Benoît R Kloeckner

CURVATURES AND ANISOMETRY OF MAPS by

come

Introduction

One of the basic facts of Riemannian geometry is that curvatures are isometry invariants: this explains for example why one cannot design a perfect map of a region on the earth. In this article, we shall be interested in quantifying this fact: how far from being an isometry a map from a region of a manifold to another manifold must be, when the source and target manifolds satisfy incompatible curvature bounds?

When the source manifold is the round 2-sphere and the target manifold is the Euclidean plane, this question is a cartography problem: a round sphere is a relatively good approximation of the shape of the Earth. It has been considered by Milnor [Mil69] who described the best map when the source region is a spherical cap. Surprisingly, it seems like no other cases of the general question above have been considered.

1.1. Distortion and anisometry. -To fill this gap, one has first to ask how we should measure the isometric default of a map ϕ : D ⊂ M → N from a domain in a manifold M to a manifold N , assumed to be a diffeomorphism on its image. Milnor uses the distortion, defined as follows. Let σ 1 = σ 1 (ϕ) and σ 2 = σ 2 (ϕ) be the Lipschitz constants of ϕ, i.e. σ 1 d(x, y) d(ϕ(x), ϕ(y)) σ 2 d(x, y) ∀x, y ∈ D and σ 1 , σ 2 are respectively the greatest and least numbers satisfying such an inequality. Then the distortion of ϕ is the number dist(ϕ) = log(σ 2 /σ 1 ).

However, when the target manifold is not Euclidean, the distortion is illsuited: it is zero for maps that are not isometries, but mere homotheties. More disturbing is the case when M is positively curved and N is negatively curved: to minimize distortion, one is inclined to take ϕ with a very small image, so that the curvature of N barely matters. To make this case more interesting, we propose the following definition of anisometry:

aniso(ϕ) = | log σ 1 | + | log σ 2 |.
This quantity generalizes distortion in the sense that when N = R n , inf ϕ aniso(ϕ) = inf ϕ dist(ϕ).

1.2. Azimuthal maps. -To describe our results we will need to introduce a specific family of maps between model spaces. All considered manifolds will be of the same fixed dimension n; we set X κ for the simply connected manifold of constant curvature κ (thus a sphere, the Euclidean space or a hyperbolic space).

Given a point x ∈ X κ , we have polar coordinates (t, u) (t a positive real, u a unit tangent vector at x) given by the exponential map:

y = exp x (tu)
where t is less than the conjugate radius and y may be any point but the antipodal point to x (when κ > 0).

Definition. -An azimuthal map is a map ϕ : D ⊂ X ρ → X κ where D is a geodesic ball, which reads in polar coordinates centered at x and ϕ(x) as ϕ(t, u) = (R(t), L(u))

where L is a linear isometry from T x X ρ to T ϕ(x) X κ and R is a differentiable function. In other words, we have ϕ(exp x (tu)) = exp ϕ(x) (R(t)L(u)).

The function R is then called the distance function of ϕ.

As we consider only model spaces, L is irrelevant and the function R defines a unique azimuthal map up to isometries. The azimuthal map associated to each of the following distance functions bears a special name:

-R(t) = t: equidistant azimuthal map, -R(t) = σt with σ ∈ (0, 1): σ-contracting azimuthal map Moreover, given ρ and κ there exists exactly one family of conformal azimuthal maps and a unique volume-preserving map B ρ (α) → Xκ (see below for details).

1.3. Description of the results. -We shall not state our results in the greatest generality in this introduction, please see below for details.

Our main results have the following form: we assume M satisfies some kind of lower curvature bound associated with a parameter ρ, that N satisfies some kind of upper curvature bound (or more general geometric assumption) associated with a parameter κ < ρ, and that ϕ is a map (possibly satisfying extra assumptions) from a geodesic ball of center x and radius α in M to N .

Our methods provide half-local results, and we shall always assume that α is bounded above by some number. This bound shall be explicit most of the time and depends only on synthetic geometrical properties of M and N . In some cases (e.g. when the target is a Hadamard manifold) this bound will be completely harmless.

We then conclude that there is an azimuthal map φ : B ρ (α) → X κ (where B ρ (α) is any geodesic closed ball of radius α in X ρ ) such that aniso(ϕ) aniso( φ) with equality if and only if ϕ is conjugated to φ by isometries.

For simplicity, we shall write Ric M ρ to mean that the Ricci tensor and the metric tensor of M satisfy the usual bound

Ric x (u, u) ρ • (n -1)g x (u, u) ∀x, u.
Similarly, K N κ means that the sectional curvature of N is not greater than κ at any tangent 2-plane.

We shall always assume implicitly that M (or more generally B M (x, α)) and N are complete; recall that they have the same dimension n.

Theorem 1.1 (General maps). -Assume Ric M ρ, K N κ where ρ > κ, and α A 1 (M, N ) where A 1 (M, N ) is an explicit positive constant.

Then any map ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) aniso( φ)

where φ is:

the equidistant azimuthal map B ρ (α) → X κ when κ 0, -the σ-contracting azimuthal map B ρ (α) → X κ when κ < 0, where σ is such that the boundaries of B ρ (α) and B κ (σα) have equal volume.

Moreover in case of equality ϕ and φ are conjugated by isometries (in particular, the source and image of ϕ have constant curvature ρ and κ).

One can write aniso( φ) explicitly, see below. This theorem is proved using a rather direct generalization of Milnor's argument who considers the constant curvature, 2-dimensional case.

Remark. -1. It is interesting to see that the sign of κ has such an influence on the optimal map: when κ > 0 the best map is isometric along rays issued from the center, and increases distances in the orthogonal directions, while when κ < 0 the best map induces an isometry on the boundaries but contracts the radial rays. Of course, when κ = 0 all σcontracting azimuthal maps are equivalent up to an homothety, and as long as σ 1 1 σ 2 their anisometries are equal. 2. The hypothesis on N can be relaxed thanks to the generalized Günther inequality proved with Greg Kuperberg [START_REF] Kloeckner | A refinement of Günther's candle inequality[END_REF]. In particular, K N κ can be replaced by mixed curvatures bounds like

K N ρ and Ric N (n -1)κ -nρ
see Section 2.3 and Theorem 3.2 for the most general hypothesis and the above reference for various classical assumptions that imply this general hypothesis. 3. The precise expression of A 1 is given page 13. In many cases one can adapt the result and its proof to larger α but we favored clarity over exhaustivity. For example, what happens for α close to π √ ρ is that the boundary of B ρ (α) becomes very small, and one can improve the equidistant azimuthal map by making it dilating along the rays.

We shall then consider maps satisfying special conditions. Two prominent examples are volume-preserving maps and conformal maps. In cartography, both make sense: area is obviously a relevant geographic information, and for many historical uses (e.g. navigation) measurement of angles on the map have been needed. Moreover, asking a map to be conformal means that zooming into the map will decrease arbitrarily the distortion of a smaller and smaller region. We therefore ask whether in general, asking ϕ to be volume-preserving or conformal increases the anisotropy lower bound by much.

In the theorems below, we shall make the assumption that N satisfies the best isoperimetric inequality holding on X κ , meaning that for all smooth Ω ⊂ N , Vol(∂Ω) I κ (Ω) where I κ is the isoperimetric profile of X κ defined by

I κ (V ) = inf Ω⊂Xκ {Vol(∂Ω) | Vol(Ω) = V }.
This assumption can be replaced by K N κ in some cases.

One says that n is a Hadamard manifold if K N 0 and N is simply connected; it is conjectured that all Hadamard manifolds satisfy the isoperimetric inequality of X κ whenever K N κ, but this conjecture has only been proved in a handful of cases: when n = 2 [Wei26, Aub76], n = 3 [Kle92], (n = 4, κ = 0) [START_REF] Croke | A sharp four-dimensional isoperimetric inequality[END_REF] and (n = 4, κ < 0) for small enough domains [START_REF]The generalized Cartan-Hadamard conjecture and the problem of the Little Prince[END_REF].

Moreover, the similar conjecture when κ > 0 holds in dimension n = 4 for uniquely geodesic domains [START_REF]The generalized Cartan-Hadamard conjecture and the problem of the Little Prince[END_REF]. When n = 4 the curvature assumption can generally be relaxed as for Theorem 1.1, see Section 2.3 below and [START_REF]The generalized Cartan-Hadamard conjecture and the problem of the Little Prince[END_REF].

This means that in most dimensions, our results below hold under a curvature assumption only conditionally to a strong conjecture; but note that even in the case when N = X κ these results are new.

Theorem 1.2 (Volume-preserving maps). -Assume Ric M ρ, N satisfies the best isoperimetric inequality holding on X κ for some κ < ρ, and α inj(x).

Then any volume-preserving map ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) aniso( φ

)
where φ is the unique volume-preserving azimuthal map B ρ (α) → X κ . Assume further that the only domains in N satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in X κ . Then whenever aniso(ϕ) = aniso( φ), the domain of ϕ has constant curvature ρ and its range is isometric to a constant curvature ball B κ (R(α)). However, there are uncountably many different maps achieving equality.

Remark. -Here we have put little restriction on α (we only restrict it below the injectivity radius at x for simplicity), but in fact stronger restriction can appear when one wants to apply the result. Indeed, if one is only able to show that small enough domains of N satisfy the desired isoperimetric inequality, then one can still use Theorem 1.2 for small enough α: then a map B(x, α) → N either has a small image, or a large σ 2 .

Theorem 1.3 (Conformal maps). -Assume Ric M ρ, N satisfies the best isoperimetric inequality holding on X κ for some κ < ρ, and α A 3 (M, N ) where A 3 (M, N ) is an explicit positive constant.

Then any conformal map ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) aniso( φ)

where φ is:

the conformal azimuthal map B ρ (α) → X κ with R ′ (0) = 1 when κ 0, -the conformal azimuthal map B ρ (α) → X κ that induces an isometry on the boundaries when κ < 0. Assume further that the only domains in N satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in X κ . Then whenever aniso(ϕ) = aniso( φ), the maps ϕ and φ are conjugated by isometries (in particular, the domain and range of ϕ have constant curvature ρ and κ), except that when κ = 0 one can compose φ with any homothety such that we still have σ 1 1 σ 2 , and still get an optimal map.

Remark. -We shall see that A 3 can in fact be chosen independently of N (but depending on κ). Moreover, when κ 0 we can take A 3 = inj(x).

Conformal maps are rare in higher dimension, so we also tackle quasiconformal maps, whose angular distortion is controlled. Recall that a smooth map ϕ is said to be Q-quasiconformal if at each point x in its domain, we have dist(Dϕ x ) Q, i.e. its infinitesimal distortion is uniformly bounded; conformal maps are precisely the 1-quasiconformal maps.

Theorem 1.4. -Assume Ric M ρ, N satisfies the best isoperimetric inequality holding on X κ for some κ < ρ, let Q be a number greater than 1 and assume α A 4 (M, N, Q) where A 4 (M, N, Q) is some positive constant.

Then any

Q-quasiconformal map ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) aniso( φ)
where φ is an explicit Q-conformal azimuthal map, which is

C 1 but not C 2 .
Assume further that the only domains in N satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in X κ . Then whenever aniso(ϕ) = aniso( φ), the maps ϕ and φ are conjugated by isometries (in particular, the domain and range of ϕ have constant curvature ρ and κ), except that when κ = 0 one can compose φ with any homothety such that we still have σ 1 1 σ 2 , and still get an optimal map.

Remark. -Here the constant A 4 is less explicit than in the other result, but it is still perfectly constructive. Moreover we shall see that when κ 0, we can take A 4 = inj(x).

It is also interesting to compare what we obtain from the above inequalities when α is small.

Corollary 1.5. -If Ric M ρ and K N κ, any map ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) 1 6 (ρ -κ)α 2 + o(α 2 ). If ϕ is conformal, then aniso(ϕ) 1 4 (ρ -κ)α 2 + o(α 2 ). If ϕ is volume-preserving, then aniso(ϕ) n 2(n + 2) (ρ -κ)α 2 + o(α 2 ).
Remark. -1. In this Corollary, one can easily replace the curvature assumptions by scalar curvature bounds, since only small balls are considered. Note that the isoperimetric inequality needed in Theorems 1.2 and 1.3 has been proved to be true for small enough domains under the curvature assumption K N < κ (or even K N κ in some cases) by Johnson and Morgan [START_REF] Morgan | Some sharp isoperimetric theorems for Riemannian manifolds[END_REF] and under Scal N < κ by Druet [START_REF] Druet | Sharp local isoperimetric inequalities involving the scalar curvature[END_REF]. To obtain a Taylor series, these strict assumptions are sufficient (but then the remainder term cannot be made explicit). 2. In all our results, one consider maps from the higher-curvature manifold to the lower-curvature one. These results imply similar estimates for maps ϕ : B(y, α) ⊂ N → M , because either such a map contracts some distances by much (hence has large anisometry), or its image contains a ball of radius bounded below, allowing us to apply the results above to ϕ -1 . However, the estimates one gets that way are certainly not sharp, and we do not know whether φ-1 is optimal in any of the situation treated above; it seems that even the case of a map from a ball in the plane to a round 2-sphere is open. One might want to perturb the equidistant azimuthal map to enlarge the boundary of its image, so as to limit the distortion along the boundary. It is not clear whether this can be achieved without increasing distortion too much anywhere else.

Organization of the paper. -Next section gives notations and some background. We prove our main results in the following three sections (general maps, then volume-preserving maps, then conformal and quasi-conformal maps). The technique we use in the conformal and quasiconformal cases turns out to have been used by Gromov to generalize the Schwarz-Pick-Ahlfors lemma. In the final Section 6, we shall state and prove a result of this flavor that seems not to be in the literature (but certainly is in its topological closure).

Acknowledgments. -It is a pleasure to thank Charles Frances, Étienne Ghys and Pierre Pansu for interesting discussions related to the content of the present article.

Toolbox

2.1. Notations. -Let X κ be the model space of curvature κ and dimension n, i.e. a round sphere when κ > 0, the Euclidean space when κ = 0, and a hyperbolic space when κ < 0. We denote by B M (x, t) (respectively S M (x, t)) the geodesic closed ball (respectively sphere) of radius t and center x in M . When there is no ambiguity, we let B(t) = B M (x, t) and S(t) = S M (x, t). To simplify notation, we set B κ (t) (respectively S κ (t)) for any geodesic closed ball (respectively sphere) of radius t in X κ .

The volumes of manifolds, submanifolds and domains shall be denoted either by Vol(•) or | • |. We let ω n-1 = |S 0 (1)| be the (n -1)-dimensional volume of the unit sphere in

X 0 = R n .
When there is no ambiguity, σ i shall denote σ i (ϕ). When x is a point in a manifold and u a tangent vector at x, we let γ u (t) = exp x (tu) be the time t of the geodesic issued from x with velocity u.

We shall denote by T 1 M the unit tangent bundle of a Riemannian manifold M , by inj(x) the injectivity radius at x ∈ M and by inj(M ) the injectivity radius of M .

Geometry

sin κ (a) =        sin( √ κa) √ κ if κ > 0 a if κ = 0 sinh( √ -κa) √ -κ if κ < 0
We then set

cos κ (a) := sin ′ κ (a) =      cos( √ κa) if κ > 0 1 if κ = 0 cosh( √ -κa) if κ < 0 and tan κ (a) := sin κ (a) cos κ (a) =        tan( √ κa) √ κ if κ > 0 a if κ = 0 tanh( √ -κa) √ -κ if κ < 0
We shall also use occasionally

arctan κ (x) := tan -1 κ (x) =        arctan( √ κx) √ κ if κ > 0 x if κ = 0 argth( √ -κx) √ -κ if κ < 0
and we have the derivatives tan ′ κ = 1 + κ tan 2 κ and arctan ′ κ (x) = 1 1+κx 2 . A trigonometric formula that will prove useful is

sin κ (2 arctan κ x) = 2x 1 + κx 2 .
We shall need the following Taylor series:

sin κ (t) = t - κ 6 t 3 + κ 2 24 t 5 + O(t 5 ) cos κ (t) = 1 - κ 2 t 2 + κ 2 24 t 4 + O(t 6 ) tan κ (t) = t + κ 3 t 3 + 2κ 2 15 t 5 + O(t 5 ) arctan κ (x) = x - κ 3 x 3 + κ 2 5 x 5 + 0(x 7 ).
2.2.2. Volumes. -Let x be a point on X κ , t be a positive real and u be a unit tangent vector at x; then setting y = exp x (tu) we can express the volume measure dy on X k by the formula dy = sin n-1 κ (t) dt du where dt is Lebesgue measure on [0, +∞) and du is the volume measure on the unit tangent sphere T 1

x X κ naturally identified with the unit round sphere S n-1 .

In this volume formula, one can decompose the density into factors 1 (in the direction of the ray from the pole) and sin κ (t) (in the n -1 orthogonal directions). This shows that up to isometry there exists exactly one azimuthal, conformal map ϕ from the ball B κ (x, α) such that Dϕ x is a homothety of ratio σ (i.e. R ′ (0) = σ), whose distance function is driven by the following differential equation:

R ′ (t) = sin κ (R(t)) sin ρ (t) .
Moreover the (n-1)-dimensional volume of a geodesic sphere S κ (t) of radius

t is A κ (t) := |S κ (t)| = ω n-1 sin n-1 κ (t)
where ω n-1 is the volume of S n-1 ; when κ > 0 we only consider t below the conjugate radius π/ √ κ. We also name the volume of geodesic balls of X κ :

V κ (t) := |B κ (t)| = ω n-1 t 0 sin n-1 κ (s) ds.
Given ρ and κ, there is exactly one volume-preserving azimuthal map, defined by the distance function

R(t) = V -1 κ (V ρ (t)
). That R is as above is clearly necessary for an azimuthal map to be volumepreserving, but the local volume formula shows that it is also sufficient.

It is known that in X κ the least perimeter domains of given domains are balls, so that the isoperimetric profile of X κ is given by

I κ (V κ (t)) = A κ (t).
Note that the lesser is κ, the greater is I κ and the more stringent is the corresponding isoperimetric inequality.

Using the above Taylor series, we get:

A κ (t) = ω n-1 t n-1 1 - (n -1)κ 6 t 2 + O(t 4 ) V κ (t) = ω n-1 n t n 1 - n(n -1)κ 6(n + 2) t 2 + O(t 4 ) I κ (v) = n n-1 n ω 1 n n-1 v n-1 n - (n -1)κ 2(n + 2) • n n+1 n ω 1 n n-1 v n+1 n + O(v n+3 n ).

Candle functions and comparison.

-To study anisometry of maps under curvature bounds of the domain and range, we will need some tools of comparison geometry, relating the geometry of M and N to the geometry of X ρ and X κ . We will notably rely on Bishop and Günther's inequality, which in their common phrasing compare volume of balls. It will be useful to discuss their more general form, which is about comparing Jacobians of exponential maps.

Given a point x ∈ M , a vector u ∈ T 1 x M and a real number t, let y = exp x (tu) and define the candle function j x (tu) as a normalized Jacobian of the exponential map by dy = j x (tu) dt du where dy denotes the Riemannian volume and du is the spherical measure on T 1

x M . In the case of X κ , this function does not depend on x nor on u and is equal to sin n-1 κ (t).

Definition 2.1. -The manifold M is said to satisfy the candle condition Candle(κ, ℓ) if for all x, u and all t ℓ it holds j x (tu) sin n-1 κ (t). The manifold M is said to satisfy the logarithmic candle derivative condition LCD(κ, ℓ) if for all x, u and all t ℓ it holds (1)

j ′ (t) j(t) s ′ (t) s(t)
where j(t) := j x (tu) and s(t) := sin n-1 κ (t).

The name "candle condition" is motivated by the fact that j x describes the fade of the light of a candle (or of the gravitational fields generated by a punctual mass) in M .

By integration, Candle(κ, ℓ) implies that spheres and ball of radius at most ℓ have volume at least as large as the volume of the spheres and balls of equal radius in X κ .

The candle condition is an integrated version of the logarithmic candle derivative condition, which itself follows for ℓ = inj(N ) from the sectional curvature condition K κ: this is known as Günther's theorem, see [START_REF] Gallot | Lafontaine -Riemannian geometry[END_REF]. With Greg Kuperberg, we proved in [START_REF] Kloeckner | A refinement of Günther's candle inequality[END_REF] that it also follows from a weaker curvature bound, involving the "root-Ricci curvature". In particular, we proved that manifolds satisfying a relaxed bound on K and a suitably strengthened bound on Ric still satisfy a LCD condition, and therefore a Candle one.

The strong form of Bishop's Theorem is that the reversed inequality in (1) holds under the curvature lower bound Ric κ (for ℓ the conjugate time of X κ ). The corresponding comparison on the volumes of spheres and balls follow and are also referred to as Bishop's inequality.

We shall establish Theorem 1.1 using the comparison of spheres; the assumption K N κ can therefore be relaxed to Candle(κ, ℓ) where ℓ can be taken to be e.g. ∞ when N is a Hadamard manifold or chosen suitably otherwise, see the proof below.

2.4. Volume of ellipsoids and hyperplanes. -A couple of our arguments will rely on a simple and classical lemma, which we state and prove for the sake of completeness.

Let q be a scalar product in Euclidean space of dimension n, endowed with the standard inner product •, • . We shall denote by σ 1 (q) and σ 2 (q) the largest, respectively smallest numbers such that σ 1 (q) u, u q(u, u) σ 2 (q) u, u ∀u ∈ R n and say that q is at most Q-distorted if σ 2 /σ 1 Q. We shall also denote by |q| the determinant of q, that is the ratio of the volume of its unit ball to the volume of Euclidean unit ball (both volumes computed with respect to the Lebesgue measure associated to •, • ).

Lemma 2.2. -Let q 0 be the restriction of q to any hyperplane. Then we have

|q| |q 0 |σ 1 (q) and |q| 1 Q |q 0 | n n-1 .
There is equality in this second inequality if and only if q has eigenvalues λ and Qλ, with respective multiplicity 1 and n -1, and the hyperplane defining q 0 is the Qλ eigenspace of q.

Proof. -Let λ 1 . . . λ n be the eigenvalues of q and µ 1 . . . µ n-1 be the eigenvalues of q 0 . In particular, λ n = σ 1 (q). Then Rayleigh quotients show that µ i λ i for all i < n. It follows

|q| = λ 1 λ 2 . . . λ n |q 0 |λ n = |q 0 |σ 1 (q)
But by the distortion bound, we have

Qσ 1 λ 1 . . . λ n σ 1 so that (Qσ 1 ) n-1 |q 0 | and σ 1 |q 0 | 1 n-1
Q and the desired inequality follows. The equality case is straightforward.

General maps

Assume that Ric M ρ and that N satisfies Candle(κ, ℓ 0 ) for some ℓ 0 (on which we shall put some restriction latter on).

Let ϕ : B M (x, α) → N be a diffeomorphism on its image, where α < inj(x), the injectivity radius of M at x.

In what follows, we shall assume bounds involving 1/ √ κ: our convention is that this number is +∞ whenever κ 0.

Lemma 3.1. -If σ 2 (ϕ)α ℓ 0 π 2 √ κ , we have σ 2 (ϕ) sin κ (σ 1 (ϕ)α) sin ρ (α) .
The proof is a mere generalization of Milnor's argument in [START_REF] Milnor | A problem in cartography[END_REF].

Proof. -Denote by S(α) the geodesic sphere of center x and radius α. Bishop's inequality ensures that |S(α)| ω n-1 sin n-1 ρ (α). On the other hand, ϕ(S(α)) encloses the ball of N of radius σ 1 α centered at y = ϕ(x). Given a unit vector u ∈ T y N , let ℓ(u) be the first time at which γ u hits ϕ(S(α)), and β(u) be the angle between γu (ℓ(u)) and the outward normal to ϕ(S(α)). We have ℓ(u) σ 1 α and obviously cos(β(u)) 1. Moreover when κ > 0, we have

ℓ(u) σ 2 α π 2 √ κ so that the comparison candle function sin n-1 κ is increasing on [0, ℓ(u)].
Then, letting j be the candle function of N at x and du be the usual measure on the unit Riemannian sphere, we get

|ϕ(S(α))| T 1 y N j u (ℓ(u)) cos β(u) du T 1 y N sin n-1 κ (ℓ(u)) du ω n-1 sin n-1 κ (σ 1 α).
There is at least one point on S(α) at which the Jacobian of the restriction of ϕ to S(α) is at least

|ϕ(S(α))| |S(α)| sin n-1 κ (σ 1 (ϕ)α) sin n-1 ρ (α)
and the lemma follows.

Let us now define the α-bound A 1 .

Definition. -Let A 1 = A 1 (M, N ) be the greatest number such that for all α A 1 we have α inj(M ), if κ > 0:

(2)

α sin ρ (α) sin κ (α) min inj(N ), π 2 √ κ and if κ 0 (3) α 2 sin -1 κ • sin ρ (α) inj(N ).
Remark. -1. A 1 depends on M and N only through their curvature/candle bounds ρ and κ and their injectivity radii, 2. if we do not insist on a uniform bound over possible centers, we can replace inj(M ) by inj(x), 3. if N is a Hadamard manifold, then A 1 = inj(M ) (or inj(x)), 4. if inj(M ) and inj(N ) are large enough, when κ > 0 we have A 1

π 2 √ ρ
and A 1 → π √ ρ when κ → 0, 5. in some cases (e.g. when one can apply Klingenberg's Theorems, see [START_REF] Cheeger | Ebin -Comparison theorems in Riemannian geometry[END_REF] Theorems 5.9 and 5.10), the curvature bound on N is sufficient to get an estimate on inj(N ), and therefore to get a bound A ′ 1 that does not depend on the injectivity radius of the range.

Theorem 3.2. -Assume Ric M ρ, N satisfies Candle(κ, inj(N )) for some κ < ρ (e.g. K N κ) and α A 1 (M, N ) defined above. Let ϕ : B M (xα) → N be any smooth map.

If κ 0 then aniso(ϕ) log sin κ (α) sin ρ (α) and there is equality if and only if ϕ is conjugated via isometries to the equidistant azimuthal map from B ρ (α) to B κ (α) (in particular, B M (x, α) and its image must have constant curvatures ρ and κ).

If κ < 0 then, letting σ 0 = σ 0 (κ, ρ, α) be the number in (0, 1) such that sin κ (σ 0 α) = sin ρ (α), we have aniso(ϕ) log 1 σ 0 and there is equality if and only if ϕ is the σ 0 -contracting azimuthal map from B ρ (α) to B κ (σ 0 α) (in particular, B M (x, α) and its image must have constant curvatures ρ and κ).

Notice that σ 0 is the dilation coefficient that makes the volumes of the spheres S κ (σ 0 α) and S ρ (α) coincide; it makes the σ 0 -contracting azimuthal map a non-dilating map, i.e. σ 2 ( φ) = 1 when κ < 0.

Proof. -We can assume σ 2 α is small enough to apply Lemma 3.1, otherwise the way we designed A 1 ensures that σ 2 is so large that aniso(ϕ) is a least the claimed lower bound.

Let us start with the κ 0 case. From Lemma 3.1 we have (4) aniso(ϕ) | log σ 1 | + log sin κ (σ 1 α) -log sin ρ (α).

The derivative of the right-hand side with respect to σ 1 is

- 1 σ 1 + α tan κ (σ 1 α) < 0 when σ 1 < 1 and 1 σ 1 + α tan κ (σ 1 α) > 0 when σ 1 > 1.
This shows that the right-hand side of (4) achieves its minimum when σ 1 = 1, so that aniso(ϕ) log sin κ (α) -log sin ρ (α).

In case of equality, one must have σ 1 = 1 and σ 2 = sin κ (α)/ sin ρ (α), therefore there is equality in Lemma 3.1. This forces B M (x, α) and its image to have constant curvatures ρ and κ and S(α) must be mapped to the geodesic sphere of radius α and center ϕ(x) in N . Since σ 1 = 1, ϕ must then map S(a) to S κ (a) for all a. Each ray must be mapped to a curve of length at its α that connects ϕ(x) to the boundary of the image of ϕ, therefore unit rays are mapped to unit rays. The whole map ϕ then depends only on its derivative, which must preserve the norms. It follows that ϕ is azimuthal equidistant, up to isometries.

In the κ < 0 case, (4) also holds but is not optimal anymore. Indeed, the derivative of its right-hand side is positive both when σ 1 > 1 and when σ 1 < 1 since tan κ (x) x. But when σ 1 < σ 0 , the lower bound on σ 2 given by Lemma 3.1 is less than 1. It follows aniso(ϕ) log 1 σ 0 Which is achieved by φ. The case of equality is treated as above.

Corollary 3.3. -In the above setting,

aniso(ϕ) ρ -κ 6 α 2 + O(α 4 )
where the implied constant in the remainder term only depends on the curvature bounds.

Area-preserving maps

Let us now prove Theorem 1.2 in the following form.

Theorem 4.1.

-Assume Ric M ρ, N satisfies the best isoperimetric inequality holding on X κ , and α inj(x). Then any volume-preserving map

ϕ : B(x, α) ⊂ M → N satisfies aniso(ϕ) n n -1 log I κ • V ρ (α) A ρ (α)
and equality is achieved by the unique volume-preserving azimuthal map φ :

B ρ (α) → X κ .
Assume further that the only domains in N satisfying the equality case in the isoperimetric inequality are balls isometric to geodesic balls in X κ . Then whenever aniso(ϕ) = aniso( φ), the domain of ϕ has constant curvature ρ and its range is isometric to a constant curvature ball B κ (R(α)). However, there are uncountably many different maps achieving equality.

Proof. -The key point is the following Lemma, which is a direct adaptation of Theorem 3.5 in [START_REF] Morgan | Some sharp isoperimetric theorems for Riemannian manifolds[END_REF].

Lemma 4.2. -Under the assumption Ric M ρ and for all κ ρ, we have

I κ (|B(α)|) |S(α)| I κ • V ρ (α) A ρ (α)
If there is equality, then B(α) is isometric to B ρ (α).

Proof of Lemma. -Setting δ 0 := |S(α)| Aρ(α) , the strong form of Bishop's inequality yields for all t α

δ 0 |S(t)| A ρ (t) 1.
By integration, it comes

δ 0 δ 1 := |B(α)| V ρ (α) 1.
Since I κ is concave, we have

I κ (δ 1 V ρ (α)) δ 1 I κ (V ρ (α)); therefore I κ (|B(α)|) δ 0 I κ (V ρ (α)) = |S(α)| I κ (V ρ (α)) A ρ (α) .
In case of equality, we must have δ 0 = δ 1 , which implies δ 0 = δ 1 = 1. The equality case in Bishop's inequality then implies that B(α) is isometric to B ρ (α). Now, since ϕ is volume-preserving and N satisfies an isoperimetric inequality, we have

|ϕ(S(α))| I κ (|B(α)|) |S(α)| I κ (V ρ (α)) A ρ (α) .
Then, there must be a point x on S(α) such that the Jacobian of ϕ |S(α) is at least

Iκ(Vρ(α))
Aρ(α) so that

σ 2 I κ (V ρ (α)) A ρ (α) 1 n-1
but also, since jac ϕ x = 1, Lemma 2.2 shows that a direction transverse to the boundary must be contracted by ϕ and

σ 1 I κ (V ρ (α)) A ρ (α) -1
.

These two bounds combined imply the desired inequality on aniso(ϕ).

It is straightforward to see that the unique volume-preserving azimuthal map φ : B ρ (α) → X κ realizes equality. Moreover, if there is equality then there must be equality in the Lemma, so that B(α) is isometric to B ρ (α), and there must be equality in the isoperimetric inequality on N .

However, φ is far from being the only optimal map: both σ 1 and σ 2 are realized on the boundary, and for all t < α, σ 1 ( φ|Bρ(t) ) > σ 1 ( φ) and σ 2 ( φ|Bρ(t) ) < σ 2 ( φ).

If we compose φ with any diffeomorphism of B ρ (α) close to identity and supported on some B ρ (t), we get another optimal map.

Conformal and quasiconformal maps

The following result is the heart of our results for quasiconformal maps; it's formulation has been chosen to avoid repetition of arguments while keeping as much flexibility as we shall need, and it is therefore rather technical.

Theorem 5.1 (Main quasiconformal inequality). -Assume ϕ is a Qquasiconformal maps from B M (x, α) to N , where Ric M ρ and N satisfies the isoperimetric inequality of X κ .

Let G κ be the function defined by

G κ (x) = sin κ (2 arctan κ (x)) = 2x 1 + κx 2
If κ > 0, assume further that the volume of the image of ϕ is not greater than the volume 1 2 |X κ | of an hemisphere of curvature κ. Then, for all β < α we have

σ 2 (ϕ) G κ tan κ ( r(β) 2 ) • tanρ( α 2 ) tanρ( β 2 ) 1 Q sin ρ (α)
where r(β) is the radius of a ball in X κ that has the same volume as the image of B M (x, β).

The proof of this inequality follows a simple idea: at each time t, the isoperimetric inequality forces the image of the sphere of radius t to have large volume, and the quasiconformality then translate this into a large increase in the volume of the image of the ball. These two effects therefore amplify one another. At t = α, we get a lower bound on V (α), and using the isoperimetric inequality again we bound from below the perimeter of the image of the α-ball. Comparing with the perimeter of the ball, we get a lower bound on σ 2 .

Proof of the main quasiconformal inequality. -For convenience, for all t ∈ (0, α) set V (t) = |ϕ(B(t))|. In particular, r(t) = V -1 κ (V (t)). Using Hölder's inequality we get

V ′ (t) = S(t) | jac ϕ|(y) dy S(t) | jac ϕ| n-1 n (y) dy n n-1 |S(t)| 1 n-1
where jac ϕ(y) is the Jacobian of ϕ at y. Let ϕ 0 be the restriction of ϕ along S(t): using Lemma 2.2, Bishop's inequality and the isoperimetric inequality on N it comes

V ′ (t) 1 Q S(t) | jac ϕ 0 |(y) dy n n-1 |S(t)| 1 n-1 |ϕ(S(t))| n n-1 Q|S ρ (t)| 1 n-1 V ′ (t) I κ (V (t)) n n-1 Qω 1 n-1 n-1 sin ρ (t) . (5) Let F = F κ,Q be defined by F • V κ (t) = Q log tan κ (t/2)
and let us compute F ′ :

d dx (F • V κ (x)) = d dx (Q log tan κ (x/2)) V ′ κ (x)F ′ (V κ (x)) = Q sin κ (x) F ′ (V κ (x)) = Q A κ (x) sin κ (x) = Qω 1 n-1 n-1 A κ (x) 1+ 1 n-1 = Qω 1 n-1 n-1 (I κ • V κ (x)) n n-1 F ′ = Qω 1 n-1 n-1 I n n-1 κ (6)
From ( 5) and (6) it comes

F ′ (V (t))V ′ (t) 1 sin ρ (t)
As above, log(tan ρ (t/2)) defines an antiderivative of 1/ sin ρ (t) and integrating we conclude (7)

F (V (α)) -F (V (β)) log tan ρ (α/2) tan ρ (β/2)
since F ′ is a positive function, F is increasing and invertible, so that the above inequality gives a lower bound on V (α); using |ϕ(S(α))| I κ (V (α))

and proceeding as in the proof of Theorem 3.2, we get

σ 2 (ϕ) I κ (V (α)) |S(α)| 1 n-1 (I κ • F -1 ) 1 n-1 F (V (β) + log( tanρ(α/2) tanρ(β/2) ) ω 1 n-1 n-1 sin ρ (α) (beware that exponent 1
n-1 is a multiplicative power while exponent -1 stands for inverse function). Now

(I κ • F -1 ) 1 n-1 = (I κ • V κ • (F • V κ ) -1 ) 1 n-1 = (A κ • (F • V κ ) -1 ) 1 n-1 = ω 1 n-1 n-1 sin κ •(F • V κ ) -1
and the desired inequality follows from the identity

sin κ (2 arctan κ x) = 2x 1 + κx 2 .
Remark. -In the above proof, two small difficulties are hidden.

1. When κ > 0, I κ is decreasing beyond the volume of an hemisphere; this is why we assumed an upper bound on V (α). 2. When κ < 0, F has bounded image so that F -1 is not defined on the whole positive axis. Our proof shows that any quasiconformal map must map small balls to domains of relatively small volume (bounded in terms of α, κ, ρ and the radius of the considered ball), for otherwise the differential inequality on V (t) would blow up in time less than α and the map would not have compact image. This is the base to a generalization of the Schwarz-Pick-Ahlfors lemma by Gromov, see the appendix.

Definition. -Let A 3 = A 3 (M, κ, ρ, n) be defined as the greatest real number such that for all α A 3 it holds

-α inj(M ), -if κ > 0, |X κ | 2V ρ (α) 1 n 1 + (ρ -κ) tan 2 ρ (α/2) 1 + κ tan 2 ρ (α/2)
.

Let us now prove Theorem 1.3 which we restate as follows.

Theorem 5.2. -Assume Ric M ρ, N satisfies the best isoperimetric inequality holding on X κ and α A 3 (M, κ, ρ, n). For the equality case below, assume further that any domain Ω ⊂ N such that |∂Ω| = I κ (|Ω|) is isometric to a geodesic ball in X κ . Let ϕ : B(x, α) ⊂ M → N be a conformal map.

If κ > 0, then

aniso(ϕ) log 1 + (ρ -κ) tan 2 ρ (α/2) 1 + κ tan 2 ρ (α/2)
with equality when ϕ is conjugated by isometries the conformal azimuthal map B ρ (α) → X κ with R ′ (0) = 1.

If κ = 0, then aniso(ϕ) log 1 + ρ tan 2 ρ (α/2) with equality when ϕ is conjugated by isometries to a conformal azimuthal map B ρ (α) → X 0 = R n with R ′ (0) 1 and σ 2 1 (e.g. R ′ (0) = 1).

If κ < 0, then

aniso(ϕ) log -2κ sin 2 ρ ( α 2 ) 1 -κ sin ρ α -1
with equality when ϕ is conjugated by isometries the conformal azimuthal map B ρ (α) → X κ that induces an isometry on the boundaries (or, equivalently, that preserves volumes along the boundary).

Remark. -In the case κ = 0 it is easy to compare the bound for general maps and conformal ones. When α → 0, this will be done more generally below; when α → π, both lower bounds go to infinity, but in the conformal case it does so twice as fast (after taking logs!) in the sense that aniso( φc ) aniso( φ)

→ 2

where φ and φc denote the optimal azimuthal maps for radius α in the general and conformal cases, respectively. Conformality thus appears to have a significant effect on anisometry.

Proof. -The bound A 3 has been designed so that either B(α) is mapped to a domain so large that at some point y Dϕ y itself must have anisometry at least equal to the claimed bound, or the volume of ϕ(B(α)) is at most 1 2 |X κ | and we can use the main quasiconformal inequality with Q = 1 and β → 0.

Since r(β) σ 1 β + o(β), when β → 0 we have

tan κ ( r(β) 2 ) tan ρ (β/2) σ 1 + o β (1)
and we obtain

σ 2 (ϕ) σ2 := G κ (σ 1 tan ρ (α/2)) sin ρ (α) .
We would like to optimize in σ 1 the corresponding bound

f (σ 1 ) := | log σ 1 | + | log σ2 (σ 1 )|
on aniso(ϕ).

For this, we observe that for all positive σ, the number f (σ) is the anisometry of a conformal map with co-Lipschitz coefficient equal to σ. For this, let Φ σ,κ : B ρ (α) → X κ be the unique conformal azimuthal map such that σ 1 (Φ σ,κ ) = σ (i.e., its distance function satisfies R ′ σ,κ (0) = σ). Then following the proof of the main quasiconformal inequality with Q = 1, we see that all inequalities are equalities so that indeed f (σ) = aniso(Φ σ,κ ).

Moreover, if aniso(ϕ) = f (σ 1 ) (recall that σ 1 stands for σ 1 (ϕ)) then we must have equality in all inequalities in the proof of the main quasiconformal inequality, and this implies that ϕ and Φ σ 1 ,κ are conjugated by isometries.

Observe that aniso(Φ σ,κ ) is decreasing with κ, and increasing with σ whenever (8) σ 1 σ 2 (Φ σ,κ ).

It is clear that the minimum of f (σ) occurs in this range. Observe further that aniso(Φ σ,κ ) = aniso(Φ σ/λ,λ 2 κ ) whenever σ and σ/λ both are in the range (8), since Φ σ/λ,λ 2 κ is the composition of Φ σ,κ with a homothety of ratio λ. When κ > 0, if σ < 1 is in the above range then we get

aniso(Φ σ,κ ) = aniso(Φ 1,σ 2 κ ) > aniso(Φ 1,κ ) so that f (σ 1 ) f (1) with equality if and only if σ = 1. When κ < 0, if 1 < σ2 (σ) = σ 2 (Φ σ,κ ), then aniso(Φ σ,κ ) = aniso(Φ σ/σ 2 ,σ 2 2 κ ) > aniso(Φ σ/σ 2 ,κ
) so that f (σ) reaches its unique minimum for the value of σ such that σ 2 (Φ σ,κ ) = 1.

When κ = 0, f is constant on the range (8).

where β > 0 is such that sin

κ (β) Q sin ρ (β) = 1.
There is a positive number A 4 = A 4 (M, N, Q) such that if α A 4 then aniso(ϕ) aniso( φ)

and there is equality if and only if ϕ and φ are conjugated by isometries (except in the case κ = 0 where the conjugating map on the range can be a homothety). If κ < 0, let φ : B ρ (α) → X κ be the azimuthal map whose distance function satisfies

R(t) = σt when t β R ′ (t) = sin κ (R(t)) Q sin ρ (t) when t β where β > 0 is such that sin κ (σβ) Q sin ρ (β) = σ
and σ is such that σ 2 ( φ) = 1 (in particular, φ induces an isometry on the boundary). Then whenever α inj(x) we have aniso(ϕ) aniso( φ)

and there is equality if and only if ϕ and φ are conjugated by isometries.

Proof. -The proof follows exactly the same lines as the proof of Theorem 5.2, using the quasi-conformal inequality with the chosen β and Q. Fixing and α, for all given σ and κ we construct a comparison map φσ,κ : B ρ (α) → X κ as in the conclusion of the Theorem. We remark that, restricted to B ρ (β), φ is both the least anisometrical map and Q-conformal. Then for larger radii, A 4 is designed so that either the anisometry bound is true or the main quasiconformal inequality shows that φσ 1 (ϕ),κ has lesser anisometry than ϕ.

To get the desired conclusion, we only have left to optimize the anisometry of these comparison maps in σ. This does not differ from the conformal case.

We do not give explicit values for the lower anisometry bounds, but they can be obtained explicitly from the above computations (though probably not in closed form).

Remark 5.5. -It can be checked that in Theorem 5.4 the optimal azimuthal map is C 1 but not C 2 when Q > 1.

Appendix: the generalized Schwarz-Pick-Ahlfors lemma

The method used to prove the main quasiconformal inequality was already used by Gromov [Gro81, Gro85] and Pansu [START_REF] Pansu | Quasiconformal mappings and manifolds of negative curvature[END_REF] in relation with generalizations of Ahlfors lemma.

The classical Schwarz lemma says that a holomorphic map f : ∆ → ∆ from the unit disc to itself, such that f (0) = 0 must satisfy |f ′ (0)| 1 and, in case of equality, f must be a rotation. Pick reinterpreted this result by endowing the disc with its hyperbolic metric: the lemma then amounts to say that any conformal map from H 2 to H 2 must be non-dilating in the hyperbolic metric, and if at any point its Jacobian has modulus 1 then the map must be a hyperbolic isometry. Then, Ahlfors extended this result to conformal maps from a surface with curvature bounded below by -1 to a surface with curvature bounded above by -1. This had a lasting impact on several fields of mathematics. Among possible generalization to higher dimensions, one that fits particularly well with the content of the present article is the following. Theorem 6.1. -Let M and N be complete manifolds of the same dimension (at least 2) with Ric M -1 and K N -1, and let ϕ : M → N be a smooth conformal map. If the Cartan-Hadamard conjecture holds, then | jac ϕ(x)| 1 for all x ∈ M , and if there is equality at any one point, then ϕ lifts to an isometry of the universal coverings φ : M → Ñ (in particular, M and N have constant curvature -1).

The current knowledge gives us the conclusion unconditionally when N is the real hyperbolic space, and when the dimension is 2 or 3.

The above result can hardly be considered new, but we could not find a written proof; we therefore provide one.

Proof. -Let φ : M → Ñ be the lift to Ñ of the composition of the universal covering map π : M → M with ϕ. Then φ is a smooth conformal map with the same local behavior as ϕ.

We apply to φ inequation (7) from the proof of the main quasiconformal inequality. As in the beginning of the proof of Theorem 5.2, with Q = 1 and β → 0 we get for all x ∈ M and all α > 0:

tanh( r(α) 2 ) σ 0 tanh( α 2 )
where r(α) is the radius of a ball in hyperbolic space whose volume equals | φ(B M (x, α))|, and σ 0 is the conformal dilation factor at x (i.e. σ n 0 = | jac φ(x)|). If we had σ 0 > 1, then for large enough α the above inequality would yield r(α) ∞, a contradiction. Therefore, σ 0 1 independently of x. Together with the conformality of φ, this implies that φ is distance-nonincreasing.

If σ 0 = 1 (for one given x), then we have from the above inequality r(α) α for all α, so that φ maps balls of volume at most V -1 (r) to balls of volume at least V -1 (r), while not increasing distances. This implies that we have equalities in the Bishop and Günther inequalities, so that M and Ñ both have constant curvature -1 and φ is an isometry.

Remark. -1. The above result may seem weak in the sense that it asks for a conformal map, which may not exist for given M and N . However the hypothesis cannot be weakened to quasiconformal as there are local Q-quasiconformal diffeomorphisms of arbitrarily high supremum of the Jacobian. Using the above method one can only get bounds on averaged Jacobians, i.e. volume of balls. 2. Theorem 6.1 can be interpreted as follows: given a manifold M , if one can find in the same conformal class two complete metrics g and σg such that Ric g -1 and K σg -1, then σ is uniformly bounded above by 1, and if there is a point at which σ(x) = 1 then σ ≡ 1.

  of model spaces. -The model spaces X κ are well understood, let us recall a few facts about them. 2.2.1. Trigonometric functions. -It will be convenient to use the functions sin κ defined by

Note that we could also have proceeded via calculus: setting x = σ 1 tan ρ (α/2) we then have

Therefore, if κ > 0 then f has its only minimum when σ 1 = 1, and if κ < 0 then f has its only maximum when σ2 (σ 1 ) = 1. When κ = 0, any value of σ 1 between this two cases yields the same result.

We only have left to compute min f . When κ 0, we get

.

Then, using

we easily get the claimed inequality.

When κ < 0, the minimum of f is attained when σ2 = 1 and, therefore,

Since at this point we have f (σ 1 ) = -log(σ 1 ), we only have to inverse G κ to get the desired inequality.

where the remainder depends on the curvature bounds and N .

As mentioned above, we can even release the assumption on M and N to be Scal M ρ and Scal N κ: the infinitesimal Bishop inequality holds true under a scalar curvature bound, and so does the isoperimetric inequality as proved by Druet [START_REF] Druet | Sharp local isoperimetric inequalities involving the scalar curvature[END_REF].

Theorem 5.4. -Assume Ric M ρ and N satisfies the best isoperimetric inequality holding on X κ . For the equality case below, assume further that any domain

If κ 0, let φ : B ρ (α) → X κ be the azimuthal map whose distance function satisfies R(t) = t when t β R ′ (t) = sin κ (R(t)) Q sin ρ (t) when t β


Corentin Léna

Eigenvalues variations for Aharonov-Bohm operators

We study how the eigenvalues of a magnetic Schrödinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential. We consider a magnetic potential defined everywhere in R 2 except at a finite number of singularities, so that the associated magnetic field is zero. On a fixed planar domain, we define the corresponding magnetic Hamiltonian with Dirichlet boundary conditions, and study its eigenvalues as functions of the singularities. We prove that these functions are continuous, and in some cases even analytic. We sketch the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain.

Introduction

Aharonov-Bohm operators have been introduced in [START_REF] Aharonov | Significance of electromagnetic potentials in the quantum theory[END_REF] as models of Schrödinger operators with a localized magnetic field. In addition to their physical relevance, it has been shown in [START_REF] Helffer | On a magnetic characterization of spectral minimal partitions[END_REF] that these operators appear in the theory of spectral minimal partitions (see [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] for a definition of the latter object). In [START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF][START_REF] Bonnaillie-Noël | Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions[END_REF][START_REF] Bonnaillie-Noël | Spectral minimal partitions of a sector[END_REF], the eigenvalues and eigenfunctions of an Aharonov-Bohm operator with Dirichlet boundary condition have thus been studied numerically to find minimal partitions. One of the aim of the present work is to support and generalize some observations made in these papers.

We define Aharonov-Bohm operators as follows. Let ω be an open and connected set in R 2 . As usual, we denote by C ∞ c (ω) the set of smooth functions compactly supported in ω. Generally speaking, let us consider a magnetic potential, that is to say a vector field A ∈ C ∞ (ω, R 2 ). We define the sesquilinear form s A by

s A (u, v) = ω (-i∇ -A)u • (-i∇ -A)v dx the quadratic form q A by q A (u) = s A (u, u)
and the norm • A by

u 2 A = u 2 L 2 (ω) + q A (u)
for u and v in C ∞ c (ω) .

The form domain Q A associated with A is defined as the completion of C ∞ c (ω) under • A . According to Friedrichs' Theorem, there exists a unique self-adjoint extension -∆ A of the densely defined, symmetric, and positive differential operator (-i∇ -A) 2 acting on C ∞ c (ω) whose domain D A is contained in Q A (see e.g. [START_REF] Reed | Methods of Modern Mathematical physics. II. Fourier Analysis, Self-Adjointness[END_REF]). We call it the magnetic Hamiltonian on ω associated with A (note that we are imposing a Dirichlet boundary condition).

In the following, we denote by Ω an open, bounded, and connected set in R 2 with a Lipschitz boundary ∂Ω . For v = (v 1 , v 2 ) ∈ R 2 , we note v ⊥ = (-v 2 , v 1 ) . For X ∈ R 2 , α ∈ R , and x = (x 1 , x 2 ) ∈ R 2 \ {X}, we define

A X α (x) = α |x -X| 2 (x -X) ⊥ , (1.1)
and note A X α (x) = (A X α,1 (x), A X α,2 (x)) . In the rest of the paper, N is an integer, X = (X 1 , . . . , X N ) an N -tuple of points in R 2 , with X i = X j for i = j (at least for now), and α = (α 1 , . . . α N ) ∈ R N . Let us denote by Ω X the open set Ω \ {X 1 , . . . , X N }. We define the Aharonov-Bohm potential associated with X and α as the vector field

A X α = N i=1 A Xi αi . (1.2)
Let us point out that the X i 's can be in R 2 \ Ω, and in particular in ∂Ω. The Aharonov-Bohm operator associated with X and α is the magnetic Hamiltonian -∆ A X α on Ω X . We denote it by -∆ X α , and the associated form domain and quadratic form by Q X α and q X α respectively. Along the paper, we make frequent references to the Dirichlet realization of the Laplacian on Ω , and to the sequence of its eigenvalues. We denote them by -∆ D Ω and (λ D k (Ω)) k≥1 respectively. Let us conclude by a few remarks of a more physical nature. To an Aharonov-Bohm potential is associated a magnetic field

B = Curl A X α = ∂ x1 A X α,2 -∂ x2 A X α,1 = N i=1 α i δ Xi , (1.3)
which is a measure. If B i is a small disk centered at X i , such that X j / ∈ B i for j = i ,

α i = 1 2π Bi B .
The coefficient α i can therefore be called the normalized (magnetic) flux at X i . Let finally note that for any closed loop

γ in R 2 \ ∪ N i=1 {X i } , γ A X α (s) ds = 2π N i=1 ind γ (X i )α i , (1.4)
where ind γ (X i ) is the winding number of γ around X i . We use in this paper the following characterization of the form domain Q X α , which follows from Hardy-type inequalities proved in [START_REF] Laptev | Hardy inequalities for magnetic Dirichlet forms[END_REF][START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF][START_REF] Melgaard | Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians[END_REF].

Proposition 1.1. Let H X α be defined as

H X α = {u ∈ L 2 (Ω); (-i∇ -A X α )u ∈ L 2 (Ω)}
with the natural scalar product. Then H X α is a Hilbert space compactly embedded in L 2 (Ω). Furthermore, there exists a continuous mapping γ 0 :

H X α → L 2 loc (∂Ω) such that for any u ∈ C ∞ c (Ω \ {X 1 , . . . , X N }), γ 0 u = u |∂Ω . A function u in L 2 (Ω) belongs to Q X α if, and only if, u ∈ H X α and γ 0 u = 0 .
As a consequence of the compact embedding, the spectrum of -∆ X α consists in a sequence of real eigenvalues converging to +∞. Let (λ k (X, α)) k≥1 be this sequence, the eigenvalues being arranged in non-decreasing order and counted with multiplicity. We are interested in the functions X → λ k (X, α). Let us note that according to our definition of A X α , λ k (X, α) is defined only for X ∈ R 2N such that X i = X j for i = j. To state the following result, it is convenient to extend the function X → λ k (X, α) to R 2N . We define λ k (X, α) as the k-th eigenvalue of the operator -∆ Y β , where the M -tuple Y = (Y 1 , . . . , Y M ) contains once, and only once, each point appearing in X = (X 1 , . . . , X N ) and where β = (β 1 , . . . , β M ) with

β j = i, Xi=Yj α i , ∀1 ≤ j ≤ M .
For instance, with this definition,

λ k ((X 1 , X 2 , X 3 ), (α 1 , α 2 , α 3 )) , with X 1 = X 2 and X 1 = X 3 , is λ k ((X 1 , X 3 ), (α 1 + α 2 , α 3 )) . Theorem 1.2. For any given k ≥ 1 and α ∈ R N , the function X → λ k (X, α) is continuous in R 2N .
Let us note that this result has more implications than it may appear at first glance. In particular, it implies continuity of the eigenvalues in the case of one point tending to the boundary of Ω, or in the case of coalescing points.

Additional regularity is easily obtained when the poles are distinct and far from the boundary. We prove the following result.

Theorem 1.3. Let e 1 = (1, 0) and e 2 = (0, 1) be the vectors of the canonical basis of R 2 . Assume that X ∈ R 2N and k ≥ 1 are such that X i / ∈ ∂Ω for 1 ≤ i ≤ N , X i = X j for i = j , and λ k (X, α) is a simple eigenvalue. Then the function

(t 1 , t 2 , . . . , t 2N -1 , t 2N ) → λ k ((X 1 + t 1 e 1 + t 2 e 2 , . . . , X N + t 2N -1 e 1 + t 2N e 2 ), α) is analytic in a neighborhood of 0 in R 2N .
In general, an eigenfunction u of the operator -∆ X α is complex-valued and the set u -1 ({0}) consists in isolated points of Ω. However, in the case where α i ∈ Z + 1/2 for all 1 ≤ i ≤ N , we can define a class of eigenfunctions for which the notion of nodal set is meaningful. To this end, as in [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains[END_REF], we define a suitable unitary antilinear operator K X (see Section 4) and call K X -real a function u such that K X u = u . If u is a K X -real eigenfunction, the set u -1 ({0}) is locally a regular curve, except at the poles and maybe at a finite number of points where curves cross. By analogy with the terminology used for real eigenfunctions of the operator -∆ D Ω , we call u -1 ({0}) the nodal set and its component curves the nodal lines. Following a suggestion of Susanna Terracini, we have studied the derivatives of an eigenvalue with respect to a pole when at least three nodal lines of an associated K X -real eigenfunction meet at this pole. It has already been shown in [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF] that if the function Y → λ k (Y, 1/2) has a critical point at X and if λ k (X, 1/2) is simple, an associated K X -real eigenfunction u has at least three nodal lines meeting at X. By modifying our proof of Theorem 1.3 and using results from [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF] on the structure of the nodal set, we prove the converse.

Theorem 1.4. Let α ∈ R N and X ∈ R 2N . Let us fix 1 ≤ i ≤ N with X i ∈ Ω. For v ∈ R 2 , t ∈ R , and k ≥ 1, we define X(t) = (X 1 , . . . , X i + tv, . . . , X N )
and λ k (t) = λ k (X(t), α) . Let us assume that λ k (0) is simple and has a K X -real eigenfunction u with at least three nodal lines that meet at X i . Then λ k (0) = 0 .

During the writing of this work, S. Terracini showed us a preliminary version of the paper [START_REF] Bonnaillie-Noël | On the eigenvalues of Aharonov-Bohm operators with varying poles[END_REF]. It contains a similar continuity result, restricted to the case of one pole and assuming Ω to be simply connected with a C ∞ -boundary. It also contains a stronger version of Theorem 1.4. The main contribution of our paper is in the generality of Theorem 1.2, which allows us to treat coalescing poles, and in the use of the Kato-Rellich regular perturbation theory to give a simpler proof of Theorem 1.4.

The paper is organized as follows. We first recall the definition of a gauge transformation of the magnetic potential, and the fact that it preserves the spectrum of the associated magnetic Hamiltonian. We then give a criterion for -∆ X α to be unitarily equivalent to -∆ D Ω . We recall a Hardy-type inequality from [START_REF] Laptev | Hardy inequalities for magnetic Dirichlet forms[END_REF][START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF], some of its consequences, and a non-concentration inequality for functions in Q X α . We give some indications on the proof and the usefulness of Proposition 1.1. In a second part, we show the continuity of the eigenvalues with respect to the poles (Theorem 1.2). We then apply this theorem to some examples: we study the limits of the eigenvalues when the operator has a pole moving toward the boundary of the domain (Corollary 3.4) or two poles moving toward one another (Corollary 3.5). We show the analyticity of the eigenvalues with respect to the poles under some restrictive conditions (Theorem 1.3), and consider their critical points (Theorem 1.4). In the last section, we discuss the connection of these results with spectral minimal partitions.

I would like to thank my advisors, Virginie Bonnaillie-Noël and Bernard Helffer, for directing me toward this subject, and for many suggestions and corrections. Virginie Bonnaillie-Noël also sent me the numerical computations and pictures used in Section 5. I benefited greatly from exchanges with Susanna Terracini and Luc Hillairet. In particular, Lemma 2.7 and the method used to prove Theorem 1.2 are taken from unpublished notes kindly communicated by Luc Hillairet.

Gauge invariance and form domain

This section consists in a review of some useful results. We first recall how Aharonov-Bohm operators are modified under a gauge transformation. We only need a few classical properties of these transformations (see for instance [START_REF] Leinfelder | Gauge invariance of Schrödinger operators and related spectral properties[END_REF]), that we state without proof. We recall a Hardy-type inequality taken from [START_REF] Laptev | Hardy inequalities for magnetic Dirichlet forms[END_REF][START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF]. We prove that functions in the from domain satisfy a non-concentration property. Finally, we sketch the proof Proposition 1.1 and describe how it is used in proving the continuity of the eigenvalues.

Gauge transformation

Definition 2.1. A gauge transformation acts on pairs vector field-function as (A, u) → (A * , u * ) , with

A * = A -i ∇ψ ψ , u * = ψu ,
where ψ ∈ C ∞ (ω, C) satisfies |ψ| = 1. The function ψ is called a gauge function on ω. Two magnetic potentials are said to be gauge equivalent when the second one can be obtained from the first by a gauge transformation.

We see immediately from the definition that a gauge transformation does not change the magnetic field B = curl A , nor the probability distribution |u| 2 .

As a consequence of [13, Theorem 1.2], we also have the following result.

Proposition 2.2. If A and A * are two gauge equivalent magnetic potentials in C ∞ (ω, R 2 ) , the operators -∆ A * and -∆ A are unitarily equivalent.

Due to the possible multi connectedness of the open set ω, the equation curl A = 0 is a necessary but not sufficient condition for the magnetic potential A to be gauge equivalent to 0 . This is the basis of the so-called Aharonov-Bohm effect, studied in [START_REF] Aharonov | Significance of electromagnetic potentials in the quantum theory[END_REF]. We now give a criterion for the gauge equivalence, taken from [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains[END_REF].

Lemma 2.3. Let A ∈ C ∞ (ω, R 2 ). It is gauge equivalent to 0 if, and only if, 1 2π γ A X α (s) ds (2.1)
is an integer for any loop γ contained in ω .

A Hardy-type inequality

Let us now recall a useful result from [START_REF] Laptev | Hardy inequalities for magnetic Dirichlet forms[END_REF], which we express in the formulation of [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF].

Lemma 2.4. Let X ∈ R 2 and α ∈ R \ Z . For every 0 ≤ ρ 1 < ρ 2 and u ∈ C ∞ c (R 2 \ {X}), C(X,ρ1,ρ2) |u| 2 |x -X| 2 dx ≤ dist(α, Z) -2 C(X,ρ1,ρ2) -i∇ -A X α u 2 dx , (2.2) with C(X, ρ 1 , ρ 2 ) = {x ∈ R 2 ; ρ 1 < |x -X| < ρ 2 } .
We can prove the preceding statement by using polar coordinates, expanding the integrand in Fourier series in the angular variable, and applying Parseval's formula. The details can be found in [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF].

Corollary 2.5. If α i / ∈ Z for each 1 ≤ i ≤ N , then H X α is continuously embedded in H 1 (Ω). Proof. Let u ∈ Q X α .
Far from the poles, u is locally in H 1 . In the neighborhood of one pole, we can bring the flux of all the other poles to zero thanks to a local gauge transformation, and prove that u is locally in H 1 with the help of Inequality (2.2). Using these ideas, we obtain an explicit control on the H 1 -norm of u and thus prove the embedding.

A non-concentration inequality

We now establish a non-concentration result. It is a slight variant of an inequality that was communicated to us by Luc Hillairet. A similar result is proved in [START_REF] Hillairet | The eigenvalues of the Laplacian on domains with small slits[END_REF]Lem. 3.2.]. Let us first give it for H 1 -functions.

Lemma 2.6. For any 0 < ν < 1, there exists a constant C ν ≥ 0 such that, for all u ∈ H 1 (R 2 ) , x 0 ∈ Ω , and r > 0 ,

u L 2 (B(x0,r)) ≤ C ν r ν u H 1 (R 2 ) (2.3) where B(x 0 , r) = {x ∈ R 2 ; |x -x 0 | < r} .
Proof. The proof is a direct application of the continuous embedding

H 1 (R 2 ) ⊂ L p (R 2 ) , valid for any 2 ≤ p < ∞. Let u ∈ H 1 (R 2 )
. According to the Hölder inequality,

B(x0,r) |u| 2 dx ≤ |B(x 0 , r)| 1/q B(x0,r) |u| 2q dx 1/q ≤ |B(x 0 , r)| 1/q u 2 L 2q (R 2 )
for any q and q greater than 1 such that 1/q + 1/q = 1 . We chose q = 1/ν , and obtain

B(x0,r) |u| 2 dx ≤ C 1 r 2ν u 2 L 2 1-ν (R 2 )
,

where C 1 is a constant depending only on ν. Since H 1 (R 2 ) is continuously embedded in L 2 1-ν (R 2 ), we finally get B(x0,r) |u| 2 dx ≤ C 2 r 2ν u 2 H 1 (R 2 ) ,
where C 2 is a constant depending only on ν .

We now extend this result to functions in the form domain Q X α in the case ν = 1/2 . Lemma 2.7. There exists a constant C ≥ 0 such that, for any x 0 ∈ Ω, r > 0,

and u ∈ Q X α , u 2 L 2 (B(x0,r)) ≤ C r u 2 L 2 (R 2 ) + (-i∇ -A X α )u 2 L 2 (R 2 ) . (2.4)
Proof. Let us first consider the case where

u ∈ C ∞ c (Ω X ) . Then |u| ∈ H 1 (R 2 ) and inequality (2.3) yields u 2 L 2 (B(x0,r)) ≤ C r u 2 L 2 (R 2 ) + ∇|u| 2 L 2 (R 2 )
for some constant C independent of u and r. According to the diamagnetic inequality, |∇|u|(x)| ≤ (-i∇ -A X α )u(x) for almost every x ∈ R 2 . We obtain

u 2 L 2 (B(x0,r)) ≤ Cr u 2 L 2 (R 2 ) + (-i∇ -A X α )u 2 L 2 (R 2 ) . Since Q X α is defined as the closure of C ∞ c (Ω X ) for the norm • α X , the inequality still holds for functions in Q X α .

Characterization of the form domain

Since the characterization of Proposition 1.1 is standard, we do not give the details of its proof. The reader can refer to [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF]Lemma 2.1.] for the case of one pole with a non-integer flux. The poles having integer flux can be dealt with using a local gauge transformation, and the fact that a point has H 1 -capacity zero in R 2 . The operator γ 0 appearing in the statement of Proposition 1.1 is simply the classical boundary trace operator, or possibly its conjugate by a gauge transformation.

Let us comment in more details on the condition (-i∇ -

A X α )u ∈ L 2 (Ω) for a function u ∈ Q X α .
Far from the poles, it simply means that u is locally in H 1 . In a neighborhood of a pole X i , its meaning depends on the normalized flux

α i . If α i / ∈ Z , Inequality (2.
2) shows that u is locally in H 1 (as seen in the proof of Corollary 2.5), with the additional integrability condition u/|x -X i | ∈ L 2 (Ω) . If α i ∈ Z , the condition means that there is a local gauge transformation ψ such that u * = ψu is locally in H 1 , but does not mean that u is locally in H 1 . Indeed this is not the case as soon as α i = 0, because then the function (r, θ) → e 2iπαiθ (expressed in polar coordinates) is not H 1 in a neighborhood of 0.

Most of the quoted authors have restricted themselves to the case where no α i is an integer, and the form domain is contained in H 1 (Ω) . This is for example the case in [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF][START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF][START_REF] Bonnaillie-Noël | On the eigenvalues of Aharonov-Bohm operators with varying poles[END_REF]. This does not introduce any restriction for fixed poles, since an integer flux can be brought to zero by a gauge transformation defined on Ω X . However we want to prove in this paper, among other results, the continuity of the eigenvalues when several poles, each with a non-integer flux, coalesce into a single pole whose flux that can be an integer (see for instance Corollary 3.5). The formulation of Proposition 1.1, which does not distinguish between integer and non-integer fluxes, allows us to do this.

3 Continuity with respect to the poles

Main statement

Let us first restate Theorem 1.2.

Proposition 3.1. Let (X n ) n≥1 be a sequence of points in R 2N such that X n → X when n → +∞. Then, for each k ≥ 1, λ k (X n , α) → λ k (X, α) as n → +∞.
The proof is based on two key lemmas which will be stated and proved in the next subsections.

An extraction lemma

The following lemma is the central part of the proof of Proposition 3.1. Its proof uses the weak compactness of the unit ball of H 1 , the non-concentration result of Lemma 2.7, and the characterization of the form domain Q X α given in Proposition 1.1. Lemma 3.2. Let (λ n ) n≥1 be a sequence of eigenvalues associated with -∆ X n α and (u n ) n≥1 a sequence of corresponding normalized eigenfunctions. Assume that λ n → λ when n → +∞. Then there is a strictly increasing sequence of integers (n p ) p≥1 and a function u ∈ Q X α such that i) u is an eigenfunction of -∆ X α associated with the eigenvalue λ,

ii) u np → u when p → ∞ in L 2 (Ω) (strongly), iii) u np (x) → u(x) when p → ∞ for almost every x in Ω. Proof. For m ≥ 1, we define S m = ∪ N i=1 B(X i , 1 m ) and Ω m = Ω \ S m . Let us fix m ≥ 1.
Then, for any integer n large enough so that Ω m does not contain any point of

X n , u n L 2 (Ωm) ≤ 1 and (-i∇ -A X α )u n 2 L 2 (Ωm) ≤ 2 (-i∇ -iA X n α )u n 2 L 2 (Ωm) + 2 A X n α -A X α 2 L ∞ (Ωm) u n 2 L 2 (Ωm) .
We have

(-i∇ -A X n α )u n 2 L 2 (Ωm) ≤ (-i∇ -A X n α )u n 2 L 2 (Ω) = λ n ≤ sup q≥1 λ q , lim n→∞ A X n α -A X α L ∞ (Ωm) = 0 , and A X α L ∞ (Ωm) < ∞ . As a consequence, the sequence (u n ) n≥1 is bounded in H 1 (Ω m ).
Due to the weak compactness of the unit ball of H 1 (Ω m ), we can extract from (u n ) n≥1 a subsequence that converges weakly to u ∈ H 1 (Ω m ). According to Rellich's Theorem, H 1 (Ω m ) is compactly embedded in L 2 (Ω m ). We can therefore extract from the preceding subsequence a subsequence that converges strongly to u in L 2 (Ω m ), and also (up to one further extraction) that converges to u almost everywhere in Ω m .

After applying a diagonal extraction procedure, we obtain a subsequence (u np ) p≥1 of (u n ) n≥1 and a function u defined almost everywhere in Ω such that for each m ≥ 1, u np → u when p → +∞ weakly in H 1 (Ω m ), strongly in L 2 (Ω m ), and almost everywhere in Ω m . We have, for each m ≥ 1, each p ≥ 1,

Ωm |u np | 2 dx ≤ Ω |u np | 2 dx = 1 and Ωm |(-i∇ -A X α )u np | 2 dx ≤ 2 sup q≥1 λ q + 2 sup Ωm |A X n α -A X α | 2 .
Passing to the limit p → 0, we get

Ωm |u| 2 dx ≤ 1 and Ωm |(-i∇ -A X α )u| 2 dx ≤ 2 sup q≥1 λ q .
We deduce that u ∈ L 2 (Ω), with u L 2 (Ω) ≤ 1, and that (-i∇ -A X α )u , as a distribution on Ω X , belongs to L 2 (Ω).

For m ≥ 1, let us note Γ m = ∂Ω∩∂Ω m . For p ≥ 1, we have γ 0 u np = 0 on Γ m . Since (u np ) p≥1 converges weakly to u in H 1 (Ω m ) , (γ 0 u np ) p≥1 converges weakly to γ 0 u in L 2 (Γ m ) and therefore

γ 0 u = 0 on Γ m . Since ∪ ∞ m=1 L 2 (Γ m ) = L 2 (∂Ω), we have shown γ 0 u = 0 . According to Proposition 1.1, u ∈ Q X α .
Let us check that u is a weak solution of the eigenvalue equation. Let ϕ be a C ∞ -function with support in Ω X . There exists m ≥ 1 such that supp(ϕ) ⊂ Ω m . For p large enough,

Ω u np (-i∇ -A X np α ) 2 ϕ dx = λ np Ω u np ϕ dx . Since (-i∇ -A X np α ) 2 ϕ converges uniformly on Ω to (-i∇ -A X α ) 2 ϕ, we get, Ω u (-i∇ -A X α ) 2 ϕ dx = λ Ω u ϕ dx
and therefore

Ω (-i∇ -A X α )u • (-i∇ -A X α )ϕ dx = λ Ω u ϕ dx .
By definition of Q X α , the preceding relation can be extended to any ϕ ∈ Q X α . It remains to check that u is not the trivial solution. To this end, we use the non-concentration property of Lemma 2.7 to show that (u np ) p≥1 converges in L 2 (Ω) to u. Indeed, if ε > 0, there exists m large enough so that, for all p ≥ 1,

Sm |u np (x)| 2 dx ≤ CN m sup q≥1 λ q + 1 ≤ ε ,
where C is the constant appearing in Inequality (2.4), N is the number of balls composing S m , and 1/m the radius of these balls.

Let us pick such a m. Then

u -u np L 2 (Ω) ≤ u -u np L 2 (Ωm) + u -u np L 2 (Sm) so u -u np L 2 (Ω) ≤ u -u np L 2 (Ωm) + u L 2 (Sm) + u np L 2 (Sm) .
Hence we get u -

u np L 2 (Ω) ≤ u -u np L 2 (Ωm) + 2 ε .
Since lim p→∞ u -u np L 2 (Ωm) = 0, we have, for p large enough,

u -u np L 2 (Ω) ≤ 3 ε .

End of the proof of the continuity

We now prove Proposition 3.1, using the min-max characterization of the eigenvalues of -∆ X α . We still need a simple lemma.

Lemma 3.3. Let (X n ) n≥1 be such that X n → X when n → ∞ . Then, for each k ≥ 1 , lim sup n→∞ λ k (X n , α) ≤ λ k (X, α) .
Proof. According to the Min-Max Formula, for any Y ∈ R 2N ,

λ k (Y, α) = inf ϕ1,...,ϕ k ∈C ∞ c (Ω Y ) max u∈vect(ϕ1,...,ϕ k ) (-i∇ -A Y α )u 2 u 2 .
Let us pick ε > 0. We fix a family ϕ 1 , . . . ,

ϕ k ∈ C ∞ c (Ω X ) such that max u∈vect(ϕ1,...,ϕ k ) (-i∇ -A X α )u 2 u 2 ≤ λ k (X, α) + ε .
For every n ≥ 1, large enough that supp(ϕ i ) ⊂ Ω X n for each 1 ≤ i ≤ k , let us set

µ n = max u∈vect(ϕ1,...,ϕ k ) (-i∇ -A X n α )u 2 u 2 .
Let us fix v n ∈ vect(ϕ 1 , . . . , ϕ k ) such that v n = 1 and (-i∇ -A X n α )v n 2 = µ n . Let µ be a limit point of (µ n ) n≥1 , with µ = lim p→∞ µ np . Since the space vect(ϕ 1 , . . . , ϕ k ) has finite dimension, we can, up to a subsequence, assume that there exists v ∈ vect(ϕ 1 , . . . , ϕ k ) such that v np → v in L 2 (Ω) when p → ∞ . Since the ϕ i 's are supported away from X, it is easily seen that in that case

(-i∇ -A X np α )v np → (-i∇ -A X α )v in L 2 (Ω) when p → ∞ . Therefore µ = lim p→∞ µ np = (-i∇ -A X α )v 2 v 2 ≤ λ k (X, α) + ε .
We have proved that lim sup

n→∞ µ n ≤ λ k (X, α) + ε .
Since λ k (X n , α) ≤ µ n for all n ≥ 1 , we obtain

lim sup n→∞ λ k (X n , α) ≤ λ k (X, α) + ε .
We get the desired result by letting ε tend to 0.

We now complete the proof of Proposition 3.1 by induction. Let us first consider the case k = 1. According to Lemma 3.3,

lim sup n→∞ λ 1 (X n , α) ≤ λ 1 (X, α) .
The sequence (λ 1 (X n , α)) n≥1 is therefore bounded. According to Lemma 3.2, its limit points are eigenvalues of -∆ X α . This implies

λ 1 (X, α) ≤ lim inf n→∞ λ 1 (X n , α) .
We conclude that lim

n→∞ λ 1 (X n , α) = λ 1 (X, α) .
Let k ≥ 2 . We assume that for 1 ≤ j ≤ k -1 , lim n→∞ λ j (X n , α) = λ j (X, α) . For each n ≥ 1 , 1 ≤ j ≤ k , we fix u j n an eigenfunction of -∆ X n α associated with λ j (X n , α) such that u j n = 1 . Lemma 3.2 tells us that, up to a subsequence, we can assume that for each 1 ≤ j ≤ k -1 , u j n → u j when n → ∞ , where u j is an eigenfunction of -∆ X α associated with the eigenvalue λ j (X, α) . Now let us assume that λ is a limit point for the sequence (λ k (X n , α)) n≥1 , with λ = lim p→∞ λ k (X np , α) . Up to a subsequence, u k np → u k in L 2 (Ω) when p → ∞, where u k is an eigenfunction of -∆ X α associated with λ (always according to Lemma 3.2). For each 1 ≤ j ≤ k -1, each n ≥ 1, we have

u j np , u k np = 0 .
Passing to the limit in L 2 (Ω), we get u j , u k = 0 , and therefore λ k (X, α) ≤ λ .

We have proved that

λ k (X, α) ≤ lim inf n→∞ λ k (X n , α) ,
which, together with Lemma 3.3, gives

lim n→∞ λ k (X n , α) = λ k (X, α) .

Some applications

In the following, we fix k ≥ 1 . Our first result deals with one point moving toward the boundary of the domain. Let us assume that the set R 2 \ Ω can be written as the reunion of a finite number K + 1 of closed and connected sets D 1 , D j , . . . , D K , with D j bounded for 1 ≤ j ≤ K and D 0 unbounded. Let us consider an Aharonov-Bohm operator with only one pole X ∈ R 2 . We have already noticed that the winding number of any path γ contained in Ω is zero around any point in D 0 and, for each 1 ≤ i ≤ K , is the same around all the points in D i . According to Lemma 2.3, this implies that the function X → λ k (X, 1/2) is constant on each of the D i . For each 1 ≤ i ≤ K, let us note λ i k (Ω) the value of λ k (X, α) for X ∈ D i . Theorem 1.2 gives the following convergence result. Corollary 3.4. Let X be a point in ∂Ω and (X n ) n≥1 be a sequence of points in Ω converging to X. Then,

1. if X ∈ ∂D 0 , lim n→∞ λ k (X n , α) = λ D k (Ω) , 2. if X ∈ ∂D i with 1 ≤ i ≤ K, lim n→∞ λ k (X n , α) = λ i k (Ω) .
Let us now consider an Aharonov-Bohm operator with two poles X and Y , whose respective normalized flux are α and β. We restate Theorem 1.2 in that case.

Corollary 3.5. If X ∈ Ω, and (X n ) n≥1 , (Y n ) n≥1 are sequences of points in Ω converging to X, then,

lim n→∞ λ k ((X n , Y n ), (α, β)) = λ k (X, α + β) . If α = β = 1/2, Lemma 2.3 implies λ k (X, α + β) = λ D k (Ω)
4 Analyticity with respect to the poles

We now turn to the proof of Theorem 1.3. Let us first give a brief outline. For t = (t 1 , t 2 , . . . , t 2N -1 , t 2N ) ∈ R 2N , we note X 1 (t) = X 1 + t 1 e 1 + t 2 e 2 , . . . , X N (t) = X N + t 2N -1 e 1 + t 2N e 2 , and X(t) = (X 1 (t), . . . , X N (t)) . We construct a suitable unitary mapping U (t) : Let us first construct U (t). For r > 0 and 1 ≤ i ≤ N , we note B i = B(X i , r) and B i = B(X i , r/2) . We choose r small enough so that B i ∩ ∂Ω = ∅ and B i ∩ B j = ∅ for i = j. Next we fix, for each

L 2 (Ω) → L 2 (Ω) such that q X α • U (t)
1 ≤ i ≤ N , a smooth function χ i satisfying i. 0 ≤ χ i ≤ 1 ; ii. χ i = 1 on B i ; iii. χ i = 0 on R 2 \ B i .
We then note

V t (x) = N i=1 χ i (x)(t 2i-1 e 1 + t 2i e 2 ) and Φ t (x) = x + V t (x) .
Using the inverse function theorem and the mean value inequality, it is easy to check that, for |t| small enough, Φ t is a smooth diffeomorphism that sends Ω onto itself. Furthermore, we have Φ t (X i ) = X i (t) for 1 ≤ i ≤ N . We now define U (t) : L 2 (Ω) → L 2 (Ω) by

U (t)u = J((Φ t ) -1 ) 1/2 u • (Φ t ) -1 , with J((Φ t ) -1 ) = | det(D((Φ t ) -1 ))| . The map U (t) is unitary and satisfies U (t)(C ∞ c (Ω X )) = C ∞ c (Ω X(t) ) . For u ∈ C ∞ c (Ω X ), we define r t (u) = q X(t) α ((U (t))u) . We note A = A X α and A t = A X(t) α
for better readability. A straightforward computation gives the formula

r t (u) = Ω -iJ(Φ t ) 1/2 (D(Φ t ) T ) -1 ∇(J(Φ t ) -1/2 u) -A t • Φ t u 2 dx .
We can then write

r t (u) = Ω |(-i∇ -A)u + E(Φ t )(-i∇ -A)u + F (Φ t )u| 2 dx , (4.1) with E(Φ t ) = (D(Φ t ) T ) -1 -Id and
F (Φ t ) = (D(Φ t ) T ) -1 A -A t • Φ t -iJ(Φ t ) 1/2 (D(Φ t ) T ) -1 ∇(J(Φ t ) -1/2 ) .
Let us note that lim t→0 sup Ω |E(Φ t )| = 0 . Let us also note that |F (Φ t )| is uniformly bounded when t tends to 0. The difficulties in the proof of this latter assertion only appear in the neighborhood of a pole X i , where A has a singularity. Since the poles are far from each other, it is enough to prove the assertion in the case of one pole. We assume

A(x) = α (x -X) ⊥ |x -X| 2 .
We note

B = B(X, r), B = B(X, r/2) . Let us note v(t) = t 1 e 1 + t 2 e 2 for concision. If x ∈ B , Φ t (x) = x + v(t)
, and therefore

F (Φ t (x)) = A(x)-A t (Φ t (x)) = α (x -X) ⊥ |x -X| 2 - (x + v(t) -X -v(t)) ⊥ |x + v(t) -X -v(t)| 2 = 0 . If x / ∈ B , we have |x -X| ≥ r/2.
We can chose t = (t 1 , t 2 ) small enough so that, for all y ∈ R 2 , |Φ t (y) -y| ≤ r/8 . In that case,

|Φ t (x) -Φ t (X)| ≥ |x -X| -|x -Φ t (x)| -|Φ t (X) -X| ≥ r 2 - r 8 - r 8 = r 4 .
This shows that both |A(x)| and |A t (x)| are bounded by 4|α|/r for t small enough. Therefore, there exists a constant C, independent of x and t, such that |F (Φ t )(x)| ≤ C for t small enough. Expanding Equation (4.1), we find

r t (u) = q X α (u) + Ω |E(Φ t )(-i∇ -A)u| 2 dx + Ω |F (Φ t )u| 2 dx + 2Re Ω (-i∇ -A)u T E(Φ t )(-i∇ -A)u dx + 2Re Ω (-i∇ -A)u • F (Φ t )u dx + 2Re Ω F (Φ t )u T E(Φ t )(-i∇ -A)u dx .
Applying the Cauchy-Schwarz and Young inequalities, we find

|r t (u) -q X α (u)| ≤ 2 sup Ω E(Φ t ) + 2 sup Ω E(Φ t ) 2 + 1 2 q X α (u) + 4 sup Ω |F (Φ t )| 2 u 2 L 2 (Ω) .
We conclude that there exist 0 < a < 1 and b ≥ 0, independent of u, such that, for t small enough,

|r t (u) -q X α (u)| ≤ aq X α (u) + b u 2 L 2 (Ω) .
The family of forms t → r t is therefore uniformly q X α -bounded for t in a neighborhood of 0. According to standard results on the perturbation of sesquilinear sectorial forms (c.f. [11, Th. VI-1.33]), this means that the form domain of r t (u) is equal to Q X α for all t small enough. It is obvious that for u ∈ Q X α , r t (u) depends analytically on t . The family t → r t therefore depends analytically on t in the sense of Kato. Theorem 1.3 is a consequence of the general perturbation theory.

Let us now give a proof of Theorem 1.4. We first give a precise definition of a K X -real eigenfunction, as announced in the introduction. We assume in the rest of this section that α i ∈ Z + 1/2 for all 1 ≤ i ≤ N . According to Lemma 2.3, there exists a gauge function ψ defined on Ω X such that -iψ∇ψ = 2A X α . Let us fix such a ψ . We define K X by

K X u = ψ u ,
and say that a function

u is K X -real if K X u = u . A direct calculation shows that -∆ X α • K X = K X • (-∆ X α ) .
We can therefore choose a basis of K X -real eigenfunctions for -∆ X α . We will use the local description of the nodal lines of a K X -real eigenfunction of -∆ X α given in [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF] (see also [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF]).

Theorem 4.1. Let α ∈ (Z + 1/2) N and X ∈ R 2N . Let us fix 1 ≤ i ≤ N with X i ∈ Ω. Let k ≥ 1
and assume that u is a K X -real eigenfunction associated with λ k (X, α) . Let (r, θ) be the polar coordinates centered at X i in a neighborhood of X i . There exist a non-negative integer m and C 1 functions f and g such that

f (0) = 0 , u(x) = r m+1/2 f (x) ,

and

(-i∇ -A X α (x))u(x) = r m-1/2 g(x) . Furthermore, 2m + 1 is the number of nodal lines meeting at X i .

The basic idea of our proof of Theorem 1.4 is the following. We construct a family of diffeomorphisms Φ h that depends on a parameter h > 0. Using the Feynman-Hellmann formula, we compute λ k (0) (which does not depend on h) as an integral I(h) depending on h. We then use Theorem 4.1, with m ≥ 1 , to show that lim h→0 I(h) = 0 .

To simplify notation, let us assume X i = 0 . Let us first construct a family of diffeomorphisms Φ h,t that allows us to write the eigenvalue problem on the fixed domain Ω X . We fix r > 0 such that B(0, r) ⊂ Ω and X j / ∈ B(0, r) for j = i. We then fix a vector field

V ∈ C ∞ (R 2 , R 2 ) such that 1. V(x) = v if x ∈ B(0, r/2) , 2. V(x) = 0 if x / ∈ B(0, r) ,
We then define, for t > 0 , Φ h,t (x) = x + tV(h -1 x) , and X(t) = Φ h,t (X) . It is easy to see that for any fixed h > 0, for |t| small enough, Φ h,t is a family of diffeomorphisms such that Φ h,t (Ω) = Ω. As in the proof of Theorem 1.3, we define the family of unitary operators U (h, t) :

L 2 (Ω) → L 2 (Ω) by U (h, t)u = J(Φ -1 h,t ) 1/2 u • Φ -1 h,t ,
and the family of sesquilinear forms r h,t = q X(t) α

• U (h, t) . The same reasoning as in the proof of Theorem 1.3 shows that t → r h,t is analytic in the sense of Kato in a neighborhood of 0 . We can therefore apply the Feynman-Hellmann formula:

λ k (0) = ∂ t r h,t (u) |t=0 , (4.2)
where u is an eigenfunction associated with λ k (0) such that u L 2 (Ω) = 1 . We

set A = A X α , A t = A X(t)
α , and keep the notation used in the proof of Theorem 1.3. We have

r h,t (u) = q X α (u) + Ω |E(Φ h,t )(-i∇ -A)u| 2 dx + Ω |F (Φ t )u| 2 dx + 2Re Ω (-i∇ -A)u T E(Φ h,t )(-i∇ -A)u dx + 2Re Ω (-i∇ -A)u • F (Φ h,t )u dx + 2Re Ω F (Φ h,t )u T E(Φ h,t )(-i∇ -A)u dx .
We apply Equation (4.2) to the preceding expression. Since Φ h,t (x) = x + tV x h , we find

DΦ h,t (x) = Id + t h DV x h .
We deduce

∂ t E(Φ h,t ) |t=0 (x) = ∂ t (D(Φ h,t ) T ) -1 |t=0 (x) = - 1 h (DV) T x h .
We also have

J(DΦ h,t )(x) = det Id + t h DV x h , which gives ∂ t (J(DΦ h,t )(x)) |t=0 = 1 h Tr (DV) x h .
From this we deduce

∂ t ∇ x J(DΦ h,t ) -1/2 x h |t=0 = ∇ x ∂ t J(DΦ h,t ) -1/2 x h |t=0 = ∇ x - 1 2h Tr(DV) x h = - 1 2h 2 ∇ x Tr(DV) x h .
We obtain

∂ t F (Φ h,t )(x) |t=0 = ∂ t (D(Φ h,t ) T ) -1 A(x) -A t • Φ h,t (x) |t=0 - i 2h 2 ∇ x Tr(DV) x h .
For |x| ≤ hr/2, we have Φ h,t (x) = x + v. We therefore have

(D(Φ t ) T ) -1 A(x) -A t • Φ h,t (x) = 0 and ∂ t (D(Φ t ) T ) -1 A(x) -A t • Φ h,t (x) |t=0
= 0 .

We now do the computations for |x| > hr/2 . We write

A = α i x ⊥ |x| 2 + A ,
where A is smooth on a neighborhood of 0 . We have

∂ t (x + tV(x/h)) ⊥ |x + tV(x/h)| 2 |t=0 = 1 |x| 2 - x • V(x/h) |x| 4 x ⊥ .
We conclude that there exists a constant C, such that for all h > 0 and all x ∈ Ω,

∂ t E(Φ h,t )(x) |t=0 ≤ C h and ∂ t F (Φ h,t )(x) |t=0 ≤ C h 2 . We have ∂ t r h,t (u) |t=0 = 2 Re Ω (-i∇ -A)u T ∂ t E(Φ h,t ) |t=0 (-i∇ -A)u dx + 2 Re Ω ∂ t F (Φ h,t ) |t=0 u • (-i∇ -A)u dx .
We deduce immediately

|∂ t r h,t (u) |t=0 | ≤ 2C h (-i∇ -A)u 2 L 2 (B(0,hr)) + 2C h 2 (-i∇ -A)u L 2 (B(0,hr) u L 2 (B(0,hr)) .
According to Theorem 4.1, there exist an integer m ≥ 0 and C 1 -functions f and g such that in a neighborhood of 0 ,

u(x) = |x| m+1/2 f (x) and (-i∇ -A)u = |x| m-1/2 g(x) ,
where f (0) = 0 and 2m + 1 is the number of nodal lines meeting at 0 . From this we deduce the estimates B(0,hr)

|u| 2 dx = O h 2m+3 , B(0,hr) |(-i∇ -A)u| 2 dx = O h 2m+1 ,
and finally

|∂ t r h,t (u) |t=0 | = O h 2m .
Since we have assumed that at least three nodal lines meet at 0, we have in fact m ≥ 1 and we deduce

|∂ t r h,t (u) |t=0 | = O h 2 .
Since this holds for all h > 0 and since the left-hand side of Equation (4.2) does not depend on h, we obtain λ k (0) = 0 , which concludes the proof.

The Kato-Rellich perturbation theory gives some information in the case where λ k (X, α) is not simple. Let us denote by d the multiplicity of λ k (X, α) . Let us fix an N -tuple (v 1 , . . . , v N ) of vectors in R 2 . For any t in R , we note

X(t) = (X 1 + tv 1 , . . . , X N + tv N ) .
There exist d functions t → λ 1 (t), . . . , t → λ d (t) (not necessarily distinct), analytical in a neighborhood of 0, such that for 1 ≤ j ≤ d , λ j (0) = λ k (X, α) and λ j (t) is an eigenvalue of the operator -∆

X(t)

α . There also exist d analytical functions t → u j (t) from a neighborhood of 0 to L 2 (Ω) such that u j (t) L 2 (Ω) = 1 and u j (t) is an eigenfunction associated with λ j (t) . They can be chosen so that u j (t) is K X -real. Furthermore, the Feynman-Hellmann formula then tells us that

λ j (0) = ∂ t (-∆ X(t)
α ) |t=0 u j (0) , u j (0) .

Let us now fix 1 ≤ i 0 ≤ N and consider the case where v i = 0 for all i = i 0 (we only move the point X i0 ) and α i0 ∈ Z + 1/2. Assume that there exists 1 ≤ j 0 ≤ d such that u j0 (0) is K X -real and has at least three nodal lines meeting at X i . The proof of Theorem 1.4 applies; we obtain λ j0 (0) = 0 . However, the existence of a K X -real eigenfunction associated with λ k (X, α) with at least three nodal lines meeting at X i0 is a priori not enough to ensure the existence of 1 ≤ j 0 ≤ d such that λ j0 (0) = 0 .

Minimal partitions

Magnetic characterization and critical points

One motivation for this work was the investigation of the properties of minimal partitions. A k-partition P of Ω, with k a positive integer, is a set P = {D 1 , . . . , D k } of k disjoint, connected, and open sets contained in Ω. It is said to be minimal if the quantity

max 1≤i≤k λ D 1 (D i )
is minimal among all the k-partitions. This problem has recently been the subject of various studies. In particular, the paper [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], by B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, shows that a minimal partition of Ω exists for any integer k provided ∂Ω is sufficiently regular. It also proves regularity properties of minimal partitions. The paper [START_REF] Helffer | On a magnetic characterization of spectral minimal partitions[END_REF] by B. Helffer and T. Hoffmann-Ostenhof, establishes a connection between minimal partitions and nodal partitions for an Aharonov-Bohm operator. We can summarize it in the following statement. We recall that a nodal domain of a K X -real eigenfunction u of -∆ X α is a connected component of the set Ω \ u -1 ({0}) .

Theorem 5.1. Let us assume that Ω is a connected (possibly with holes) regular open set. Let k ≥ 1 be an integer and P = {D 1 , . . . , D k } be a minimal kpartition. There exist a finite number of points X 1 , . . . , X N in R 2 such that P is the nodal partition (i.e. the set of all nodal domains) of a K X -real eigenfunction u of -∆ X α , with X = (X 1 , . . . , X N )

and α = (1/2, . . . , 1/2) .
Furthermore, u is associated with the eigenvalue λ k (X, α) .

Roughly speaking, this result shows that a minimal partition is nodal, provided that we add enough poles with a normalized flux of 1/2 . Let us note that we may have to add poles both in Ω and in R 2 \ Ω . One can indeed define the boundary of a sufficiently regular partition (which can be informally described has the union of the boundaries of all the D i 's), and it has almost the same properties as the nodal set of a real eigenfunction of -∆ D Ω . However, there may be singular points where an odd number of lines meet. We have to add a pole at these points. In the case where Ω is not simply connected, we may also have to add a pole in some holes (i.e. in some bounded connected components of R 2 \ Ω).

Theorem 5.1 allows us to give a spectral characterization of minimal partitions. Let us note

L k (Ω) = inf max 1≤i≤k λ i (D i ) ; {D 1 , . . . , D k } k-partition of Ω .
If X ∈ R 2N , with distinct X i 's, and α = (1/2, . . . , 1/2) , we denote by L k (X, α) the smallest eigenvalue of -∆ X α that has a K X -real eigenfunction with k nodal domains. If there is no such eigenfunction, we set L k (X, α) = +∞ . Then

L k (Ω) = inf N ≥1 inf X∈R 2N L k (X, α) .
We can combine Theorem 5.1 with the results of Section 4 to show that L k (Ω) is a critical value for the k-th eigenvalue of an Aharonov-Bohm operator. Let us give a precise statement. Theorem 5.2. Let us assume that Ω is a connected open set, k a positive integer, and P a minimal k-partition of Ω . We denote by X = (X 1 , . . . , X N ) and α = (1/2 . . . , 1/2) the poles and fluxes defined in Theorem 5.1. Let us additionally assume that the eigenvalue λ k (X, α) is simple. The point X is then critical for the function Y → λ k (Y, α) , which is defined and analytic in a neighborhood of X .

Proof. According to Theorem 1.3, Y → λ k (Y, α) is analytic in a neighborhood of X . As allowed by Theorem 5.1, X has been chosen so that there exists a K X -real eigenfunction u whose nodal partition is P . To show that X is critical, it is enough to show that the gradient with respect to each point X i , with i ∈ {1 , . . . , N }, is zero. There are two possible cases.

• X i ∈ R 2 \ Ω . In that case λ k (Y, α) = λ k (X, α) if Y = (Y 1 , . . . , Y N ) is such that Y j = X j for j = i and Y i is in the same connected component of R 2 \ Ω as X i (This is a consequence of Lemma 2.3).
• X i ∈ Ω . In that case at least three nodal lines of u meet at X i . Therefore, according to Theorem 1.4, X i is a critical point for the function

Y → λ k ((X 1 , . . . , X i-1 , Y, X i+1 , . . . , X N ), α) .
The fact that X is a critical point is proved in the case N = 1 for the minimal 3-partitions of a simply connected domain in [START_REF] Bonnaillie-Noël | On the eigenvalues of Aharonov-Bohm operators with varying poles[END_REF]. As far as we know, our result is new for a general k-partition.

Domain with an axis of symmetry

According to Theorem 5.1, minimal partitions of an open set are nodal partitions for an Aharonov-Bohm operator. This can help us to find good candidates to be minimal partitions of a given set. Let us explain a method that was used in the papers [START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF][START_REF] Bonnaillie-Noël | Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions[END_REF][START_REF] Bonnaillie-Noël | Spectral minimal partitions of a sector[END_REF]. The authors of these papers where looking for minimal 3-partitions of rectangles and circular sectors, but we will explain the method in more general terms.

Let Ω ∈ R 2 be a simply connected open set with a regular boundary. Let us also assume that the line {x 2 = 0} is an axis of symmetry for Ω . It seems reasonable to look for minimal partitions that have only one singular point in their boundary and that admit {x 2 = 0} as an axis of symmetry. We therefore look for X = (x, 0) , a pole belonging to {x 2 = 0} , such that λ 3 (X, 1/2) admits a K X -real eigenfunction with three nodal domains.

In the case that we consider, there exists a convenient reformulation of the eigenvalue problem. We note Ω + = Ω ∩ {x 2 > 0} , Γ + = ∂Ω ∩ {x 2 > 0} , and Ω ∩ {x 2 = 0} = (A, B) , with A = (a, 0) and B = (b, 0) . We then define two eigenvalue problems for the Laplace operator in Ω + with mixed boundary condition. Proposition 5.3 motivates the study of Problems (5.1) and (5.2). More precisely, the authors of [START_REF] Bonnaillie-Noël | Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions[END_REF][START_REF] Bonnaillie-Noël | Spectral minimal partitions of a sector[END_REF] compute, by a finite element method, λ N D 2 (x) and the nodal lines of an associated eigenfunction u for different values of the parameter x ∈ (a, b) . They look for a value x 0 of x such that there is an eigenfunction u associated with λ N D 2 (x 0 ) that has one nodal line, which connects a point of Γ + with X 0 = (x 0 , 0) . In that case, the symmetrization of the nodal partition of u gives a 3-partition of Ω which is a plausible candidate to be minimal. The search for x 0 can be guided by the following result. Proof. Let us first note that according to Proposition 5.3, and since λ N D k (x 0 ) is a simple eigenvalue of -∆ X0 1/2 , there is a positive integer such that λ N D k (x) = λ (X, 1/2), with X = (x, 0) , for x in a neighborhood of x 0 . According to Theorem 1.3, the function x → λ N D k (x) is therefore analytic in a neighborhood of x 0 . The hypothesis on the nodal line, together with Proposition 5.3, implies that there is a K X -real eigenfunction associated with λ (X 0 , 1/2) that has at least three nodal lines meeting at X 0 . According to Theorem 1.4, this implies (λ N D k ) (x 0 ) = 0 . But, as seen on Problem 5.2, the function x → λ N D k (x) is non-increasing. This implies (λ N D k ) (x 0 ) = 0 .

There exists of course a similar result for x → λ DN k (x). The following example illustrates this proposition when looking for a 3-partition of a circular sector of radius 1 and angular opening π/31 . Figure 1 shows a point of inflexion x 0 0.64 of the function x → λ N D 2 (x) . Figure 2 shows the 3-partition obtained after symmetrization of the nodal partition of an eigenfunction associated with λ N D 2 (x 0 ) . Let us note that the point of inflexion in Figure 1 is hard to determine precisely in practice. We only obtain an approximate value. To draw Figure 2, we look for x 0 around this approximate value by computing the nodal set for different values of x .

Proposition 5 . 3 .

 53 = λu in Ω + , u = 0 on [A, X] ∪ Γ + , ∂ n u = 0 on (X, B) ; = λu in Ω + , ∂ n u = 0 on (A, X) , u = 0 on [X, B] ∪ Γ + . (5.2) We denote by (λ DN k (x)) k≥1 (resp. (λ N D k (x)) k≥1) the eigenvalues of Problem (5.1) (resp. (5.2)), counted with multiplicity. The following result is proved in[START_REF] Bonnaillie-Noël | Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions[END_REF]. The spectrum of -∆ X 1/2 is the reunion (counted with multiplicities) of the sequences (λ DN k (x)) k≥1 and (λ N D k (x)) k≥1 . Moreover, if u is a real eigenfunction associated with λ DN k (x) (resp. λ N D k (x)), the symmetrization of N (u) with respect to {x 2 = 0} is the nodal set of a K X -real eigenfunction of -∆ X 1/2 associated with λ DN k (x) (resp. λ N D k (x)).

Proposition 5 . 4 .

 54 Let k be a positive integer. Let us assume that x 0 ∈ (a, b) is such that the eigenvalue λ N D k (x 0 ) has a real eigenfunction with one nodal line hitting {x 2 = 0} at X 0 = (x 0 , 0) . Let us additionally assume that λ N D k (x 0 ) is a simple eigenvalue of -∆ X0 1/2 . Then x 0 is a point of inflexion for the function x → λ N D k (x) .

Figure 1 :

 1 Figure 1: Eigenvalue λ N D 2 (x) as a function of x.

Figure 2 :

 2 Figure 2: Nodal set of a third eigenfunction of an Aharonov-Bohm operator.

  is an analytic family of type (a) with respect to the variable t in the sense of Kato (cf[START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] Chap. 7, Sec. 4]). The theorem is then a direct application of the Kato-Rellich regular perturbation theory.

We thank V. Bonnaillie-Noël for the computations and the figures.