regular, for example the Simons cone is known to be minimal ([10]); moreover one might lose the symmetry of the solution in general. Despite this, more can be said about the solution, in the case it exists, if the fixed volume V approaches the zero.

In this case it is known that the solution to isoperimetric problem are close to geodesic spheres with small radius ([22]); moreover, Druet has shown in [12] that these solutions tend to concentrate at maximum points of the scalar curvature function, exactly as one can expect. In order to describe deeplier the problem in the regime V ∼ 0, Ye in [45] relaxes the hypothesis of being a minimizer for (1.1), by considering the problem of finding closed hypersurfaces, whose mean curvature is constant and whose volume enclosed is small; in fact it is well known that a solution of the isoperimetric problem has a constant mean curvature, where it is regular (see Corollary 6.2.3 and Theorem 6.2.4).

In [45] Ye proved that near each point p, which is a non degenerate critical point for the scalar curvature, there exists a branch of constant mean curvature hypersurfaces, each of which is a normal graph over a small geodesic sphere centered in p. Furthermore, the hypersurfaces in this branch constitute a local foliation of a neighbourhood of p. These results suggest once more the important role played by the critical points of the scalar curvature in this kind of problem.

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The proof of Ye consists in perturbing normally small geodesic spheres, and trying to impose that the mean curvature of the perturbed sphere to be constant; however this turns out to be not always possible, but he showed that one can impose it if the center is a non degenerate critical point of the scalar curvature.

In the same spirit, there is a work by Pacard and Xu ([33]), in which the authors provide the existence of constant mean curvature hypersurfaces without assuming anything on the scalar curvature. This result includes important cases left uncovered by the result of Ye, such as the case in which M has constant scalar curvature, which is a very large class of manifold thank to the solution of the Yamabe problem.

The proof in Pacard and Xu [33] is a modification of the argument of Ye; the difference is that they use a variational characterization of constant mean curvature hypersurfaces that let them to introduce a function on M , whose critical points give rise to a branch of constant mean curvature hypersurfaces. They proved that this function is close to the scalar curvature function (in the Ck _{topology), and from this they recover the existence result of Ye; nevertheless, they cannot}

prove the foliation property in general. Moreover, they also get expansion for the isoperimetric profile in the regime v ∼ 0.

We conclude this section by quoting tha article of Nardulli [32]. In this article, Nardulli proved the uniqueness of some almost constant mean curvature hypersurfaces, called pseudobubbles, once fixed their (small) radius and their baricenter (see Chapter 3). It turns out that these pseudobubbles are the same perturbed spheres constructed by Pacard-Xu and Ye. By this uniqueness, he was able to show that each solution to the isoperimetric problem, with sufficiently small volume constraint, is in fact a pseudobubble, and he got expansions for the isoperimetric profile too; thus he solved the problem of characterize the solutions of the isoperimetric problem fo small volume.

1.2 Double Bubbles

A natural generalization of the isoperimetric problem regards the case of several volumes: given k volumes in Rn_{, which is the least-area way to enclose and separate them? In the sequel we refer}

to any of this ”hypersurfaces” enclosing these volumes and separating them as bubble clusters. The corresponding minimization problem is:

inf V ol(Ωi)=Vi {Area ([ i ∂Ωi) \ [ i6=j (∂Ωi∩ ∂Ωj) | Ωi∩ Ωj= ∅ if i 6= j}.

First of all, one might wonder if the problem is well posed. The answer is yes, since Almgren proved in [1] the existence and the almost-everywhere regularity of area minimizer bubble clusters enclosing these k volumes. The main drawback of the existence proof, is that one cannot expect the region associated to a volume to be connected. In fact, (at least for n ≥ 3), one can consider the minimization problem restricted to open connected sets Ωi, i = 1, ..., k, without changing the

value of the infimum: it suffices to connect each connected component of Ωiby a small thin tube

and erase a corresponding small volume in the interior, and this can be done increasing the area in a controlled way (see [20]); nevertheless, in the minimization procedure it might happen that one loses the connectedness, for instance, by a small thin tube that tends to disappear! This particularity makes the problem of characterizing the minimizers hard to achieve, and really little has been done in such generality.

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1.2. DOUBLE BUBBLES 5 A first step in this direction was the resolution of the so-called Double Bubble Conjecture by Hutchings, Morgan, Ritor´e and Ros in the case n = 3 ([21]), and then by Reichardt for general dimension n ([36]). This conjecture relates the case in which we have only two volumes V1and V2, so we call double bubble the bubble clusters in this case. We define a standard double

bubble, a piecewise smooth hypersurface that consists of two spherical caps, separated by a third hypersurface, which is either spherical (if V16= V2) or planar (if V1= V2), meeting each other in

an equiangular way along an Sn−2. We postpone the theory about this bubbles to Section 2.4. The conjecture reads as follow:

Is a minimizer double bubble a standard double bubble?

The planar case, i.e. n = 2, was solved in positive by the ”SMALL” undergraduate group of the William College in 1990 ([14]). The case n = 3 requires more work, and the main turning point was when, in the 1997 Hutchings proved a structure theorem ([20]): any minimal double bubble in Rn _{that is not a standard double bubble, is a surface of revolution about some line,}

and consists of a topological sphere with a three of annular bands attached. The root of the tree has just one branch, and the ”top” and the ”bottom” are spherical caps. For a precise statement we refer to [20]. Actually the symmetry part of the statement holds in more generality: an area minimizer bubble cluster enclosing m volumes in Rn _{is symmetric about some (m − 1)-plane.}

Having this strong result, the conjecture becomes equivalent to exclude all the possible trees. In [21], Hutchings et. al solved this problem: their argument is based on the so called Basic Estimate ([20]), which let them to bound the number of connected component of each region, and on a very hard stability argument ([21]).

A refinement of this stability argument permits Reichardt, Heilmann, Lai and Spielman to prove the conjecture in R4 _{and for some cases in higher dimensions ([35]). Finally, Reichardt in}

[36] improved again the stability argument to prove the conjecture in total generality.

Something is done also in the case of space form, for example in [9] Cotton and Freeman proved that the standard double bubble is the least-area way to enclose and separate two regions of equal volume in H3 _{, and in S}3 _{when the exterior is at least ten percent of S}3_.

Is worth mentioning that the double bubbles arise in the formation of certain natural patterns in inhibitory system: an example of inhibitory systems is the block copolymer, that is a material characterized by fluid-like disorder on the molecular scale, and a high degree of order at a longer length scale. A molecule in a block copolymer consists of several monomers of different type, bonded covalently each other.

The free energy of an inhibitory ternary system, with a self-organization property, admits a mathematical model that includes an interface energy that describe a short range interaction (this term is a perimeter functional, so local), and an inhibitory interaction energy that acts at long range (this term is a square root of the laplacian, thus non local). The interface energy push the system to grow indefinitely, while the long range energy forces the system to remain bounded. One can expect that if these energies are comparable, then some self-organization patterns might arise at the equilibrium. In [37], [38] and [39], Ren and Wei showed that this is true in the case of a planar domain; they showed that if these two energies balance each other, and two of the three constituents are present in a small fraction with respect to the third, the free energy admits a local minimizer that is shaped like a perturbed double bubble.

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1.3 Our work

Inspired by the works quoted above, we would like to provide a characterization similar to that obtained by Nardulli, but for the solutions of the isoperimetric problem associated to two small given volumes, in an ambient compact manifold. Of course this requires a lot of work, and this thesis is an attempt to make the first step: in fact, in this work we want to obtain an existence result for small smooth constant mean curvature double bubble, in an ambient compact manifold M , so we would like to extend the argument in [33]. We expect these bubble to exist and concentrate near critical points of a suitable function, as in [33], but we do not expect that they are unique once fixed the radii and the baricenter: since the symmetry of the standard double bubble are less than the ones of the sphere, we expect that there should exist more such bubbles, with a new degree of freedom: the asymptotic axis of symmetry. However, once obtained this result, one should adapt the result of Nardulli to conclude the characterization.

The method used to do extends the construction in [33], i.e. we get the existence of such double bubble by perturbing small double bubble in the tangent space of M in a fixed point p, then we impose the perturbed surface to be of constant mean curvature by varying the point p; we emphatise the latter, because also in this case it turns out that we cannot impose the CM C-condition directly in every point of the ambient manifold M , but we have to deal with the Kernel of the Jacobi operator associated to the standard double bubble, which we treat in the Chapter 5. The characterization of this kernel is the main effort in this thesis, since it requires to deal with some special functions, with the singular set of the standard double bubbles and to use strongly the geometric properties of them. Another difficulty is that, in perturbing a double bubble, one has to allow the presence of a tangential component in the perturbation too, so we have to adapt all the expansion for the geometric quantities of a perturbed sphere obtained in [33].

This work is divided as follows: in Chapter 2 we resume briefly some preliminary results on elliptic regularity and Riemannian geometry; then we recall the definition of a standard double bubble following [21], and we establish some geometric properties of them.

In Chapter 3, we describe the argument used in [33], and we cite the results in [32].

In Chapter 4 we exploit the perturbation argument by introducing normal coordinates in our manifold M , in order to get the expansion of the mean curvature of the perturbed bubble.

In Chapter 5, we prove that the Kernel of the Jacobi operator associated to a standard double bubble consists of normal perturbations generated by traslations and rotations, in some cases.

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Chapter 2

Notations and Preliminaries

In this chapter we fix our notations and we briefly recall some basic facts about elliptic regularity and Riemannian geometry.

2.1 Notation

We start with the notation.

• Rn _{denotes the (n)-dimensional space endowed with the euclidean scalar product h·, ·i and}

the euclidean norm k·k; • Sn

denotes the subset of Rn+1_{defined by S}n

:= {x ∈ Rn+1_{| kxk = 1};}

• B(x0, R) (resp. S(x0, R)) is the ball (resp. the sphere) of radius R and center x0 in some

Rn;

• given an open subset Ω of Rn _{or of a riemannian manifold M , and given a manifold N ,}

Ck_{(Ω; N ) is the space of the k-differentiable maps from Ω to N (if k = ∞ then of smooth}

maps). Ck,α_{(Ω; R}n) is the space of the k-differentiable functions, whose k-derivative is α-H¨_{olderian; if Ω ⊂ R}n, the space S0_(Rn) denotes the space of tempered distribution supported in Ω;

• given an open subset Ω ⊂ Rn_{, L}p_{(Ω) denotes the Lebesgue space of p-integrable functions}

(if p = ∞ of essentially bounded functions);

• given a Riemannian manifold (M, g), its Riemann tensor, Ricci tensor and scalar curvature at a point p will be denoted respectively by Rp, Ricp and s(p);

• given a submanifold M of a manifold N , its second fundamental form and mean curvature will be denoted by h and H;

• Σ will be a double bubble and Γ will be its singular set;

• N will be the normal of an immersed surface with the sign convenction to be fixed case by case; in the case of double bubble, ν will be the inner cohonormal of the singular set, i.e. the normal to the singular set which is both tangent to the smooth sheet considered and points towards its interior;

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• w will be a normal component of a considered perturbation of a surface; in the case of double bubble, Y will be a tangential component of a considered perturbation, and u will be a inner cohonormal component at the singular set; since the singular set will be a triple junction, the values of u and w will depend on the sheet considered;

• the Laplace-Beltrami operator of the several manifold M considered, will be denoted by ∆M; in the case of Bn, it will be denoted simply by ∆;

• the tensorial vector bundle of p-controvariant q-covariant tensors will be denoted by Tp q (M );

the unitary tangent bundle U T M is the fiber bundle over M whose fiber at p is given by the unit sphere in TpM ;

• a CM C surface is a surface of costant mean curvature;

2.2 Some Basic Facts about Elliptic Regularity Theory

In this section we recall some basic facts about elliptic regularity theory. There is a very large licterature about this subject, we mainly follow [4], [17] and [43]. Since we will work in Holder space, we remind only the Schauder theory. Throughout this section the Einsten convenction of summing over repeated indices is understood.

Let Ω ⊂ Rn _{be an open set. Consider a linear differential operator of the second order}

Lu = ai,j(x)Di,ju + bi(x)Diu + c(x)u,

where u : Rn_{, D} i= _∂x∂

i and Di,j=

∂2

∂xi∂xj.

Definition 2.2.1. We say that the operator L is elliptic at a point p ∈ Ω if there exists a costant λ > 0 such that it holds

ai,j(p)ξiξj≥ λkξk2

for every ξ ∈ Rn. If there exists a costant λ > 0 such that this inequality holds for every p ∈ Ω, we say that the operator L is uniformly elliptic.

Recall the definition of weak solution:

Definition 2.2.2. Suppose L is as above. Given f ∈ L2

(Ω), we say that a function u ∈ H1_(Ω)

is a weak solution for the equation −L(u) = f if: ˆ

Ω

ai,j(x)DjuDiφ − bi(x)uDiφ + c(x)uφ = 0

for every φ ∈ C_c∞(Ω).

We state next some existence and regularity theorems on elliptic PDEs.

Theorem 2.2.3. Let Ω be bounded, of class C2,α_{. Let L be as above and uniformly elliptic.}

If ai,j ∈ C0,α( ¯Ω), bi(x), c(x) ∈ L∞(Ω) and c(x) ≤ 0, then for every f ∈ L2(Ω), and for every

g ∈ L1/2_{(∂Ω), there exists a unique u ∈ H}1_{(Ω) weak solution for the problem}

(

−Lu = f in Ω

u = g on ∂Ω (2.1)

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2.3. SOME BASIC FACTS ABOUT RIEMANNIAN GEOMETRY 9 Is worth mentioning that one can weak a lot the regularity of the coefficients (up to L∞) if one assumes, for istance, L in divergence form. We can now state the Schauder estimates: Theorem 2.2.4. Let Ω be bounded, of class C2,α. Let L be as above and uniformly elliptic. If ai,j∈ C0,α( ¯Ω), bi(x), c(x) ∈ C( ¯Ω) and c(x) ≤ 0, then for every f, g ∈ C0,α there exists a unique

u ∈ C2,α( ¯Ω) classical solution to (

−Lu = f in Ω

u = g on ∂Ω (2.2)

and there exists a c > 0 such that

kukC2,α_(Ω)≤ c(kf k_C0,α_(Ω)+ kgk_C0,α_(Ω)).

These results can be adapted to the case of an ambient compact manifold (in case there is no boundary one impose only the equation).

We conclude by stating the results we will need; remark that one can use a bootstrap argument in the following case in order to get smoothness:

Corollary 2.2.5. For every f ∈ C0,α_(Bm+1_{) there exists a unique smooth solution of}

(

∆u = 0 in Bm+1

u = f on Sm

Corollary 2.2.6. Let Σ ⊂ S(x0, R) be a cap of sphere and denote by Γ its boundary. Then for

every f ∈ C0,α_{(Ω) there exists an unique smooth solution of}

( R2_∆

Σu + mu = 0 in Σ

u = f on Γ

2.3 Some Basic Facts about Riemannian Geometry

In this section we recall briefly some basic theorems on the geometry of immersed manifolds. We will mainly follow [11], but one can finds a clear exposition of these arguments also in, for instance, [8], [16], [27], [28] and [44]. For this purpose, let us consider an immersion f : Mm−→ ¯Mm+n=k, where the apices are the dimension of these manifolds. Since f is an immersion, for each point p ∈ M , there exists a neighborhood U ⊂ M of p such that f (U ) ⊂ ¯M is a submanifold of ¯M . So there exists a neighborhood ¯U ⊂ ¯M of f (p) and a diffeomorphism φ : ¯_{U −→ V ⊂ R}k, where V is an open subset of Rk _{such that φ maps diffeomorphically f (U ) ∩ ¯}_{U onto an open set of the}

subspace Rm

× {0} ⊂ Rk_{. We identify U with f (U ) and each tangent vector v at a point q ∈ M}

with its image via df .

The metric h·, ·i of ¯M induces for each point p ∈ M the splitting TpM = T¯ pM ⊥ TpM⊥, i.e.

in the tangent space of M and its orthogonal space. So given any v ∈ TpM , where p ∈ M , we¯

denote by vT _{its tangential (to M ) component, and by v}N _{its normal component.}

Let ¯∇ be the Levi-Civita connection on ¯M ; it is easy to show that it induces a Levi-Civita connection ∇ on M , defined in the following way: given two local vector fields X, Y ∈ T1

0(U )

for some open set U , consider any local extension ¯X, ¯Y ∈ T1

0( ¯U ) for a (sufficiently) small ¯U and

define:

∇XY := ( ¯∇X¯Y )¯ T;

it can be proved that the connection depends only on the value of ¯X(p) = X(p) in the point of evaluation p ∈ M , and on the behaviour of ¯Y along a curve at p tangent to X(p), so this quantity does not depend on the choice of the extensions ¯X and ¯Y .

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Define now B(X, Y ) = ¯∇_X¯Y − ∇¯ XY = ( ¯∇_X¯Y )¯ N. It turns out that B is well defined,

symmetric and tensorial (i.e. C∞-linear) in each argument. Tensoriality implies that B depends only on the value of X(p), Y (p) ∈ TpM .

We can now define the second fundamental form of M ; given a point p ∈ M and a vector η ∈ TpM⊥, define hη∈ T20(M ) by:

hη(x, y) := hB(x, y), ηi

for each x, y ∈ TpM . This map is bilinear and symmetric; moreover hη ∈ T20, i.e. it is a

2-form.

Definition 2.3.1. The quadratic map hη(x, x) for x ∈ TpM is the second fundamental form of

M (actually f ) at p, along the direction η ∈ TpM⊥.

Being a symmetric bilinear mapping, the second fundamental form has an associate linear self-adjoint operator Sη: TpM −→ TpM such that

hη(x, y) = hSη(x), yi.

The operator Sη is known as the shape operator, or the Weingarten map. The following

proposition holds:

Proposition 2.3.2. Let p ∈ M , x ∈ TpM and η ∈ TpM⊥; let N be a local extension of η,

normal to M . Then

Sη(x) = −( ¯∇xN )T.

In the case in which M is an hypersurface, i.e. the codimension is 1, we have only one second fundamental form up to costants. Since Sη is a symmetric linear operator, it has m real

eigenvalues λi, which are called principal curvatures of f ; the mean curvature of f is by definition

H =P

iλi. Remark that some authors define H = _m1 Piλi, e.g. [16]. We note that

H = trace(Sη(·)) = trace(hη(·, ·)) =

X

i

hη(ei, ei),

where the trace of a bilinear operator is calculated by raising an index and, in the latter, ei

is an orthonormal basis of TpM .

After the basic notions of the immersed geometry, we recall some definition regarding calculus on manifolds; throughout the rest of this section, let us consider a fixed m-dimensional manifold M , with metric g.

Definition 2.3.3. Let w be a smooth (say) R-valued function on M , and let Y ∈ T1

0(M ). The

gradient of w is the vector fields ∇w such that, for each point p ∈ M and each vector v ∈ TpM

it holds h∇w, vi = dwp[v]. Its espression in local coordinates xi is

∇w = (gi,k∂w

∂xk

) ∂ ∂xi

.

The divergence of the vector field Y , is the scalar function such that at each p it holds LXµ = div(Y )µ; in local coordinates its espression is given by

div(Y ) = ∂Y

i

∂xi

+ Γj_i,jYi. Another equivalent definition is div(Y ) = trace(V 7→ ∇VY ).

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2.3. SOME BASIC FACTS ABOUT RIEMANNIAN GEOMETRY 11 The hessian of w (resp. of Y ) is the 2-tensor defined by Hess(w)(u, v) = ∇u(dw)(v) (resp.

Hess(Y )i_{(u, v) = ∇}

u(dYi)(v)); its espression in local coordinates is

Hess(w) = ∂ 2_w ∂xj∂xk + Γl_j,k∂w ∂xl , and respectively Hess(Y )i= ∂ 2_Yi ∂xj∂xk + Γlj,k ∂Yi ∂xl .

The Laplace-Beltrami operator on w (resp. the rough laplacian of Y ) is the trace of the respective hessian, and it is denoted by ∆w (resp. ∆Y ). In the case of a function w, it turns out that ∆w = div(∇w). Moreover one has another espression in local coordinates:

∆w = √1 g ∂ ∂xi (√ggi,j ∂ ∂xj ),

where g = det(gi,j).

Remark 2.3.4. The Laplace-Beltrami operator on an oriented compact manifold M , with bound-ary ∂M (might be void), is a symmetric operator in the space L2_{(M, g); in fact, given two}

functions u, v ∈ C∞(M ), a simple application of the Green identity yields ˆ M u∆v − v∆u = ˆ ∂M u∂v ∂ν − v ∂u ∂ν,

where ν is the exterior unit normal to ∂M ; since the right hand side is symmetric in u and v, this concludes the proof. Moreover, being self-adjoint, any of its eigenvalue is real.

Let us remark that, if M is closed (i.e. compact and without boundary), the operator −∆ has non negative eigenvalues. In fact, given an eigenvalue λ and a corresponding eigenfunction f 6= 0, a simple application of the Green identity gives

λ ˆ M f2= ˆ M −∆f f = ˆ ∂M g(∇f, ∇f ) ≥ 0.

We easily conclude that λ ≥ 0; furthermore, in the case the latter inequality is an equality, we conclude that λ = 0 and also that ∇f = 0, i.e. f is constant in each connected component of M .

We will use the following property of the Laplace-Beltrami operator, that express how it behaves under conformal changes of the metric: suppose ˜g = e2φ_{g for some function φ on M .}

Then we have: ∆˜gw = e−2φ(∆gw − (m − 2) X k ∂φ ∂xk ∂w ∂xk ),

that reduces, in the case of ˜g = λ2

g, λ ∈ R, to: ∆˜gw = λ2∆gw.

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Lemma 2.3.5. (Gauss Lemma) Let p ∈ M and let v, w ∈ TpM such that Expp(v) is defined;

identify TpM ∼= Tv(TpM ). Then

hd(Expp)v[v], d(Expp)v[w]i = g(v, w).

There is a natural generalization of the Gauss Lemma, the so called normal coordinates; since we will use them in the sequel of this work, we introduce them here.

In order to do that, let us consider a fixed point p ∈ M ; fix an orthonormal basis Ei_{, i = 1, ..., m,}

of TpM , and introduce coordinates x := (x1, ..., xm) in Rm. From the basic riemannian geometry

theory, we know that the exponential map Exppat the point p is a local diffeomorphism between a

(small) neighborhood U of 0 ∈ TpM , and a neighborhood ˜U of p in M . Let S be the isomorphism

from Rm _{and T}

pM which sends xi in Ei. The chart φ := S−1◦ Exp−1p : ˜U −→ V ⊂ Rm, where

V := φ( ˜_{U ) (is a neighborhood of 0 ∈ R}m_{), induces coordinates y}i_{(q) := x}i_{(φ(q)) on ˜}_{U which are}

called geodesic normal coordinates centered at p.

These coordinates have a lot of interesting properties; for istance, in this coordinates the equation on a geodesic passing through p at the time t = 0 has the form yi _{= a}i_{t, there t is the}

arch-lenght, and a = ai∂/∂yi is its initial velocity. Writing the equation of the geodesics d2_yi ds2 + Γ i j,k dyj ds dyk ds = 0

we easily derive Γi_j,k(p)ajak= 0 at p, and hence, by the arbitrariety of a, it holds Γi_j,k(p) = 0. Define the local coordinate vector fields Yi:= ∂/∂yiand let gi,j= g(Yi, Yj) be the coefficients of

the matrix associated to de metric g in this coordinates. Clearly, we have gi,j|p= δi,j; moreover,

by the compatibility of the metric and the vanishing of the Christoffel symbols one gets ∂gi,j

∂yk = 0

for all i, j, k = 1, ..., m, but we can say more:

Proposition 2.3.6. At the point q = Expp(xiEi), the following expansion holds for all i, j =

1, ..., m g(Yi, Yj) = δi,j+ 1 3g(Rp(Ξ, Ei)Ξ, Ej) + 1 6g(∇ΞRp(Ξ, Ei)Ξ, Ej) + O(|x| 4₎ _(2.3)

where Rp is the Riemann curvature tensor at the point p, and Ξ = xiEi∈ TpM .

In the chapters 3 and 4 we will adapt this formula (a more precise one) in order to get expansion for some geometric quantities associated to several submanifold.

We conclude this section by stating other properties of the normal coordinates. Let ω = pdet(gi,j)dy1∧ ... ∧ dymbe the volume form associated to the metric g. Then defining ω1,...,m :

= ω(Y1, ..., Ym) it holds ω1,...,m= 1 − 1 6Ricp(Ξ, Ξ) − 1 12∇ΞRicp(Ξ, Ξ) + O(|x| 4_).

Furthermore, denote by S(p, r) := {Expp(x) | kxk = r} and B(p, r) := {Expp(x) | kxk ≤ r}

respectively the geodesic sphere and the geodesic ball of radius r centered at p; let s(p) be the scalar curvature of M at p. There holds

V olm−1(S(p, r)) = rm−1V olm−1(Sm−1) 1 −

1 6mr

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2.4. GEOMETRY OF THE DOUBLE BUBBLES 13

V olm(B(p, r)) = rmV olm(Bm) 1 −

1 6(m + 2)r

2_{s(p) + O(r}4_),

and for the mean curvature of the geodesic sphere H(S(p, r)) = m − 1

r −

1

3Ricp(Ξ, Ξ)r + O(r

2_).

2.4 Geometry of the Double Bubbles

In this section we introduce the concept of double bubble and study some geometric properties of a special kind of double bubbles, the standard double bubble. We follow the article [21] by Hutchings, Morgan, Ritor´_{e and Ros in a special case, i.e. in R}3.

Definition 2.4.1. A double bubble in Rm+1 _{is the union of the topological boundaries of two}

disjoint regions of prescribed volumes. A smooth double bubble Σ ⊂ Rm+1_{is a piecewise smooth}

oriented hypersurface consisting of three compact pieces Σ1, Σ2 and Σ0 (smooth up to the

boundary), with a common (m − 1)-dimensional smooth boundary Γ, such that Σ1∪ Σ0 (resp.

Σ2∪ Σ0) encloses a region R1 (resp. R2) of prescribed volume v1 (resp. v2).

None of these objects is assumed to be connected. The unit normal vector field N along Σ will be always chosen according to the following criterion: N points into R1 along ∂R1, and

points into R2 along Σ2. We denote by h and H the second fundamental form and the mean

curvature of Σ. Note that these objects are not univalued along the singular set Γ but they depend on the sheet Σi we use to compute them. We will also use the notation Ni, hiand Hi to

denote their restriction to Σi, i = 0, 1, 2.

Definition 2.4.2. A standard double bubble in Rm+1_{is a smooth double bubble, which consists}

of two exterior spherical pieces and a separating surface (which is either spherical or planar) meeting in an equiangolar way along a given (m − 1)-dimensional sphere Γ.

We have the following uniqueness result.

Proposition 2.4.3. Given two volumes, there exists a unique standard double bubble (up to rigid motions) enclosing them. Furthermore, the mean curvatures satisfy H1= H0+ H2.

Proof. Consider a unit sphere through the origin and a congruent or smaller sphere intersecting it at the origin at 120 degrees. Note that the intersection of this two spheres is a (m − 1)-sphere Γ. There is a unique completion to a standard double bubble, in fact imposing the third sphere to intersect the others in Γ, we are left with one degree of freedom; finally, by the condition on the angle of intersection determines uniquely the third sphere. Varying the size of the smaller sphere yields all volume ratios precisely once. Scaling yields all pairs of volumes precisely once. The condition on the curvatures follows by plane geometry for R2.!!!!!!!!!!!!!!!!!mettere il disegno For our purpose we will deal only with standard double bubbles; however is remarkable that in their article [21] they solve in positive (in ambient dimension 3) the celebrated Double Bubble Conjecture:

Is an area minimizing double bubble a standard one?

Actually there is a wide literature regarding this conjecture, and their proof relies on many articles, among others [14], [20], [30], [34] and [40].

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standard double bubble Σ in Rm+1_{, whose symmetry axis is the last coordinate axis Span(x}m+1_).

By definition, Σ = Σ0∪ Σ1∪ Σ2 where Σi⊂ S(Ci, Ri) are spherical caps which meet each other

along Γ, (recall Γ is an Sm−1_{), in an equiangular way. By hypothesis, the centers C} i are of

the form Ci = (0, .., 0, hi); since the two enclosed volumes are fixed, the radii Ri’s are uniquely

determined. If we impose Γ ⊂ {xm+1_{= 0}, we determine uniquely the h} i’s too.

Let us now recall some properties of this standard dubble bubble. Remark that, if p ∈ Γ, we have the splitting TpΣi = TpΓ ⊥ (TpΓ)⊥, where the ⊥ is respect to the metric of Σi; let us

denote by νi∈ TpΣi the unit normal to TpΓ, which points towards Σi (we will call it the inner

cohonormal ). Fixing each normal Ni to be toward the respective center. Remark that by the

choice V1 ≤ V2, N0 points toward V1, so we rescue Hutchings et al. notation. We have the

following:

(

N1= N0+ N2,

ν0+ ν1+ ν2= 0,

(2.4) at every point in Γ and also:

1 R1 = 1 R0 + 1 R2 ⇐⇒ H1= H0+ H2. (2.5)

From now on we suppose (w.l.o.g.) that the region enclosing V1 is in the positive hemispace

w.r.t. the plane {xm+1 = 0}. Suppose now that V1 = V2. In this case, the bubble Σ is

symmetric with respect to the plane {xm+1 = 0}, so Σ0 is a disk B(0, R0). Moreover, by the

simmetry and the equiangolarity of Σ, it is easy to show that R1 = R2 = R, R0 = √

3 2 R and

C1= −C2= (0, ..., 0,R₂).

In the case in which V1 6= V2, we have to fix some notations: let us denote by α the angle

between the pole in Σ0and any point of Γ, measured in such a way that it is less than π; define

in the same way β for Σ1 and γ for Σ2. From the fact that Γ ⊂ {xm+1 = 0} we easily derive

h0= R0cos(α), h1= −R1cos(β) and h2= R2cos(γ). Furthermore, writing each point of Γ w.r.t.

the different pieces we get the first ”structural” equations:

R0sin(α) = R1sin(β) = R2sin(γ). (2.6)

Analyzing the two 2/3π angles in the two regions enclosing V1 and V2, it is easy to prove

that:

β + γ = 4

3π (2.7)

γ − α = 2

3π. (2.8)

From this equations we trivially get α + β = 2/3π.

Remark 2.4.4. Note that in the simmetric case we used that, in this notation, β = γ = 2/3π. We will call all these equations structural equations in the rest of this work.

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Chapter 3

Pacard and Xu’s work

In this chapter we present Pacard and Xu’s work [33] which we would like to extend to the double bubble case.

In their work they get the existence of small CM C-spheres in a compact manifold M , as perturbations of small geodesic spheres. This idea of perturbing small geodesic spheres was already used in [45] by Ye, in order to get the existence of CM C-spheres centered in the non-degenerate critical points of the scalar curvature function; he also proved that such spheres constitute a smooth foliation of a neighbourhood of these critical points. However, Pacard and Xu obtain the existence without any assumption on the scalar curvature, so for example their result holds also in the case of constant scalar curvature manifolds, which is known to be a wide class of manifold by the resolution of the Yamabe problem.

Throughout all the chapter the Einstein convection of summing over repeated indices is understood.

3.1 Perturbation of Geodesic Spheres

The authors proceed as follows: first of all, we make the following convection: the Greek index letters range from 1 to (m + 1), while the Latin ones range from 1 to m. Fix a compact (m + 1)-dimensional riemannian manifold (M, g). Then, consider coordinates x = (x1_{, ..., x}m+1

) in Rm+1_,

fix a point p in M , and an orthonormal basis Eµ∈ TpM , µ = 1, .., m + 1. Set

F (x) := Expp(xµEµ),

where Exp is the exponential map of M . The map F is a parametrization which induces normal coordinates centered in p. This choice of coordinates induces coordinate vector fields Xµ= F∗(∂xµ). Clearly g_µ,ν:= g(X_µ, X_ν) = δ_µ,νat p. Moreover, we have the following expansion:

Proposition 3.1.1. At the point q = F (x), the following expansion holds for all µ, ν = 1, ..., m+1 g(Xµ, Xν) = δµ,ν+ 1 3g(Rp(Ξ, Eµ)Ξ, Eν) + 1 6g(∇ΞRp(Ξ, Eµ)Ξ, Eν) (3.1) + 1 20g(∇Ξ∇ΞRp(Ξ, Eµ)Ξ, Eν) + 2 45g(Rp(Ξ, Eµ)Ξ, Eι)g(Rp(Ξ, Eν)Ξ, Eι) + O(|x| 5₎ _(3.2)

where Rp is the Riemann curvature tensor at the point p, and Ξ = xµEµ ∈ TpM . Furthermore,

the symbol O(|x|k_{) indicates a smooth function depending on p such that it and its partial}

deriva-tives of any order, with respect to the vector fields xµ_X

µ, are bounded by a costant times |x|k in

some fixed neighbourhood of 0, uniformly in p. 15

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We can derive from this expansion, analogous expansion for several geometric quantities of normal perturbations of geodesic spheres, as well as the metric, the enclosed volume, the area, the second fundamental form and the mean curvature. In order to do so, we fix a small parameter ρ > 0, a small C2,α _{function w on S}m _{and we use a local parametrizion S}m _{3 z 7→ Θ(z) of}

Sm_{⊂ T}

pM . Now define the map

G(z) := Expp(ρ(1 − w(z))Θ(z)),

and denote its image by Sp,ρ(w); the latter is the perturbed sphere we want to be CM C.

We have some remarks:

Remark 3.1.2. First of all, the unspecified smallness of w seems to be too restrictive. Despite this, it is not the case, as they proved in their article. However there are two natural reasons to impose such a smallness: firstly, one does not want the perturbed sphere to intersect itself; secondly, remind we want the perturbed sphere to be CM C, and recall that for a geodesic sphere Sp,ρ it holds: H(Sp,ρ) = m ρ − 1 3Ricp(Θ, Θ)ρ + O(ρ 2_),

so one expects the wanted perturbation to be of order ρ2.

Finally, the idea of perturbing small geodesic sphere in order to get CM C hypersurfaces was already used by Ye in [45] to show that every non degenerate critical point for the scalar curvature function has a neighbourhood foliated by CM C hypersurfaces. However they do not require such a non degeneracy condition to get the existence, but they lose the foliation property.

We want to find conditions on w that garantee the perturbed surface to be CM C. We have to fix some notations: the coordinates of Θ in {E1, .., Em+1} are given by Θ1, .., Θm+1, so that

Θ = ΘµEµ. Define Θi:= ∂ziΘµEµ, which are vector fields along Sm⊂ TpM , while Υ := ΘµXµ

and Υi := ∂ziΘµXµ are vector fields along Sp,ρ(w). For brevity, we also write

wj:= ∂ziw, w_i,j:= ∂_zi∂_zjw.

In this notations, the tangent space of Sp,ρ(w) at any point is spanned by

Zj:= G∗(∂zj) = ρ((1 − w)Υ_j− w_jΥ), (3.3)

for j = 1, ..., m.

Imposing the perturbed sphere to be of costant mean curvature, we connect this problem to a fixed point argument, so it is quite natural, due to the unspecified smallness of w, to make the following convenctions:

Any expression of the form Lp(w) denotes a linear combination of the function w, together with

its derivatives (up to order 2), with respect to the tangent vector fields induced by G on Sp,ρ

(we will denote them by Θi). The coefficients of Lp might depend on ρ and p but, for all k ∈ N,

there exists a constant c > 0 independent of ρ ∈ (0, 1) and p ∈ M such that:

kLp(w)kCk,α_(Sm₎≤ ckwk_Ck+2,α_(Sm₎. (3.4)

Similarly, given a ∈ N, any expression of the form Q(a)p (w) denotes a nonlinear operator in

the function w, together with its derivatives (up to order 2), with respect to the tangent vector fields induced by G on Sp,ρ. The coefficients of the Taylor expansion of Q

(a)

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3.1. PERTURBATION OF GEODESIC SPHERES 17 and their partial derivatives might depend on ρ and p but, given k ∈ N, there exists a constant c > 0 independent of ρ ∈ (0, 1) and p ∈ M such that Q(a)p (0) = 0 and

kQ(a)p (w2) − Q(a)p (w1)kCk,α_(Sm₎≤ c(kw1kCk+2,α_(Sm₎+ kw2kCk+2,α_(Sm₎)a−1· kw2− w1kCk+2,α_(Sm₎,

(3.5) provided kwikCk+2,α are small enough.

Using this convenction, we exploit the various geometric quantities we list above. First of all we get the expansion for the metric of Sp,ρ(w); fix a point p ∈ M , and set q = G(p). The derive

from (3.1.1), taking Ξ = ρ(1 − w)Θ, that it holds: g(Xµ, Xν) = δµ,ν+ 1 3ρ 2_{(1 − w)}2_g(R p(Θ, Eµ)Θ, Eν) + 1 6ρ 3_{(1 − w)}3_g(∇ ΘRp(Θ, Eµ)Θ, Eν) + 1 20ρ 4_{(1 − w)}4_g(∇ Θ∇ΘRp(Θ, Eµ)Θ, Eν) + 2 45ρ 4_{(1 − w)}4_g(R p(Θ, Eµ)Θ, Eι)g(Rp(Θ, Eν)Θ, Eι) + O(ρ5) + ρ5Lp(w) + ρ5Q(2)p (w),

where the Riemann tensor is calculated at p. In order to exploit ˚gi,j := g(Zi, Zj), i.e. the

coefficients of the first fundamental form of Sp,ρ(w), we notice that

g(Υ, Υ) = 1 and g(Υ, Υj) = 0 j = 1, ..., m (3.6)

by the Gauss Lemma. Using (3.3) it is easy to show the following Lemma 3.1.3. For the metric we have the following expansion:

(1 − w)−2ρ−2˚gi,j = g(Θi, Θj) + (1 − w)−2wiwj+ 1 3ρ 2_{(1 − w)}2_g(R p(Θ, Θi)Θ, Θj) +1 6ρ 3_{(1 − w)}3_g(∇ ΘRp(Θ, Θi)Θ, Θj) + 1 20ρ 4_{(1 − w)}4_g(∇ Θ∇ΘRp(Θ, Θi)Θ, Θj) + 2 45ρ 4_{(1 − w)}4_g(R p(Θ, Θi)Θ, Eι)g(Rp(Θ, Θj)Θ, Eι) + O(ρ5) + ρ5Lp(w) + ρ5Q(2)p (w).

From this formula we get expansion for the volume and the area of Sp,ρ(w). However, since

we want to calculate the mean curvature of Sp,ρ(w), let us consider a less precise formula:

ρ−2(1 − w)−2˚gi,j= g(Θi, Θj) + Q(2)p (w) + 1 3ρ 2_{(1 − w)}2_g(R p(Θ, Θi)Θ, Θj) + ρ3Lp(w) + O(ρ3). (3.7) We consider now the geometry of Sp,ρ(w) as embedded manifold.

Since the perturbation is small, one expects the unit normal vector fields to be near the one of the geodesic sphere, i.e. −Υ (it points towards the 0). Therefore we define N = −Υ + a˚˜ j_Z

j,

and we choose the coefficients aj _{so that}_{N is orthogonal to all the Z}˚_˜

i’s. Using (3.6) we get this

condition:

˚gi,jaj= −ρwi. (3.8)

Moreover

g(N,˚˜ N ) = g(−Υ + a˚˜ iZi, −Υ + ajZj) = 1 + aiajg(Zi, Zj) = 1 − ρaiwi= 1 + ρ2˚gi,jwiwj. (3.9)

Thus the unit normal vector field is ˚N := g(N,˚˜ N )˚˜ −1/2N .˚˜ From these, we derive for its second fundamental form

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Proposition 3.1.4. For the second fundamental form the following expansion holds ˚_h_i,j_{= ρ(1 − w)g(Θ}_i_{, Θ}_j_{) + ρ(Hess}_Sm(w))_i,j+

2 3ρ

3_{(1 − w)}3_g(R

p(Θ, Θi)Θ, Θj) (3.10)

+ ρQ(2)_p (w) + ρ3Lp(w) + O(ρ4). (3.11)

Proof. We first obtain an expansion of˚˜hi,j= −g(∇Zi

˚_˜

N, Zj). In order to get it, we use (3.3) and

the definition ofN ; it is worth having in mind that, since˚˜ N is near the radial vector, one tries˚˜ to recover the radial vector field Υ (more precisely a multiple):

˚_˜ hi,j = g(∇ZiΥ, Zj) − g(∇Zi(a k_Z k), Zj) = 1 1 − wg(∇Zi((1 − w)Υ), Zj) + 1 1 − wwig(Υ, Zj) − g(∇Zi(a k_Z k), Zj) = 1 1 − wg(∇Zi((1 − w)Υ), Zj) − ρ 1 − wwiwj− g(∇Zi(a k_Z k), Zj).

Using the compatibility of the metric we get g(∇Zi(a

k_Z

k), Zj) = −ρwi,j− akg(Zk, ∇ZiZj) = −ρwi,j− a

k_˚_g

k,l˚Γli,j = −ρwi,j+ ρ˚Γli,jwl,

where the ˚Γl_i,j’s are the Christoffel symbols of Sp,ρ(w). Therefore we have

˚_˜_h i,j= 1 1 − wg(∇Zi((1 − w)Υ), Zj) − ρ 1 − wwiwj+ ρwi,j− ρ˚Γ l i,jwl.

Now we consider ρ as a variable, and we will recover a radial derivative of the metric from the first term. Define ˜F (ρ, z) = F (ρ(1 − w(z))Θ(z)), so that Zi = ˜F∗(∂zi), and hence we preserve

all the coordinate vector fields, but now we have another one Z0:= ˜F∗(∂ρ) = (1 − w)Υ.

We remark that

g(∇Zi((1 − w)Υ), Zj) = g(∇Zj((1 − w)Υ), Zi),

because by the compatibility of the metric

g(∇Zi((1 − w)Υ), Zj) = ∂zig((1 − w)Υ, Zj) − g((1 − w)Υ, ∇ZiZj)

= −ρ(1 − w)wi,j+ ρwiwj− (1 − w)g(Υ, ∇ZiZj),

which is clearly symmetric in i and j, because the connection induced on Sp,ρ(w) as a

sub-manifold is of Levi-Civita, and thus the Christoffel symbols are symmetric. Once we have this symmetry, using that [Z0, Zi] = d ˜F ([∂ρ, ∂zi]) = 0 and the metric has no torsion, we can write

2g(∇Zi((1 − w)Υ), Zj) = g(∇ZiZ0, Zj) + g(∇ZjZ0, Zi) = ∂ρ˚gi,j,

where in the latter equality we have used the compatibility of the metric. Inserting the latter in the espression founded for˚˜hi,j we get

˚_˜

h = 1

2(1 − w)∂ρ˚g − 1

1 − wρdw ⊗ dw + ρHess˚gw.

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3.2. A FIXED POINT ARGUMENT 19 1 2(1 − w)∂ρ˚gi,j= ρ(1 − w)g(Θi, Θj) + 2 3ρ 3_{(1 − w)}3_g(R p(Θ, Θi)Θ, Θj) + ρ4Lp(w) + ρQ(2)p (w) + O(ρ 4_), and also ˚_Γl

i,j= Gammali,j+ ρ2Lp(w) + ρ2Q(2)p (w) + O(ρ4),

where the Γl

i,j’s are the Christoffel symbols of Sm in the parametrization z 7→ Θ(z) and

we have used the formula for the Christoffel symbols in the case of a Levi-Civita connection. Resuming all these formulas we get

˚_h

i,j= ρ(1 − w)g(Θi, Θj) + ρ(wi,j− Γki,jwk) +

2 3ρ

3_{(1 − w)}3_g(R

p(Θ, Θi)Θ, Θj) (3.12)

+ ρQ(2)_p (w) + ρ3Lp(w) + O(ρ4). (3.13)

Finally, from (3.7) we get

g(N,˚˜ N )˚˜ −1/2= 1 + Q(2)_p (w),

so we conclude since the espression remain invariated by multiplying for such a factor.

By taking the trace with respect to ˚g we easily get that for the mean curvature

ρH(Σp,ρ(w)) = m + (∆Sm+ m)w − 2 3ρ 2_{Ric(Θ, Θ) + ρ}2_L p(w) + Q(2)p (w) + O(ρ 3_). _(3.14)

3.2 A fixed point argument

Now one would like to impose this mean curvature to be m/ρ. Unfortunately, the operator ∆Sm+ m has a non-trivial kernel, so one cannot expect to solve for

ρH(Σp,ρ(w)) = m.

Let us remark that, since the sphere is smooth and compact, every small perturbation of it can be written as a normal graph on the sphere for a suitable function, so the impossibility of solving this problem is not due to the lack of a tangential component.

Instead of solving it, if one try to solve it modulo the kernel of the operator ∆Sm+ m, one

can succeed, as they showed in their article. Since this operator is the Jacobi operator of the sphere, it is known that its kernel is generated by infinitesimal traslations, (we will prove it in the Appendix (6.1) ), so we have to solve

ρH(Σp,ρ(w)) = m − ρg(Θ, Ξ),

where Ξ is a fixed vector in TpM . So substituing the expression for the mean curvature, we

get (∆Sm+ m)w + ρg(Θ, Ξ) = 2 3ρ 2_{Ric(Θ, Θ) + ρ}2_L p(w) + Q(2)p (w) + O(ρ 3_).

Recall that this equation is solved by a fixed point argument. More precisely, we perform a Lyapunov-Schmidt reduction of the problem, by splitting the space C2,α_(Sm_{) in the direct}

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sum Ker(∆Sm + m) ⊕ Ker(∆Sm + m)⊥, where ⊥ is w.r.t. the L2 product. Let us denote

Ck,α_(Sm₎⊥ _{:= Ker(∆}

Sm + m)⊥∩ Ck,α(Sm). Project the equation in these two space. One

can assume that w ⊥ Ker(∆Sm+ m)); it easy to solve the equation obtained by projecting on

Ker(∆Sm+ m)⊥ as a fixed point problem for the (invertible!) operator:

∆Sm+ m : C2,α(Sm)⊥ −→ C0,α(Sm)⊥.

In this way one finds that there exist constants κ, ρ0> 0, which are independent of the choice

of the point p ∈ M (by its compactness), such that, for all ρ ∈ (0, ρ0) and p ∈ M , there exists a

unique couple (wp,ρ, Ξp,ρ) ∈ C2,α(Sm)⊥× TpM , solution of the equation, which belongs to the

closed ball of radius κρ2 _{in this space. More precisely, one finds that kw}

p,ρk ≤ cρ3.

Furthermore, it is easy to see (might be necessary take a smaller ρ0) that both wp,ρ and Ξp,ρ

depend smoothly on p; therefore, as p varies over M , Ξp,ρ define a smooth vector field in T M

and wp,ρdefines a function on the unitary tangent bundle U T M . Moreover, for all k ≥ 0 it holds

k∇k

pwp,ρkC2,α_{(U T M )}+ k∇k_pΞp,ρkC2,α_{(T M )}≤ ckρ2,

for some costant ck which does not depend on ρ ∈ (0, ρ0) nor on p ∈ M . Denote Sp,ρ[ the

surface Sp,ρ(wp,ρ).

3.3 A variational argument

Clearly, S[

p,ρ might not be a CM C surface for every point p, in fact in order to be CM C it

is necessary that it holds Ξp,ρ= 0. We use the following variational characterization of CM C

surfaces: given any hypersurface S embedded in Rm+1, which bounds a compact domain BS, set

Eeucl(S) := V olm(S) − H0V olm+1(BS),

where H0is a fixed costant. Given any vector field Ξ, one can flow the hypersurface S along

Ξ, obtaining a family St. With these notation, the first variation of

t 7−→ Eeucl(St) is given by d dtEeucl(St) |t=0= ˆ S (H(S) − H0)Ξ · NSdvolS,

where H(S) is the mean curvature and NS is the unit normal vector field of S, which is

assumed to point towards BS. In the case when Ξ is a Killing vector field, i.e. the flow generated

by Ξ acts by isometries, one get

Eeucl(St) = Eeucl(S),

so _ˆ

S

(H(S) − H0)Ξ · NSdvolS = 0.

This identity was already used by Kapouleas in [23] to prove the following observation: assume S is a compact embedded hypersurface in Rm+1_{whose mean curvature verify H(S) = H}

0+Ξ0·NS

for some Killing vector field Ξ0, then Ξ0= 0 and S is a costant mean curvature hypersurface.

Therefore we adapt this osservation in a Riemannian setting; let us define the function Ψρ(p) := V olm(Sp,ρ[ ) −

m

ρV olm+1(B

[ p,ρ),

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3.3. A VARIATIONAL ARGUMENT 21 where B[

p,ρis the domain enclosed by Sp,ρ[ ; they prove the following proposition, of which we

report the proof (see Proposition 3.1. in [33]):

Proposition 3.3.1. There exists ρ0 > 0 such that, if ρ ∈ (0, ρ0) and if p is a critical point of

Ψρ, then the surface Sp,ρ[ is a constant mean curvature hypersurface with mean curvature equal

to m_ρ.

Proof. Let p be critical for the function Ψρ. Since by construction the hypersurface S[p,ρ has

mean curvature equal to m_ρ − g(Ξp,ρ, Θ), it is enough to show that Ξp,ρ= 0 (for ρ small enough).

Given any Ξ ∈ TpM we know that DΨρ |p(Ξ) = 0.

Provided t is small enough, the surface S[

q,ρ where q := Expp(tΞ), can be written as a normal

graph over Sp,ρ[ for some function fp,ρ,Ξ,tthat depends smoothly on t. This defines a vector field

on S[ p,ρ by

Zp,ρ,Ξ:= ∂tfp,ρ,Ξ,t |t=0NS[ p,ρ,

where as usual NS[

p,ρ denotes the normal vector field about S

[

p,ρ. Denoting by X the parallel

transport of Ξ along geodesics issued from p, the vector field Zp,ρ,Ξ can be estimated by

kZp,ρ,Ξ− Xkg≤ cρ2kΞ2kg (3.15)

where c > 0 is a constant not depending on ρ (small enough) nor on Ξ.

In fact, given two spheres S(p, ρ) and S(q, ρ) in Rm+1, with q = Expp(tΞ) = p + tΞ, with the

same notation of above one easily finds Zp,ρ,Ξ= ΞN, where ·N denotes the normal component of

a vector with respect to the sphere S(p, ρ). The parallel transport X of Ξ is Ξ itself, so denoting Θi a local frame for Sm, so that ρΘi is a local frame for S(p, ρ), we obtain:

kZp,ρ,Ξ− Xk = kΞTk = ρ2k

X

i

hΞ, ΘiiΘik ≤ ρ2kΞk,

where ·T is the projection on the tangent space of the sphere S(p, ρ). Using the (3.7) and the fact that S[

p,ρ and S[q,ρare normal graph over the geodesic spheres Sp,ρand Sq,ρrespectively, we

get the desired estimate.

From the formulas of the first variation of the m-dimensional (6.2.3) and the (m + 1)-dimensional volumes (6.2.4), we get

0 = DΨρ|p(Ξ) = ˆ S[ p,ρ (H(S_p,ρ[ ) −m ρ)g(Zp,ρ,Ξ, NSp,ρ[ )dvolSp,ρ[ . By costruction H(S_p,ρ[ ) =m ρ − g(Ξp,ρ, Θ), hence _ˆ S[ p,ρ g(Ξp,ρ, Θ)g(Zp,ρ,Ξ, NS[ p,ρ)dvolSp,ρ[ = 0 (3.16)

so we are in a situation very similar to the euclidean one. An hard estimation will erase the dependence on the vector field Zp,ρ,Ξ, and will let us conclude the proof. Let us denote

Z := Zp,ρ,Ξ, N := NS[

p,ρ and ˜N := a

j_Z

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perturbations, a fortiori for wp,ρ. Using the the parallel transport is an isometry (in the third

equality) we can write

g(Z, N ) = g(X + (Z − X), −Υ + (N + Υ)) = −g(X, Υ) − g(Z − X, Υ) + g(X, N + Υ) + g(Z − X, N + Υ) = −g(Ξ, Θ) + g(X, N + Υ) + g(Z − X, N ),

therefore by the triangular inequality, the Cauchy-Schwartz inequality and using again that the parallel transport is an isometry

|g(Z, N ) + g(Ξ, Θ)| ≤ kZ − Xkg+ kXkgkN + Υkg= kZ − Xkg+ kΞkgkN + Υkg

≤ cρ2_kΞk

g+ kΞkgkN + Υkg,

where in the last inequality we have used (3.15). In order to estimate kN + Υkg, let us recall

that N = k ˜N k−1g N , and that we have the equations (3.8) and (3.9) on ˜˜ N . Furthermore, recall

that (1 + x)−12 ∼ 1 −x

2+ o(x); thus we obtain (c is a costant which varies from line to line)

kN + Υkg= kk Ñ k−1g N + a˜ j_Z j− Ñ kg= k(k Ñ k−1g − 1) Ñ + a j_Z jkg ≤ |k Ñ k−1_g − 1|k Ñ kg+ kajZjkg≤ c| − ρ2 2˚g i,j_w iwj+ o(ρ2)| + q g(ai_Z i, ajZj),

hence using that kwkC2,α ≤ cρ2 and ˚gi,j∼ ρ−2

cρ2+ q ai_aj_˚_g i,j = cρ2+ p −ai_ρw i= cρ2+ q ρ2_˚_gi,j_w iwj≤ cρ2, thus resuming |g(Z, N ) + g(Ξ, Θ)| ≤ cρ2_kΞk g.

Therefore using (3.16) and the Cauchy-Schwartz inequality we obtain ˆ S[ p,ρ g(Ξp,ρ, Θ)g(Ξ, Θ)dvolS[ p,ρ = ˆ S[ p,ρ g(Ξp,ρ, Θ)g(Ξ, Θ) + g(Z, N ) − g(Z, N )dvolS[ p,ρ = (3.17) ˆ S[ p,ρ g(Ξp,ρ, Θ)g(Ξ, Θ) + g(Z, N )dvolS[ p,ρ ≤ cρ 2_kΞk g ˆ S[ p,ρ g(Ξp,ρ, Θ)dvolS[ p,ρ. (3.18)

From the formula ... in the Appendix we have V olm(Sm)kvk2= (m + 1)

ˆ

Sm

hv, Θi2_dvol Sm,

for every v ∈ Rm+1_{. Using the formula ... in the Appendix for the expansion of the volume form}

we get ˆ S[ p,ρ g(Ξ, Θ)2dvol_S[ p,ρ = ˆ Sm hΞ, Θi2ρ2(1 − w)2 q

det(˚gi,j)dvolSm(z) = (3.19)

ˆ Sm hΞ, Θi2_ρm+2_{(1 − w)}m+2_{(1 +} 1 2k∇Smwk 2 Sm+ O(ρ2))dvolSm = ˆ Sm

hΞ, Θi2_ρm+2_{(1 + O(ρ}2_))dvol Sm ≥ (3.20) 1 2 ˆ Sm hΞ, Θi2_ρm+2_dvol Sm= ρm+2kΞk2V olm(Sm) 1 2(m + 1) ≥ 1 4(m + 1)ρ m_kΞk2 gV olm(Sm), (3.21)

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3.3. A VARIATIONAL ARGUMENT 23 for ρ small enough. Hence, (3.18) becomes:

ˆ S[ p,ρ g(Ξp,ρ, Θ)g(Ξ, Θ)dvolS[ p,ρ ≤ cρ 2−m 2 ˆ S[ p,ρ |g(Ξp,ρ, Θ)|dvolS[ p,ρ· · ˆ S[ p,ρ |g(Ξ, Θ)|dvolS[ p,ρ 12_.

Substituting Ξ = Ξp,ρ and using a Cauchy-Schwartz inequality we get

ˆ S[ p,ρ |g(Ξp,ρ, Θ)|2dvolS[ p,ρ ≤ cρ 2 ˆ S[ p,ρ |g(Ξp,ρ, Θ)|2dvolS[ p,ρ.

For ρ small this is true only if the integral is null, but by (3.21), this is possible only if Ξp,ρ= 0, as required.

This concludes the proof of the existence of a CM C hypersurface; in fact, being M compact there exist at least cat(M ) critical points for Ψρ, where cat(M ) is the Lusternik-Schnirelman

category of M .

Given a topological space M , the Lusternik-Schnirelman category of a subset A ⊂ M with respect to M , denoted by cat(A, M ) (or cat(M ) in the case A = M , is the least integer k such that there exist k closed and contractible subset Ai⊂ M , such that their union cover A. If there

are no integers with this property we set cat(A, M ) = +∞.

These concept is well analyzed and applied to the theory of min-max critical points in [3], where it is proved a very general theorem concerning the existence of infinitely many critical points, using certain min-max classes defined through the Lusternik-Schnirelman category; re-stricted to our case, this theorem can read as

Theorem 3.3.2. Let M be a compact manifold, and let J : M −→ R be C1,1_{. Then J has at}

least cat(M ) critical points.

By the previous proposition each of these critical points gives rise to a CM C surface. Anyway, we obtain exspansion for the function Ψρ too, as well as for the area and the volume of these

hypersurfaces, which we recall here to have a comparison with ours:

ρ−mΨρ(Sp,ρ[ ) = 1 m + 1V olm(S m₎_{1 −} 1 2(m + 3)ρ 2_{s(p) +} 1 72(m + 3)(m + 5)ρ 4 _5s2_{(p) + 8kRic} pk2 − 3kRpk2− 18∆gs(p) + 1 18(m + 3)(m + 2)ρ 4 m + 6 m s 2_{(p) − 2kRic} pk2 + O(ρ5) , where s is the scalar curvature, and Rp is the Riemann tensor at p; for the area:

V olm(Sp,ρ[ ) = ρmV olm(Sm) 1 −

1 2(m + 1)ρ

2_{s(p) + O(ρ}4_);

finally, for the volume enclosed:

V olm+1(Bp,ρ[ ) = ρ m+1 1 m + 1V olm(S m_{) 1 −} 3(m + 2) 2m(m + 3)ρ 2_{s(p) + O(ρ}4_).

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In order to recover (and generalize) Ye’s existence result in [45], we define the following function: φ(p, ρ) := 2(m + 1)(m + 3) V olm(Sm) ρ−2 1 m + 1V olm(S m_{) − ρ}−m_Ψ ρ(Sp,ρ[ ).

Let us remark that this is an affine transformation of the function Ψ, so one preserve each critical point. With this notation we have the following

Theorem 3.3.3. There exists ρ0> 0 and a smooth function

φ : M × (0, ρ0) −→ R,

such that:

(i) For all ρ ∈ (0, ρ0), if p is a critical point of the function φ(·, ρ) then,there exists an embedded

hypersurface S[

p,ρ whose mean curvature is costant and equal to mρ and that is a normal

graph over Sp,ρ for some function which is bounded by a costant times ρ3 in C2,α topology.

(ii) For all k ≥ 0, there exists ck> 0 which does not depend on ρ ∈ (0, ρ0) such that

kφ(·, ρ) − s + ρ2_rk

Ck_{(M )}≤ c_kρ3.

Here r is given by the formula:

r(p) = 1 36(m + 5) 5s2(p) + +8kRicpk2− 3kRpk2− 18∆gs(p) + 1 9(m + 1)(m + 2) m + 6 m s 2_{(p) − 2kRic} pk2.

Remark 3.3.4. A simple application of the implicit function theorem shows that Pacard and Xu recover Ye’s existence result in [45]; more precisely one has that, given a non degenerate critical point p0 of s, one finds a critical point p = p(ρ) of φ(·, ρ) (and thus a CM C hypersurface) such

that dist(p, p0) ≤ cρ2. They generalize Ye’s result in the sense that, assuming for example that s

is costant, they get the existence of such CM C hypersurfaces near non degenerate critical points of the function r. Of course one can weak the non degeneracy condition on s, by assuming p0

local strict extremum for s, as well as p0 critical for s such that the Browder degree of ∇s at p

is non zero.

3.4 The Isoperimetric profile

We conclude this chapter quoting the interesting work of Nardulli [32].

In order to state this result easily we assume M is compact, neverthless his results hold in more general ambient manifold. He calls the surfaces S[

p,ρ, i.e. those with almost costant mean

curvature, pseudobubbles. He first proves an uniqueness result for pseudobubbles, with the help of the notion of center of mass. Let F2,α be the fiber bundle on M with fiber at p given by C2,α_{(U T M, R), where U T M is the unitary tangent bundle.}

Theorem 3.4.1. There exists a C∞ _{map, β : M × R → F}2,α _{such that for all p ∈ M and all}

sufficiently small ρ > 0, the hypersurface Expp β(p, ρ)(Θ)Θ

is the unique pseudobubble with center of mass p and volume ωm+1ρm+1, where as usual ωm+1 is the volume of the euclidean

ball in Rm+1_{. Moreover, if φ is an isometry of M , φ sends pseudobubble to pseudobubble and φ}

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3.4. THE ISOPERIMETRIC PROFILE 25 He also gets a result related to the isoperimetric problem, as well as an expansion for the isoperimetric profile near the zero:

Theorem 3.4.2. There exists a C∞ _{map, f : M × R → R such that for sufficiently small ρ, the} solutions of the isoperimetric problem at volume v = ωm+1ρm+1 are exactly the pseudobubbles

(uniquely) determined by β (with a possibly different ρ0), with center of mass in the minima of f . Moreover f is invariant by isometries.

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Chapter 4

Perturbed Double Bubbles

In this chapter we want to extend the strategy developed in Pacard-Xu [33] to our problem. However, due to the presence of the singular set Γ, we have to allow the perturbation to have a tangential component too, so we have to generalize their exspansions; moreover, we cannot center all the three sheets of the standard double bubble contemporarly, so we will get formulas involving the several center C of the several sheets. Furthermore, we will get an analogous expansion for the mean curvature, that is, one in which the Jacobi operator of the double bubble is involved; unfortunately, the kernel of the latter operator is larger than the one of the sphere, in fact the rotations preserve the area of our double bubble and might have a non-trivial normal component. Thus we will characterize this kernel in chapter 5.

Let us fix two volumes V1 ≤ V2 and a standard double bubble ˜Σ in Rm+1. We use the same

notation of chapter 3.

4.1 Riemannian Setting

In order to adapt the strategy in [33], it is very important to fix some notation. Firstly, we consider only one of the smooth pieces of ˜Σ without its boundary, i.e. a spherical cap ˜Σ included in a sphere S(C, R). We postpone the case in which ˜Σ is a disk.

Fix a point p ∈ M ; we introduce normal coordinate on a neighborhood U of p: take an orthonormal basis {Eµ}, µ = 1, ..., m + 1, of TpM , and set F : Rm+1 −→ M by F (x) :=

Expp(xµEµ), where Exp is the exponential map on M and summation on repeated indices

is understood. This choice of coordinates induces coordinate vector field Xµ = F∗(∂xµ). Recall

that the choice of normal coordinates implies gµ,ν := g(Xµ, Xν) = δµ,ν at p. Moreover, we have

the following expansion:

Proposition 4.1.1. At the point q = F (x), the following expansion holds for all µ, ν = 1, ..., m+1

g(Xµ, Xν) = δµ,ν+ 1 3g(Rp(Ξ, Eµ)Ξ, Eν) (4.1) +1 6g(∇ΞRp(Ξ, Eµ)Ξ, Eν) + O(|x| 4₎ _(4.2)

where Rp is the Riemann curvature tensor at the point p, and Ξ = xµEµ ∈ TpM .

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4.1.1 Notation

We denote by ι : Rm+1_{−→ T}

pM the isomorphism which sends ∂xµ to E_µ; let Σ := ι( ˜Σ) and let

us call Θ = Θµ_E

µ the parametrization of Σ induced by the parametrization z = (z1, .., zm+1)

of ˜Σ through the map ι. Moreover, as we did in the proposition, we denote by Ξ an arbitrary element in TpM , so that Ξ = ι(x) = xµEµ.

We denote for a function f (or a vector field V ) defined on ˜Σ, the first order derivative fi(z) =

∂zif (z) (resp. (V_i(z))µ= ∂_zi(V (z)µ)), and similarly for higher order derivative.

We identify every vector with its image through the map ι. Moreover, we use the same symbol for C, N and their image through Expp; in particular, it holds g(Θ − C, Θ − C) = R2, so

TΘΣ = (Θ − C)⊥ = Span(Θi) where i = 1, ..., m: Proof. g(Θi, C − Θ) = ∂zi(Θµ)(C − Θ)νδµ,ν = ∂zi((Θ − C)µ)(C − Θ)νδµ,ν = 1 2∂zi( X µ (Cµ− Θµ₎2_{) =}1 2∂zi(R 2_{) = 0.}

Since Y is tangent to Σ too, it holds g(C − Θ, Y ) = 0. Now define Ω := Yµ_X

µ, Υ := ΘµXµ and Υi := Θ µ

iXµ vector fields on U . We can not apply

the Gauss Lemma, because the center is not zero, but by Proposition 3.1.1, we have this quasi-orthogonality lemma that will be useful in the sequel:

Lemma 4.1.2. At every point of q ∈ Σp,ρ(0, 0) := Expp(ρΣ), we have the exspansions:

     g(Υ − C, Υi) = −ρ 2 3g(Rp(Θ, C)Θ, Θi) + O(ρ 3₎ g(Υ − C, Ω) = −ρ₃2g(Rp(Θ, C)Θ, Y ) + O(ρ3) g(Υ − C, Υ − C) = R2+ρ₃2g(Rp(Θ, C)Θ, C) + O(ρ3) (4.3)

Proof. Using the proposition 3.1.1, we get:

g(Υi, C − Υ) = ∂zi(Θµ)(C − Θ)νg(X_µ, X_ν) = ∂_zi(Θµ)(C − Θ)ν(δ_µ,ν+ ρ2 3g(Rp(Ξ, Eµ)Ξ, Eν) + O(ρ3)) = ρ 2 3 g(Rp(Θ, Θ − C)Θ, Θi) + O(ρ 3_{) = −}ρ 2 3 g(Rp(Θ, C)Θ, Θi) + O(ρ 3_).

Analogously for the others.

From now on we consider a fixed admissible perturbation φw,Y, and set Gρ(z) := F ◦φw,Y(z) =

Expp(ι(ρ(z − w(z)N (z) − Y (z)))) for ρ small enough, z ∈ ˜Σ.

Analogously at what is done in [33] , we adopt this convenction: any expression of the form Lp(w, Y ) denotes a linear combination of the functions w (and Y ), together with their derivatives

with respect to the vector fields Θi up to order 2 (and 1). The coefficients of Lp might depend

on ρ and p but, for all k ∈ N, there exists a constant c > 0 independent of ρ ∈ (0, 1) and p ∈ M such that (we omit the domain and codomain of the functions)

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4.2. PERTURBATIONS OF DOUBLE BUBBLES 29

Similarly, given a ∈ N, any expression of the form Q(a)p (w, Y ) denotes a nonlinear operator

in the functions w (and Y ), toghether with their derivatives with respect to the vector fields Θi

up to order 2 (and 1). The coefficients of the Taylor expansion of Q(a)p (w, Y ) in powers of w, Y

and their partial derivatives might depend on ρ and p and, given k ∈ N, there exists a constant c > 0 independent of ρ ∈ (0, 1) and p ∈ M such that Q(a)p (0, ·) = Q(a)p (·, 0) = 0 and

kQ(a)

p (w2, Y2) − Q(a)p (w1, Y1)kCk,α ≤ c(k(w1, Y1)kCk+2,α_×Ck+1,α+ (4.5)

k(w1, Y1)kCk+2,α_×Ck+1,α)a−1· k(w2− w1, Y2− Y1)kCk+2,α_×Ck+1,α, (4.6)

provided k(wi, Yi)kCk+2,α_×Ck+2,α are small enough. For our purpose, we can look at only the

Q(2)’s nonlinearity, so we will absorb Q(a)’s in it when a ≥ 2 (there is no mistake in do it, because Q(a) _{easily verify the inequality defining Q}(2) _{in that case). A typical example of Q}(a)_{, (maybe}

the only in this paper), is an homogeneous polynomial of degree a, in w, Y and their derivatives up to order 2 (e.g. wjRic(Y, Yi) = Q(3)(w, Y )).

4.2 Perturbations of Double Bubbles

In this section, we want to calculate which are the admissible perturbations for a standard double bubble. In order to do so, let us fix two volumes V1≤ V2 and a standard double bubble

˜

Σ in Rm+1, the symmetry axis of which is the last coordinate axis Span(xm+1) and such that Γ ⊂ {xm+1_{= 0}.}

Let us define a perturbation φw,Y: ˜Σ −→ Rm+1 by:

φw,Y: x 7−→ x − w(x)N (x) − Y (x) (4.7)

where is understood that in Σi, w = wi∈ C2,α(Σi), N = Niand Y = Yi ∈ C2,α(Σi) ∩ T (Σi).

Let us remark that the tangential component Y is due to the non regularity of ˜Σ up to the common boundary Γ. Note that not all of the above perturbations are admmissible, but we have to impose that the image of the points of Γ through φw,Y is well defined, i.e we have to impose:

x − w0(x)N0(x) − Y0(x) = x − w1(x)N1(x) − Y1(x) = x − w2(x)N2(x) − Y2(x), (4.8)

whenever x ∈ Γ. From now on we can consider x ∈ Γ to be fixed. This corresponds to the geometric idea of not unstick the smooth pieces, otherwise the bubble would burst. This is clearly equivalent to:

w0(x)N0(x) + Y0(x) = w1(x)N1(x) + Y1(x) = w2(x)N2(x) + Y2(x). (4.9)

Now let us analyze this condition. Split Y (x) as Y (x) = (Y (x))Γ + u(x)ν(x), where the ·Γ _{stands for the orthogonal projection of a vector onto the space T}

xΓ. Taking in (4.9) the

projection onto TxΓ we get:

(Y0(x))Γ= (Y1(x))Γ= (Y2(x))Γ. (4.10)

Substituting in (4.9), we are left with the following:

w0(x)N0(x) + u0(x)ν0(x) = w1(x)N1(x) + u1(x)ν1(x) = w2(x)N2(x) + u2(x)ν2(x). (4.11)

Taking the scalar product of the latter with ν1 or N1, and using (2.4), one obtain the sistem

(forget the x-dependence): ( w1=1₂w2− √ 3 2 u2= 1 2w0+ √ 3 2 u0 u1= − √ 3 2 w2− 1 2u2= √ 3 2 w0− 1 2u0, (4.12)

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from which we have the condition: (

w1= w0+ w2

u0+ u1+ u2= 0.

(4.13)

Moreover, taking the norm in (4.9) we get the mixed condition:

u20+ w02= u21+ w21= u22+ w22. (4.14)

Let us note that (4.13) and (4.14) determine uniquely the ui’s and the wi’s, so that they are

equivalent to (4.11). Furthermore, (4.11) and (4.10) imply (4.9); we will then use (4.13), (4.14) and (4.10) because they are more suitable condition for a Dirichlet problem. We are led to the following:

Definition 4.2.1. We define the class of admissible couple as

Camm:= {(w, Y ) ∈ C2,α(Σ) × C2,α(Σ; Rn+1) ∩ T (Σ) | w, Y verif y (4.15)

(4.13), (4.14) and (4.10), kwkC2,α, kY k_C2,α ≤ δ}, (4.16)

for some δ small enough. Moreover we define the class of admissible perturbations to be {φw,Y |

(w, Y ) ∈ Camm}.

Remark 4.2.2. We need the smallness in norm of the couple (w, Y ) to avoid self-intersection of the perturbed double bubble.

Inspired by the work of Pacard and Xu, we expect to deal with one (or more..) degenerate elliptic operator, and to characterize its kernel. Heuristically, given a degenerate elliptic operator on each sheet Σi(it will be the Laplace-Beltrami operator of the sheet minus the first eigenvalue

of it), one expects it to have a very huge kernel, because the sheets are diffeomorphic to an open ball in the euclidean space Rm. It is so also if one imposes regularity on the interior and the junction conditions above. In their article [21], Hutchings et al. solve this problem by assuming that the variation is volume preserving; actually they consider this kind of perturbations because they work in the contest of the isoperimetric problem with two fixed volumes.

Instead of doing this, we will impose the perturbation to preserve the equiangularity, more precisely a linearized version of this condition, and we will fix this problem in the chapter 5. There are two main reasons for this choice: the first is that one cannot easily integrate function defined on a cap of sphere, whose opening angle is unknown; secondly, we will use a generalized Fourier decomposition which will let us to reduce our problem to a one dimensional problem. However, we will see that also the very weak condition of preserving the equiangularity at first order will imply that our perturbation preserves the volumes (and more..). In their article [21], Hutchings et al. call this perturbations (more precisely their normal component) Jacobi functions (for a particular elliptic operator). The reason is that this normal components are the Jacobi function associated to the area functional, i.e. they vanish its second fundamental form. Remark 4.2.3. The fact that it is sufficient to impose a linearized equiangularity condition to get uniqueness of solution, is a peculiar property of the standard double bubble; for istance, this cannot hold for the ”pitchfork”: in fact, calling t the lenght variable, every perturbation that behaves like o(t) near the singular point O preserve both the point O and the equiangularity (a fortiori the linearized one). This suggest that the double bubble are ”sensible at the singularity”. Therefore we now want to write the linearized equiangularity condition. This condition is espressed in lemma 3.6. in [21], which we recall for the convenience of the reader:

Double bubble with small volume in compact manifolds

Contents

Chapter 1

Introduction

1.1

The Isoperimetric Problem

1.2

Double Bubbles

1.3

Our work

Chapter 2

Notations and Preliminaries

2.1

Notation

2.2

Some Basic Facts about Elliptic Regularity Theory

2.3

Some Basic Facts about Riemannian Geometry

2.4

Geometry of the Double Bubbles

Chapter 3

Pacard and Xu’s work

3.1

Perturbation of Geodesic Spheres

3.2

A fixed point argument

3.3

A variational argument

3.4

The Isoperimetric profile

Chapter 4

Perturbed Double Bubbles

4.1

Riemannian Setting

4.1.1

Notation

4.2

Perturbations of Double Bubbles