Simon Raulot

Alessandro Savo

Sharp bounds for the first eigenvalue of a fourth order Steklov problem *

Keywords: AMS 2000: 58J50, 35P15, 35J40 Fourth order Steklov problem, Eigenvalues, Harmonic functions, Lower bounds

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold Ω with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of its boundary and the inner radius. The proof is obtained by estimating the isoperimetric ratio of non-negative subharmonic functions on Ω, which is of independent interest. We also give a comparison theorem for geodesic balls.

1 Introduction

The biharmonic Steklov problem

Let Ω be a compact, connected Riemannian manifold of dimension n with smooth boundary ∂Ω. We consider the following fourth order eigenvalue boundary problem

   ∆ 2 f = 0 on Ω f = 0, ∆f = q ∂f ∂N on ∂Ω, (1)
where N is the inward unit normal and ∆ = -div∇ is the Laplacian defined by the Riemannian metric of Ω; the sign convention is that, in Euclidean space, ∆ =j ∂ 2 /∂x 2 j . The first eigenvalue of (1) has the following variational characterization:

q 1 (Ω) = inf Ω (∆f ) 2 ∂Ω ∂f ∂N 2 : f = 0 on ∂Ω . (2)
This problem was introduced by Kuttler and Sigillito [START_REF] Kuttler | Inequalities for membrane and Stekloff eigenvalues[END_REF] and Payne [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF]. For a review of the main facts about q 1 (Ω), also when the boundary is not smooth, we refer to [START_REF] Bucur | On the first eigenvalue of a fourth order Steklov problem[END_REF] :in this paper, we always assume smoothness of ∂Ω. For a recent lower bound of q 1 , see [START_REF] Wang | Sharp bounds for the first non-zero Stekloff eigenvalues[END_REF]. It is known that q 1 (Ω) is positive and simple, and that any first eigenfunction f does not change sign on Ω (see [START_REF] Kuttler | Remarks on a Stekloff eigenvalue problem[END_REF]). Moreover, it turns out that q 1 (Ω) has the following interesting characterization in terms of harmonic functions:

q 1 (Ω) = inf ∂Ω h 2 Ω h 2 : h is harmonic on Ω . (3)
The infimum in (3) is attained precisely when h = ∆f , where f is a first eigenfunction of (1) (see [START_REF] Kuttler | Remarks on a Stekloff eigenvalue problem[END_REF]). In particular, taking h = 1 one observes the following upper bound by the isoperimetric ratio:

q 1 (Ω) ≤ |∂Ω| |Ω| (4)
where |Ω| (resp. |∂Ω|) denotes the volume of Ω (resp. of ∂Ω) for the induced Riemannian measure. It turns out that, if Ω is a geodesic ball in a simply connected manifold of constant sectional curvature, then equality holds in (4) (see Theorem 2 below).

The main scope of this paper is to give sharp lower bounds for q 1 (Ω). Let us review some known results. Isoperimetric inequalities for plane domains were given in [START_REF] Kuttler | Remarks on a Stekloff eigenvalue problem[END_REF], [START_REF] Kuttler | Inequalities for membrane and Stekloff eigenvalues[END_REF] and [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF]. In [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF], Payne considers convex domains in R n and shows that, if w Ω is the minimal distance between two parallel hyperplanes which enclose Ω, then

q 1 (Ω) ≥ 2 w Ω . (5)
So, convex Euclidean domains which are "thin" have large first eigenvalue. We will prove below that this is a quite general principle which applies to a large class of manifolds.

For Riemannian manifolds, Wang and Xia prove in [START_REF] Wang | Sharp bounds for the first non-zero Stekloff eigenvalues[END_REF] that, if Ω has nonnegative Ricci curvature, and ∂Ω has mean curvature bounded below by H > 0 (hence positive everywhere), then

q 1 (Ω) ≥ nH, (6)
with equality if and only if Ω is a Euclidean ball. We should mention that inequality ( 6) is also a consequence of a lower bound of q 1 (Ω) by the first eigenvalue of a certain Steklov problem for differential forms, recently proved by the authors (see Theorem 10 in [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF]).

Wang and Xia proof of (6) makes use of the Reilly formula: this approach, however, seems to be hard to apply when the curvature of the domain (or the mean curvature of its boundary) assume negative values somewhere. In this paper, using a Laplacian comparison argument, and a symmetrization procedure developed in [START_REF] Savo | A mean-value lemma and applications[END_REF], we give a sharp, explicit lower bound of q 1 (Ω): see Theorem 2. The method produces a positive lower bound for all compact Riemannian manifolds with boundary, and in particular it applies also in negative curvature. Moreover, our bound implies (and actually improve) both ( 5) and ( 6): see Remarks 4 and 6.

Finally, we prove a comparison theorem for geodesic balls, analogous to Cheng comparison theorem for the first Dirichlet eigenvalue: see Theorem 9.

The main estimate

Let S be the shape operator of ∂Ω relative to the inner unit normal N: recall that if X is a vector tangent to ∂Ω, then S(X) = -∇ X N, where ∇ is the Levi-Civita connection of Ω. The mean curvature H of ∂Ω is the function defined by

H = 1 n -1 trS,
and the sign convention is that H = 1 for the boundary of the unit ball in R n .

Definition 1. Let K and H be arbitrary real numbers. We say that the n-th dimensional domain Ω has curvature bounds (K, H) if:

-the Ricci curvature of Ω is bounded below by (n -1)K, -the mean curvature of ∂Ω is bounded below by H.

Let R be the inner radius of Ω, defined as the maximal radius of a ball included in Ω.

Clearly: R = max{dist(x, ∂Ω) : x ∈ Ω}.

We then prove that q 1 (Ω) is uniformly bounded below by a positive constant depending only on K, H, n, R. To make this constant explicit, introduce the function:

s K (r) =              1 √ K sin(r √ K) if K > 0, r if K = 0, 1 |K| sinh(r |K|) if K < 0, (7)
and let

Θ(r) = (s ′ K (r) -Hs K (r)) n-1 . (8)
Note that Θ depends on K and H, and that Θ(0) = 1. In Proposition 14 it will be shown that Θ is positive on [0, R), and that Θ(R) = 0 if and only if Ω is a ball in the simply connected manifold M K of constant curvature K (we will call M K a space form).

Here is the main estimate.

Theorem 2. Assume that Ω has curvature bounds (K, H) (see Definition 1). Let Θ be the function defined in [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF]. Then:

q 1 (Ω) ≥ 1 R 0 Θ(r) dr . ( 9
)
The inequality is sharp: it reduces to an equality when Ω is a ball in the space form M K of constant curvature K (in this case, the right-hand side is the isoperimetric ratio |∂Ω|/|Ω|).

The proof will be given in Sections 1.6 and 2. It turns out that, besides balls in M K , equality holds also for other domains, like flat cylinders (see Corollary 5); however, we will not study the complete equality case of Theorem 2. The proof of Theorem 2 relies on an estimate of the isoperimetric ratio of non-negative subharmonic functions, which is of independent interest (see Theorem 10).

In Section 3 we have explicited a number of lower estimates of q 1 (Ω), which follow directly from [START_REF] Petersen | Riemannian Geometry[END_REF]. Let us see some other consequences of Theorem 2. For simplicity we assume K ∈ {0, 1, -1} so that the space form M K is, respectively, the Euclidean space R n , the sphere S n and the hyperbolic space H n .

Corollary 3. Assume that Ω has curvature bounds (K, H), and that:

a) H > 0 if K = 0; b) H ∈ R if K = 1; c) H > 1 if K = -1.
Let Ω be the unique ball in M K having boundary of (constant) mean curvature H. Then

q 1 (Ω) ≥ q 1 ( Ω) = |∂ Ω| | Ω| ,
with equality if and only if Ω = Ω.

For the proof, see Section 3.1.

Remark 4. Corollary 3 can be seen as a generalization of the Wang-Xia bound [START_REF] Kuttler | Remarks on a Stekloff eigenvalue problem[END_REF], which is the statement in (a): in fact, the Euclidean ball of mean curvature H has first eigenvalue q 1 ( Ω) = |∂ Ω|/| Ω| = nH. However, the main bound ( 9) is actually stronger and a straightforward calculation gives Corollary 3b applies to spherical domains; in particular we get that, if Ω is a domain in S n having boundary of non-negative mean curvature, then the ball Ω is just the hemisphere and one has:

q 1 (Ω) ≥ nH 1 -(1 -RH) n ≥ nH
q 1 (Ω) ≥ q 1 ( Ω) = 2|S n-1 | |S n | . ( 10
)
with equality if and only if Ω = Ω. Finally, Corollary 3c implies that, for all domains in H n with mean curvature bounded below by 1, one has the simple bound

q 1 (Ω) > n -1, (11)
and the equality is asymptotically approached by the hyperbolic ball of radius r, as r → ∞: this is seen immediately by estimating the isoperimetric ratio of hyperbolic balls (but see also Theorem 13).

The lower bound ( 11) is a kind of Mc Kean inequality for the eigenvalue q 1 (Ω) (the original Mc Kean inequality states that the first eigenvalue of the Laplacian for the Dirichlet boundary conditions satisfies the bound λ 1 (Ω) ≥ (n -1) 2 /4 for all domains in H n ).

Lower bounds by the inner radius

Perhaps, a more interesting feature of the main lower bound ( 9) is its dependence on the inner radius. We first illustrate this fact on a special case (see Theorem 12b for a proof).

Corollary 5. Assume that Ω has non-negative Ricci curvature and that ∂Ω has nonnegative mean curvature. Then:

q 1 (Ω) ≥ 1 R .
Equality holds for flat cylinders, that is, for all Riemannian products

Ω = N × [0, 2R]
where N is any closed manifold and R > 0.

Remark 6. Corollary 5 applies to mean-convex (in particular, convex) Euclidean domains, and in that case it improves the bound (5). In fact, just observe that, if the strip between two parallel planes contains Ω, it contains a ball of maximal radius inside Ω. Hence w Ω ≥ 2R which implies

q 1 (Ω) ≥ 1 R ≥ 2 w Ω .
However, very often one has 1/R > 2/w Ω : for example, if Ω is close to the equilateral triangle circumscribed to the unit disk one has w Ω ∼ 3 while 2R ∼ 2.

We then observe the following rough, general estimate.

Corollary 7.

Let Ω be a domain with curvature bounds (K, H). Assume that R ≤ 1.

Then:

q 1 (Ω) ≥ c R .
for a positive constant c depending only on K, H and n = dim Ω.

For the proof, let R ∈ (0, ∞] be the first positive zero of Θ: then we know that R ≤ R (see Proposition 14). Let C be the maximum value of Θ on the interval [0, min{1, R}] (note that C depends only on K, H and n). Since R 0 Θ(r) dr ≤ RC, the corollary follows from Theorem 2 by taking c = 1/C. In conclusion, if the Ricci curvature of Ω and the mean curvature of ∂Ω are uniformly bounded below, then q 1 (Ω) becomes larger and larger as the inner radius R tends to zero. With that in mind, the following general principle holds: Remark 8. Thin domains have large first eigenvalue.

Finally, we remark that a lower bound of the mean curvature is essential in order to have a positive, uniform lower bound of q 1 (Ω). In fact, let B r be the ball of radius r in the unit sphere S n . As r → π (the diameter of the sphere) we see that |∂B r | tends to zero while the volume of B r approaches the volume of the sphere. The isoperimetric ratio |∂B r |/|B r | tends to zero, and, by (4), so does q 1 (B r ). Note however that the mean curvature of ∂B r tends to -∞ as r → π.

An upper bound of Cheng type

Let M be a manifold with Ricci curvature of M bounded below by (n-1)K, with K ∈ R, and B(x 0 , r) be the geodesic ball in M with center x 0 and radius r. Cheng proves in [START_REF] Cheng | Eigenvalue comparison theorems and its geometric applications[END_REF] the following comparison theorem for the first eigenvalue of the Laplacian under Dirichlet boundary conditions:

λ 1 (B(x 0 , r)) ≤ λ 1 (B K (r)),
where B K (r) is any ball of radius r in M K , the space form of constant curvature K. It turns out that the same fact holds for q 1 . Precisely:

Theorem 9. Assume that the Riemannian manifold M has Ricci curvature bounded below by (n -1)K. Then, for all x 0 ∈ M and for all r less than the injectivity radius of M at x 0 , we have:

q 1 (B(x 0 , r)) ≤ q 1 (B K (r)),
where B K (r) is any ball of radius r in M K .

The assumption on the injectivity radius is done to insure that the ball B(x 0 , r) has smooth boundary. For the proof, see Section 2.3.

Boundary integral estimates

The proof of Theorem 2 is obtained by estimating the isoperimetric ratio (that is, the quantity ∂Ω h/ Ω h) of any non-negative subharmonic function h on Ω. Precisely, the following estimate holds.

Theorem 10. Let Ω be a compact domain with smooth boundary. Assume that Ω has curvature bounds (K, H). If h is any non-trivial, non-negative subharmonic function on Ω (that is, h ≥ 0 and ∆h ≤ 0 on Ω), then

∂Ω h Ω h ≥ 1 R 0 Θ(r) dr , ( 12
)
where Θ is defined in [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF].

If Ω is a ball in the space form M K and h is harmonic, then equality holds; in that case [START_REF] Wang | Sharp bounds for the first non-zero Stekloff eigenvalues[END_REF] reduces to the classical mean-value lemma for harmonic functions on balls:

∂Ω h Ω h = |∂Ω| |Ω| .
The proof of Theorem 10 is given in Section 2. In Theorem 3.1 of [START_REF] Guerini | Eigenvalue and gap estimates for the Laplacian acting on p-forms[END_REF], P. Guerini and the second author give a lower bound of ∂Ω h/ Ω h when h is non-negative and satisfies ∆h ≤ µh for µ less than a suitable positive constant. However, this bound is sharp only when K = H = 0. Let us sketch the idea of the proof of [START_REF] Wang | Sharp bounds for the first non-zero Stekloff eigenvalues[END_REF]. Introduce the function

F : [0, ∞) → R defined as F (r) = {ρ>r} h, (13)
where {ρ > r} denotes the set of points of Ω at distance greater than r from the boundary. As h is nonnegative and subharmonic, the function F is shown to satisfy the following simple differential inequality:

F ′′ - Θ ′ Θ F ′ ≥ 0
in the sense of distributions. Integrating this inequality twice one gets the desired bound. All this will be explained in detail in the next section. This symmetrization procedure has been used in [START_REF] Savo | A mean-value lemma and applications[END_REF] and [START_REF] Guerini | Eigenvalue and gap estimates for the Laplacian acting on p-forms[END_REF] to get spectral estimates in various contexts.

Proof of Theorem 2

Let f be a first eigenfunction of problem (1), and set g = ∆f . Then g is harmonic on Ω and on ∂Ω one has g = q 1 (Ω)∂f /∂N. Now:

Ω g 2 = Ω (∆f ) 2 = q 1 (Ω) ∂Ω ∂f ∂N 2 = 1 q 1 (Ω) ∂Ω g 2 ,
that is

q 1 (Ω) = ∂Ω g 2 Ω g 2 .
As g is harmonic, the function h = g 2 is subharmonic and non-negative. Then, applying Theorem 10 to h we immediately obtain the assertion:

q 1 (Ω) ≥ 1 R 0 Θ(r) dr .
Finally, if Ω is a geodesic ball in M K , then the right-hand side of the previous inequality is the isoperimetric ratio |∂Ω|/|Ω| (see ( 22)). On the other hand, by ( 4), one has also q 1 (Ω) ≤ |∂Ω|/|Ω|, hence equality holds.

2 Proofs

Known facts on the distance function

The proof of Theorem 10 is based on the calculation of the second derivative, in the distributional sense, of the function F defined in (13). This calculation was done in [START_REF] Savo | A mean-value lemma and applications[END_REF] and involves the Laplacian of the distance function to the boundary: see Proposition 11 below. In this section we state the results we need from [START_REF] Savo | A mean-value lemma and applications[END_REF]; for convenience of the reader, we recall the main arguments of the proof in the Appendix below.

Let Ω be a compact domain with smooth boundary and let ρ : Ω → [0, ∞) be the distance function to the boundary: ρ(x) = dist(x, ∂Ω). Then ρ is only Lipschitz regular, and |∇ρ| = 1 almost everywhere on Ω. The set where ρ is singular (i.e. not C 1 ) is called the cut-locus and is denoted by Cut(∂Ω): it is closed, and has measure zero in Ω. Since the boundary of Ω is assumed smooth, the function ρ is smooth on a strip near ∂Ω, precisely on ρ -1 0, inj(∂Ω) where inj(∂Ω) = dist(∂Ω, Cut(∂Ω)) is the injectivity radius of the normal exponential map. We define the distributional Laplacian of ρ in the usual way:

(∆ρ, f ) . = Ω ρ∆f,
for all smooth functions f compactly supported in the interior of Ω, where (•, •) denotes the duality between a test-function and a distribution. We want to estimate ∆ρ from below. To that end, assume that Ω has Ricci curvature bounded below by (n -1)K, that ∂Ω has mean curvature bounded below by H and let Θ be the function defined in [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF].

Then one has the following bound, proved in Lemma 3.5 of [START_REF] Savo | A mean-value lemma and applications[END_REF]:

∆ρ ≥ - Θ ′ Θ • ρ (14)
as distributions on Ω. This means that, if f is any non-negative test-function:

(∆ρ, f ) ≥ - Θ ′ Θ • ρ, f = - Ω f Θ ′ Θ • ρ . (15)
We remark that if Ω is a ball in a space form M K then the distance function ρ is smooth (except at the center of the ball), the function ∆ρ is radial (i.e. it depends only on ρ) and equality holds in (14). For r ≥ 0, we denote by {ρ > r} (respectively, {ρ = r}) the set of points of Ω at distance greater (respectively, equal) to r from ∂Ω. Let φ be a smooth function on Ω and let F : [0, ∞) → R be defined by:

F (r) = {ρ>r} φ.
Then F is Lipschitz regular and vanishes for r ≥ R, where R is the inner radius of Ω. By the formula of co-area, since |∇ρ| = 1 almost everywhere, we have:

F (r) = ∞ r {ρ=s} φ dH n-1 ds,
where dH n-1 is the (n -1)-th dimensional Hausdorff measure of the level set {ρ = s}. Then:

F ′ (r) = - {ρ=r} φ dH n-1 , (16)
a.e. on [0, ∞). Note that, as ρ is smooth near the boundary, the function F is smooth near r = 0: precisely on the interval 0, inj(∂Ω) , where inj(∂Ω) is the injectivity radius of the normal exponential map. On that interval one has F ′ (r) = -{ρ=r} φ because dH n-1 coincides with the Riemannian measure of the smooth hypersurface {ρ = r}.

Here is the main result we need.

Proposition 11. ( [START_REF] Savo | A mean-value lemma and applications[END_REF], formula (7) p. 517) Let φ be a smooth function on Ω, and F (r) = {ρ>r} φ. Then, as distributions on (0, ∞):

F ′′ (r) = - {ρ>r} ∆φ + ρ ⋆ (φ∆ρ), ( 17
)
where ρ ⋆ (φ∆ρ) denotes the distribution on (0, ∞) given by the push-forward of φ∆ρ by ρ.

This means that, if ψ is a test-function on (0, ∞), then, by definition:

(ρ ⋆ (φ∆ρ), ψ) . = (φ∆ρ, ψ • ρ) = ∆ρ, φ(ψ • ρ) ,
where on the right hand side (•, •) denotes the duality in Ω. We remark that ψ • ρ is only Lipschitz regular; but (see the Appendix below) ∆ρ is a zero-order distribution (that is, a Radon measure): then, the right-hand side is well-defined because it is just the integral of the continuous function φ(ψ • ρ) with respect to the measure ∆ρ.

Proof of Theorem 10

Fix a non-negative subharmonic function h on Ω, and consider the function

F (r) = {ρ>r} h.
Let ψ be a non-negative test-function on (0, ∞). Then, by ( 14), the co-area formula and ( 16) we see that:

(ρ ⋆ (h∆ρ), ψ) = ∆ρ, h(ψ • ρ) ≥ - Θ ′ Θ • ρ, h(ψ • ρ) = - Ω h(ψ • ρ) Θ ′ Θ • ρ = - ∞ 0 ψ(r) Θ ′ Θ (r) {ρ=r} hdH n-1 dr = ∞ 0 ψ(r) Θ ′ Θ (r)F ′ (r) dr = Θ ′ Θ F ′ , ψ that is ρ ⋆ (h∆ρ) ≥ Θ ′ Θ F ′ (18)
as distributions on the half-line. We apply Proposition 11 to φ = h; as ∆h ≤ 0 we conclude by (18) that

F ′′ - Θ ′ Θ F ′ ≥ 0 (19)
as distributions on (0, ∞).

Now set a = Ω h and b = ∂Ω h. As {ρ > r} is empty for r ≥ R, we have:

F (0) = a, F ′ (0) = -b, F (r) = 0 if r ≥ R.
We first assume that Θ(R) > 0; we know that Θ is positive on [0, R] by Proposition 14 in the Appendix and then the function

G = F ′ Θ is integrable. By (19), the distribution G ′ is non-negative: it is a simple fact that then G is non-decreasing on a set of full measure in [0, R].
As G is regular near r = 0 and Θ(0) = 1 we must have G(r) ≥ G(0) = -b a.e. on [0, R], which implies that

F ′ (r) + bΘ(r) ≥ 0 ( 20
)
almost everywhere. We integrate (20) on (0, R) to obtain b R 0 Θ(r) dr ≥ a. Finally:

∂Ω h Ω h = b a ≥ 1 R 0 Θ(r) dr , (21)
which gives the assertion. Now assume that Θ(R) = 0: then, again by Proposition 14, Ω is a ball of radius R in M K . In that case, the injectivity radius is just the radius of the ball, the function F is smooth on [0, R) and the proof carries over as well. Finally, assume that h is harmonic and Ω is a ball in M K : we have equality in ( 14) and then we have equality at every step of the proof, so that

∂Ω h Ω h = 1 R 0 Θ(r)dr .
Note that applying the above to h = 1 we obtain:

1 R 0 Θ(r)dr = |∂Ω| |Ω| . (22)
In fact, Θ is the density of the Riemannian measure in normal coordinates around ∂Ω (see the Appendix below).

Proof of Theorem 9

Assume that M is a complete Riemannian manifold with Ricci curvature bounded below by (n -1)K, and let M K be the space form of constant curvature K. For any x 0 ∈ M the well-known Bishop-Gromov inequality (see for example [START_REF] Petersen | Riemannian Geometry[END_REF], Lemma 1.6) states that the function:

V (r) = |B(x 0 , r)| |B K (r)|
is non-increasing on (0, ∞), and tends to 1 as r → 0 (here

B K (r) is any ball of radius r in M K ). If r < inj(x 0 ) we have d dr |B(x 0 , r)| = |∂B(x 0 , r)|, d dr |B K (r)| = |∂B K (r)| and as V ′ (r) ≤ 0 we obtain: |∂B(x 0 , r)| |B(x 0 , r)| ≤ |∂B K (r)| |B K (r)| .
Now, from inequality (4) (which is an equality for geodesic balls in a space form) we have:

q 1 (B(x 0 , r)) ≤ |∂B(x 0 , r)| |B(x 0 , r)| ≤ |∂B K (r)| |B K (r)| = q 1 (B K (r))
which proves the assertion.

Explicit estimates

We consider the cases where K = {0, -1, 1}. First, assume that Ω is a domain with non-negative Ricci curvature, that is, K = 0. One has: Θ(r) = (1 -Hr) n-1 .

The integral R 0 Θ can be explicitly computed, and from Theorem 2 one gets easily: Theorem 12. Let Ω n be a domain with non-negative Ricci curvature and mean curvature bounded below by H ∈ R. Let R be the inner radius of Ω. a) If H > 0 then

q 1 (Ω) ≥ nH 1 -(1 -RH) n . b) If H ≥ 0 then q 1 (Ω) ≥ 1/R. c) If H = -|H| < 0, then q 1 (Ω) ≥ n|H| (1 + R|H|) n -1 .
We already observed that a) is sharp. Observe that by Proposition 14 one has 1-RH ≥ 0, with equality only for the ball: hence q 1 (Ω) ≥ nH.

We remark that b) is also sharp. In fact, let N be a closed manifold and Ω = N × (0, 2R) with the product metric (Ω is often called a flat cylinder). Note that the inner radius of Ω is exactly R. Now |∂Ω| = 2|N| and |Ω| = 2R|N|, so that |∂Ω|/|Ω| = 1/R. By the upper bound (4) we have q 1 (Ω) ≤ 1/R and then, by b), we have equality.

We now take K = -1, so that the estimates below apply in particular to hyperbolic domains.

Theorem 13. Let Ω n be a domain with Ricci curvature bounded below by -(n -1) and mean curvature bounded below by H ∈ R.

a) If H ≥ 1 then q 1 (Ω) ≥ n -1 1 -e -(n-1)R . In particular:

q 1 (Ω) > n -1. b) If H ≥ 0 then q 1 (Ω) ≥ n -1 e (n-1)R -1 . c) If H = -|H| ≤ 0 then q 1 (Ω) ≥ n -1 (1 + |H|) n-1 (e (n-1)R -1)
.

For the proof, we observe that K = -1 and so Θ(r) = (cosh r -H sinh r) n-1 . To prove a) we use the fact that, if H ≥ 1, then cosh r -H sinh r ≤ e -r , and then we carry out integration. If H ≥ 0 we use the inequality cosh r ≤ e r , and if H = -|H| ≤ 0 we use the inequality cosh r -H sinh r ≤ (1 + |H|)e r . Finally, if the Ricci curvature is bounded below by n -1, then Θ(r) = (cos r -H sin r) n-1 , and one can get explicit estimates as well. We omit further details.

Proof of Corollary 3

Let Ω be a domain with curvature bounds (K, H) and let Θ be as in [START_REF] Payne | Some isoperimetric inequalities for harmonic functions[END_REF]. By Theorem 2 we have:

q 1 (Ω) ≥ 1 R 0 Θ . ( 23
)
Under the given assumptions on K and H, there is a unique ball Ω in M K with boundary of (constant) mean curvature H. By its definition Ω has also curvature bounds (K, H) and by the second part of Theorem 2 we have

q 1 ( Ω) = |∂ Ω| | Ω| = 1 R 0 Θ , ( 24
)
where R is the radius of Ω. Now, by Proposition 14, R is also the first zero of Θ, and one has R ≥ R with equality if and only if Ω is isometric to Ω. Comparison of ( 23) and (24) leads to the inequality q 1 (Ω) ≥ q 1 ( Ω) with equality iff Ω is isometric to Ω.

Appendix

We outline the main arguments and definitions in [START_REF] Savo | A mean-value lemma and applications[END_REF]

Cut(∂Ω) = {exp x (c(x)N x ) : x ∈ ∂Ω}
It is known that the cut-locus has measure zero in Ω; denote by

Ω reg = Ω \ Cut(∂Ω)
the set of regular points of ρ. Then, ρ is C ∞ -smooth on Ω reg and there one has |∇ρ| = 1. Consider the set

U = {(r, x) ∈ [0, ∞) × ∂Ω : 0 ≤ r < c(x)}.
The pair (r, x) gives rise to the normal coordinates of a regular point of Ω. The map Φ(r, x) = exp x (rN x ) is a diffeomorphism Φ : U → Ω reg , and if we pull back the Riemannian volume form dvol g of Ω by Φ, we can write Φ ⋆ (dvol g )(r, x) = θ(r, x)dr dx, where dx denotes, for short, the induced volume form of ∂Ω. The function θ is then the density of the Riemannian volume form in normal coordinates. Obviously θ is positive on U, and θ(0, x) = 1 for all x ∈ ∂Ω. So, for all integrable functions f on Ω:

Ω f = Ωreg f = ∂Ω c(x) 0 θ(r, x)f (r, x) dr dx (25)
where we identify a regular point of Ω with its pair of normal coordinates.

The map θ extends by continuity on Ū , and we will define

θ(c(x), x) . = lim r→c(x) - θ(r, x).
We let ∆ reg ρ = ∆(ρ| Ωreg ), be the regular part of the Laplacian of the distance function. It is an L 1 -function on Ω reg .

In normal coordinates, one has the formula (see [START_REF] Gallot | Inégalités isopérimétriques et analytiques sur les variétés riemanniennes[END_REF] p. 40):

∆ reg ρ(r, x) = - 1 θ ∂θ ∂r (r, x). (26)
Geometrically, ∆ reg ρ(r, x) is equal to (n -1)-times the mean curvature of the level set {ρ = r} at the regular point (r, x).

In what follows, we assume that Ω has curvature bounds (K, H); that is, its Ricci curvature is bounded below by (n-1)K and its boundary ∂Ω has mean curvature bounded below by H. Recall that Θ(r) = (s ′ K (r) -Hs K (r)) n-1 , where s K (r) has been defined in [START_REF] Kuttler | Inequalities for membrane and Stekloff eigenvalues[END_REF]. Then, the classical volume estimates of Heintze and Karcher (for an independent derivation using a Laplacian comparison argument see p. 41 of [START_REF] Gallot | Inégalités isopérimétriques et analytiques sur les variétés riemanniennes[END_REF]) imply that, at all regular points (r, x) ∈ U one has

∆ reg ρ(r, x) = - 1 θ ∂θ ∂r (r, x) ≥ - Θ ′ (r) Θ(r) (27)
This implies in particular that, on U, one has θ(r, x) ≤ Θ(r). As a consequence of this, and a detailed analysis of the focal points of ∂Ω, one can prove (see Theorem A of [START_REF] Kasue | Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary[END_REF]):

Proposition 14. Assume that Ω has curvature bounds (K, H), and let R ∈ (0, ∞] be the first positive zero of the function r → s ′ K (r) -Hs K (r). Then R ≥ R, where R is the inner radius of Ω. Moreover, equality holds if and only if Ω is a ball of radius R in the space form M K . Then, the function Θ is smooth and positive on the interval [0, R) and Θ(R) = 0 only when Ω is a geodesic ball in M K .

As a distribution on Ω, the Laplacian ∆ρ splits as follows:

∆ρ = ∆ reg ρ + ∆ cut ρ, (28)
where the regular part ∆ reg ρ is L 1 and satisfies (27) and where ∆ cut ρ is the distribution supported on the cut-locus and defined by: (∆ cut ρ, f ) = ∂Ω θ(c(x), x)f (c(x), x) dx.

Note that ∆ cut ρ is a non-negative distribution: as such it can be identified with a nonnegative Radon measure on Ω. Hence ∆ρ is a (signed) Radon measure on Ω, and can be tested on any continuous function. To verify the splitting in (28) (proved in Lemma 2.1 and Section 3.2 of [START_REF] Savo | A mean-value lemma and applications[END_REF]), we test ∆ρ on a smooth function, use normal coordinates, and integrate by parts. From the above splitting, the positivity of ∆ cut ρ and inequality (27) one then obtains the bound ∆ρ ≥ -Θ ′ Θ • ρ in (14). Finally, the proof of Proposition 11 is done using integrating by parts on the level domains {ρ > r}. We remark that the arguments in [START_REF] Savo | A mean-value lemma and applications[END_REF] extend to the distance function to any submanifold of a Riemannian manifold.

Remark 15. When Ω is a ball of radius R in M K , one sees that, by the symmetries of M K , the density θ(r, x) is independent on x ∈ ∂Ω and in fact one has θ(r, x) = Θ(r) for all r, x; moreover c(x) = R for all x ∈ ∂Ω, and Θ(R) = 0. The cut-locus reduces to a point; this is the unique focal point of ∂Ω and coincides with the center of the ball: hence ∆ cut ρ = 0. We have equality in (27) hence also in (14) and from (25) one verifies that

because 1 -

 1 RH ≥ 0, with equality only for Euclidean balls (see Theorem 12a).

  leading to Proposition 11. All these facts are proved in Sections 2 and 3.2 of[START_REF] Savo | A mean-value lemma and applications[END_REF]. Let N x be the inner unit normal at x ∈ ∂Ω, and consider the geodesic normal to ∂Ω and starting at x: γ x (t) = exp x (tN x ), where t ranges in a suitable interval. The cut-radius of x ∈ ∂Ω is the positive number c(x) defined in the following way: -The geodesic γ x (t) minimizes the distance to ∂Ω if and only if t ∈ [0, c(x)]. The map c : ∂Ω → [0, ∞) is continuous; moreover, since ∂Ω is smooth, c is positive (and inf ∂Ω c is called the injectivity radius of the normal exponential map). The cut-locus Cut(∂Ω) is the closed subset of Ω defined by:

Authors addresses:

Simon Raulot Laboratoire de Mathématiques R. Salem UMR 6085 CNRS-Université de Rouen Avenue de l'Université, BP.12 Technopôle du Madrillet 76801 Saint-Étienne-du-Rouvray, France E-Mail: simon.raulot@univ-rouen.fr

Alessandro Savo

Dipartimento SBAI, Sezione di Matematica Sapienza Università di Roma Via Antonio Scarpa 16 00161 Roma, Italy E-Mail: alessandro.savo@sbai.uniroma1.it


Imagined Futures: Why business studies dominate the higher education choices of second generation Turks in the Netherlands?

Keywords: Netherlands, Second Generation, higher education access, subject choice, Turks, ethnic minority students ethnic minority students, higher education access, Netherlands, second generation, subject choice, Turks

Imagined Futures: Why business studies dominate the higher education choices of second generation Turks in the Netherlands?

In terms of where, ethnic minority participation patterns seem to point towards the existence of an internal system of stratification within the higher education sector (interlinked with the compulsory system of education) which is to ensure that nontraditional students are channelled towards specific institutions. But not only are ethnic minority students enrolled into different types of higher education institutions; they also follow different subject choices. In spite of the challenges of accessing higher education, ethnic minority students are overwhelmingly driven to studying medicine, law or IT in a fairly uniform manner across countries [START_REF] Connor | Social Class and Higher Education: issues affecting decisions on participation by lower social class groups[END_REF][START_REF] Boliver | Social Inequalities of Access to Higher Status Universities in the UK: The Role of University Admissions Decisions[END_REF]Modood 2006;Kristen, Reimer and Kogan 2008). However, in contrast to the British or the German experience, Turkish students in Holland are choosing business studies in mass numbers while leaving the proportion of those enrolled to study medicine and law relatively low. As the choice of economics seems to be very particular to this group in this particular country, the aim of this paper is to investigate what are the factors behind the second generation Turks' overrepresentation in the field of business studies. As subject choices are inextricably linked to institutional choices in the Dutch educational setting one cannot explain the 'why' without looking into the accessibility of certain options for a non-traditional student body.

Access to higher education for non-traditional students. International overview

National educational systems play a significant role in determining whether one can reach the top of the educational ladder. In stratified educational systems [START_REF] Allmendinger | Educational systems and labour market outcomes[END_REF], such as the German and the Dutch, for example, tracking takes place at the beginning of compulsory schooling; whereas in more comprehensive systems (e.g. UK) selection takes place much closer to the entry to tertiary education. But how do these Even after the formal abolition of the binary divide in the British higher education system, the uneven distribution of the student body pointed at persisting inequalities within the sector. While in the UK context of comprehensive education students can 'theoretically' apply to any higher education institution, their practical chances of enrolment can be very restricted. Access here is subject to prior scholastic achievement often identified by the so called 'A levels', which together with silent factors (such as the type of secondary school attended and the cultural and social capital of parents) play a significant role in shaping higher education access. High tuition fees and the selective nature of university admissions, together with institutional racism at the most prestigious institutions often result in low rates of ethnic minority participation. The undying image of the typical university student as a white middle class male at top institutions also leads to students' self-selection. As many ethnic minority students voice their concerns over 'not fitting in' and losing their 'authenticity' at high ranking universities (Bird 1996;Reay et al. 2001;Reay, David and Ball 2005) they end up opting for the post-1992 institutions (former polytechnics), instead of the older, long established universities (Reay et al. 2001;Connor et al. 2003;2004;Reay, David and Ball 2005;Modood 2006).

By contrast in countries where tracking is an integral part of schooling, educational systems channel students into different school types envisaging full, limited or no access to higher education early on. In these contexts it is often observed that immigrant children do not do particularly well during compulsory schooling and are often penalised by their language difficulties, teacher racism and/or the lack of cultural capital and so are channelled to the least selective school types (Kristen, Reimer and Within the Dutch context, strong channelling is built into the system, directing pupils through certain routes after the first selection point which takes place at the age of ten. Higher education cannot be viewed as completely open given that direct access to universities is granted only to a limited few who attended the most selective type of secondary schools (i.e. VWO/gymnasium). The rest of the students have access to the 'second tier' of higher education, namely professional colleges, with the option of continuing into university level subsequently. Thus tracking is the main reason why the proportion of second generation Turks is relatively low at university level whereas they are well represented in professional colleges within the Dutch higher education system.

Within the German setting, tracking takes place during compulsory schooling in similar manner to the Netherlands. However, once the necessary secondary school certificates are obtained, ethnic minority students find the doors of all types of higher education institutions open to them. As a result of this there is a growing proportion of Turkish students within the German higher education system who enrol into universities instead of the 'Fachhochschulen'. The latter are seen as institutions with a rank far behind the prestige of the university as university degrees rank the highest among all institutions back in Turkey (Kristen, Reimer and Kogan 2008).

Considering the hierarchical structure of the higher education sector across countries allows us to contextualise the higher education choices of ethnic minority students within an international perspective. Although the massification of higher education opened up the access to a non-traditional student body, higher education sectors successfully managed to retain their exclusivity by opening up avenues to the least prestigious institutions only (e.g. in Holland and the UK). The binary system of higher education operating in many developed countries provided a well established avenue for channelling ethnic minority students into qualitatively different types of higher education institutions equipping them with different and separate experiences and economic returns (Reay et al. 2001). But even where higher education institutions maintained the illusion of open access (as in Germany), they were resting assured that successful enrolment would not necessarily ensure successful completion. Ethnic minority students arriving from academically less strong schools are at risk of dropping out long before graduation; thus the apparently different educational systems uniformly achieve social reproduction.

In the following I will focus on the specific case of Turkish students within the Dutch educational system in order to demonstrate the effects of institutional access on subject choices.

The Dutch higher education system

The Netherlands has a binary system of higher education consisting of (research oriented) universities and institutions of professional higher education (HBO 1 ). While universities offer a wide range of disciplines, HBOs prepare students for careers limited to certain sectors such as agriculture; engineering; economics; health care; arts; social welfare and teacher training (Marginson et al. 2008). The three tier system based on the Bologna process has been introduced in the Dutch higher education from 2002/3. In the new system university students can graduate and receive their Bachelor degree within three years. For the same qualification HBO students have to study for four years. After the first year of the HBO, students can get a certificate called 'propedeuse' which allows them to access university courses unless they wish to complete their Bachelor at the HBO level. After gaining The initial choice between university and higher professional education is made prior to the higher education application as access to the Dutch higher education system is conditional upon prior qualifications. While pre-university education (VWO 2 certificate) is strictly required for university enrolment, the enrolment into the HBO is open to all students who successfully completed their senior secondary education (VWO, HAVO or MBO certificate). However, students attending the least selective secondary school tracks (such as VBO, MAVO or VMBO in the new system) have no direct access to any higher education institution; they have to obtain a certificate from a senior secondary education institution first (MBO or HAVO) in order to qualify for enrolment.

Once the necessary prior qualifications are obtained higher education institutions are not allowed to pre-select students with regards to their prior scholastic achievement or potential. Only a few sectors have so called 'numerous fixus', such as medicine and dentistry, where the number of enrolled students is set. To all other sectors enrolment is ensured by application from the side of secondary school leavers, therefore (bearing in mind the above outlined limitations) it's up to the student which institution or degree programme he or she selects.

The situation of the Turkish second generation in the Netherlands

The descendants of Turkish guest workers are considered to be non-traditional students in the Dutch higher education system given their generally low social class background, their ethnic minority status and the low levels of parental human capital (the vast majority being the first of their family to enter higher education). Their parents arrived in Holland as labor migrants during the 1960s and 1970s and were subsequently employed in low skilled jobs requiring little or no prior schooling. Once settled in the Netherlands, it was expected, that their children, the second generation, would do considerably better in terms of labor market participation and educational attainment.

However, the descendants of Turkish guest workers often found themselves at the bottom end of the educational ladder (Pásztor 2010). While ethnic minority children from the former colonies and other Western European countries gained access to higher education in similar proportion to the native Dutch students (table 1), the children of Turkish guest workers stayed significantly under-represented at higher education level (Wolff and Pásztor 2010). In spite of the positive changes over time recent statistics see less than one third (29 per cent) of ethnic minority youth of Turkish origin (from the 18-20 year old cohort) enrolled into higher education; a figure approximately half of that for natives (54 per cent) (Jennisen 2007). The more prestigious university education accounts for only 16 per cent of the total enrolment as the vast majority of the Turkish second generation is studying at professional colleges instead (table 1).

[table 1 about here] The preference for HBOs instead of universities is not a deliberate choice for ethnic minority students, as prior schooling restricts their access to university education.

As many were approaching higher education indirectly, through the so called 'long route' (i.e. climbing up from the least selective secondary school types), professional colleges were the most accessible if not the only, option.

With respect to subject choices, higher education statistics clearly point to second generation Turks' overrepresentation in applied subjects compared to sciences or As the question of why students actually chose this particular field where left largely unexplained, in the following, I will be looking into the aspirations and career choices of these young people bearing in mind the limitations of institutional access.

Data and Methodology

The study relies on qualitative data which were gathered in the form of in-depth interviews in the city of Amsterdam. Participants were selected from the TIES survey 5 .

All respondents were second generation Turks aged between twenty and thirty five years, who were born in the Netherlands having at least one parent born in Turkey. The interviewees have been purposefully chosen in order to ensure a balanced sample regarding gender, age, parental education, prior schooling and the type of higher education institution attended. While there was an attempt to reach most respondents who were higher education students or graduates at the time of the survey, sixteen young people agreed to be re-interviewed qualitatively. The interviews ranged from one to two and half hours. They were conducted in English, were fully transcribed and were subsequently analysed in Atlas.ti. The names of the interviewed persons are kept confidential in order to ensure their anonymity. From the question on 'who goes where, and why?' the where has been discussed above when pointing to ethnic minority students' overrepresentation at professional colleges.

Why business studies? Ethnic minority options and choices

While prior tracking is the main factor behind institutional 'choices', we still need to explain why exactly business studies rank first as their subject choice.

Heath and his colleagues (Heath, Rothon and Kilpi 2008) in their comprehensive review of recent research on the second generation in ten European countries argued that the preference for applied subjects is a sign of 'risk-aversion' in the educational choices of ethnic minority students. However, they also maintained that the preference for vocational subjects is a consequence of the discriminatory practices these young people expect to face at entry into the labour market thus they limit their choices to occupations with relatively high probability of securing employment (Heath, Rothon and Kilpi 2008). In addition to this, it has been argued that the study of law and medicine would allow Turkish students to take up self-employment within their ethnic community thus boosting their employment prospects (Kristen, Reimer and Kogan

2008).

However, the explanation of discriminatory practices hasn't been proven by previous research in the Dutch context. Tesser and Dronker's (2007) study showed that the chances of being in a particular occupational class were the same for the second generation as for their native peers in Holland, with the exception of the salariat which remained more closed to ethnic minorities. The current study didn't find discrimination affecting the choices of second generation Turks either, as the interviewed young people were not giving voice to such concerns.

One has to be reminded of the fact that these young people had all been employed since they were fifteen-sixteen so they had firsthand experience of the Dutch labour market. They usually worked in low skilled part-time jobs where discrimination Still, when choosing the field, many students have to abandon the parental idea of studying medicine due to the effects of early tracking preventing them from accessing university courses. Having attended the 'right 'secondary school a selected few try to meet the parental aspirations and consider the medical profession, but discover that becoming a doctor is not the easiest way to get an education: 'To be a doctor it's a long, so long way to be. I thought, no, that's not something for me, I do something in four years not in eight or ten years. So I chose different way to do education ' [20,female,HBO,economics]. Thus having considered the costs, benefits and the probabilities of success many Turkish students abandon the field of medicine [START_REF] Breen | Explaining educational differentials: Towards a formal rational action theory[END_REF]. Still, the prestige of this field is so deeply rooted in the community that it is difficult to go for alternate choices which are often not understood or acknowledged.

They [parents] thought that I would study something like doctor. They really appreciate it in Turkish culture; they see it as the most excellent job or something. And I had the chance to do that, but I didn't, so they thought 'why doesn't she do that?' They were a bit shocked I think.

[20, female, university, economics] Not choosing medicine must be explained somehow to the wider community where fitting in is imperative and where acceptance is generally sought. The following excerpt clearly illustrates how one mother justifies her son's decision not to study medicine illustrating a case of cognitive dissonance reduction 6 . 'My father wanted me to become a doctor but my mother -now -thinks it's for the best that I didn't. She heard there was some doctor in Turkey who wasn't able to rescue someone and then the family of the deceased became very angry. "I don't want my son to end up in such a situation", my mother said, so it's for the best that I didn't go for medicine ' [23, male, university, biology].

Parental aspirations -based on a stereotyped view of doctors brought over from the home country -are resistant to change. But after failing with medicine it is economics which comes to the foreground as a plausible choice. 'My mother worked as a cleaner at a Dutch bank, and there she saw people in suits, and she wanted all her kids to also be like this ' [23, male, university, biology]. Economics with the perceived attractiveness of working in suits presents a clear contrast to the parental experience of hard physical work, late hours and minimum wage. As most interviewees experienced economic hardship in their childhood they (and their parents) perceive a job at a bank or in the government as an appealing alternative to medicine.

The popularity of economics and business studies among ethnic minority choosers is manifold. To start with, while students are very limited in their ideas for future jobs (e.g. doctor, lawyer) economics provides an alternative with a great many advantages: while it offers good labour market prospects in government, business or banking it is accessible at HBO level (doesn't require university education) and is less difficult and lengthy than medicine which would push back the time of marriage and family formation; a very important factor for second generation Turks.

Needless to say, the students also know what the subject entails. Economics is taught within a secondary school curriculum, so for many it is something they know about and could imagine studying further: 'I was good at economics so I thought this is something for me ' [20, female, university, economics]. But studying economics has a specific added value as well, it provides something extra even when compared to In spite of the apparent focus on money and status, and the instrumental approach to studying, second generation Turks are deeply family oriented; not only they are strongly attached to their parents and siblings but they regularly visit their home country and maintain strong ties with their extended family. Thus while getting an education is often defined in terms of material rewards it is frequently overlooked that the money and the 'good job' are not the end but rather the means of 'getting somewhere' for second generation Turks. Getting an education and a good job is just the way in which they can satisfy their needs for material comfort and financial security; other than that they aim for a life fulfilled in their family. The easiest way to a secure and decently paid nine-to-five job in the government, business or bank sector is through the HBO and the field of economics.

I don't have to earn a lot of money as long as I have a normal life. I mean children can be very difficult but what you get back for it is really too valuable to miss. So I want that also for myself and I want to enjoy every minute of it, that is why I want to work with a standard pattern. So that's my plan: to have a job, to have a wife, children, a home, and then life is complete. [23, male, university, economics] The notion of nine-to-five was clearly present in the narratives of young people when talking about the higher education choice. Only Serhat, who graduated in aerospace engineering, defined himself as being an outlier while embarking on the route of self-realization and not going for a nine-to-five career like most of his peers. Serhat's strive for self-realisation would indicate a post-materialist way of thinking which reminds us of the subjective nature of insecurity. Still, Serhat is more of an outlier in the sense that he spoke openly about the importance of making money as the base level of any higher education choice.

Ibrahim, an MA student in chemistry, took a contrasting point of view and was very vocal when regretting his choice of going to university. Having studied chemistry at university level he was reluctant to consider a career in research. Rather, he was longing for a nine-to-five career he could not possibly have in his chosen field.

I just want a normal life, my work is not my life, my work is only part of my life and that is why I have regrets. If I had chosen economics then I would have worked at a normal company, from nine to five.

The annotation suggests that some second generation Turks simply might have 'travelled' too far from their starting point and became scared when seeing its consequences. Not only were these young people moving from one stage of education to another but also from one class to another. While wanting to leave the life of their elders because they were filled with struggles they discovered new struggles for themselves.

The career of a researcher, which Ibrahim pictures in terms of high competition, pressure and endless lab hours not allowing him to spend enough time with his family, in many ways resembles his parents' job history which he originally aimed to avoid. I don't feel that [research] is my thing. Back at the time at VWO I was above average but now I am average student. So when first all the 'geniuses' get a good scholarship I might get a lousy research project. And I don't want that, I don't. I am tired of competing all the time, I don't want to compete anymore; I just want a normal, regular life.

While student narratives clearly indicated that Turkish families settled in Holland begun to put education first (before marriage) on their list of priorities (Pásztor 2010) apparently they did so only to satisfy their need for job -and financial -security.

Thus education didn't really become an end in itself but a means to an end for second generation Turks; namely to enable them to offer more in terms of material comfort and parental attention to their children and family.

Conclusion

The study makes a significant contribution to the debate on ethnic minority choices in showing how the limitations imposed upon accessing different types of HE institutions intertwine with intergenerational aspirations and expectations of Turks living in Holland. Here, the strong parental and community pressure towards degrees in medicine and law is counterbalanced by the lack of access to universities which are the sole providers of such prestigious courses in the receiving society. Thus not having access to university education is one of the reasons behind the differences in subject choices, forcing many Turkish students to adjust their list of preferences with courses accessible at professional colleges only. Here, economics or business comes up as a generally appealing alternative which earns parental approval based on the relatively good labour market return of this education within the Dutch economy. However, this is not the sole reason behind the mass numbers of students in the field of business. The research points to the so far neglected aspect of 'job compatibility' which plays a significant role in their decision making. While their final aim is a 'good job' to be ensured by a degree in business, it is not -only -the potential Education,vol. 19,[3][4] --2010 'Go, go on, and go higher an' higher. Second generation Turks understanding of the role of education and their route to college', British Journal of Sociology of Education, vol. 31, no. 1, pp. 59-70 REAY, DIANE et al. 2001 3 Note: economics and business are used interchangeably throughout the paper. They are meant to include all related and sub-fields such as management; marketing; accounting etc. 4 More than one third of the total Turkish HBO intake is enrolled into this field, compared to one fifth of the university intake from all areas.

5 TIES (The Integration of the European Second Generation) is a survey of second generation Turks, Moroccans, and ex-Yugoslavs across eight European countries including the Netherlands (see www.tiesproject.eu). 6 The theory of cognitive dissonance assumes that people have motivational drive to reduce dissonance by changing, justifying or rationalizing their attitudes, beliefs, and behaviours (Festinger 1957).

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Ethnic and Racial Studies

  the differing patterns of higher education participation for ethnic minority students?

  degree, students can choose between HBO and university for their Master studies.

  detailed look at the most popular degree courses points to a single field which markedly outweighs all other subject choices. Economics, business 3 and its associated fields are the number one choice of first degree courses for second generation Turks, with an astonishing forty five per cent 4 choosing this subject (compared to twenty eight per cent of the native Dutch student body) (Jennissen 2007).

  more overt than at higher levels. While a few indeed recounted such incidents none of them made the connection between these occurrences and the potential for discriminatory practices in later employment. This might be due to the 'perceived' discrepancy between the levels of the jobs they held in comparison to thesomewhat idealised -high status of jobs they were aiming to obtain. This leads us to other possible explanations for second generation Turks striving for high status jobs via careers in economics. Previous research has shown that immigrant parents tend to place great emphasis on obtaining a good education for their children as they themselves were unable to achieve this due to their difficult circumstances(Kao and Tienda 1998; Zhou and Bankston 1998;[START_REF] Brinbaum | The school careers of ethnic minority youth in France: Success or Disillusion?[END_REF] Pásztor 2010,). Correspondingly, Turkish parents expect their children to reach higher than themselves by replacing their unskilled, low paid, manual jobs for significantly better employment prospects: 'My parents expect me to reach a level where I just don't have to do hard labour and work late hours and wouldn't have financial difficulties' [22, male, university, economics].'I don't want to see you cleaning buildings or offices, you have to study and you have to get on a high level' [26, male, HBO, economics]. Understandably, while parents can clearly see a wide range of jobs where they definitely do not want to see their offspring they are much more limited in showing the array of options potentially available to them. The parental drive for qualifications is intertwined with limited knowledge of the Dutch educational system and a lack of knowledge of the job market, thus they often voice unrealistic aspirations for their children.My father came from Turkey -he didn't have any education -so he has this idea in his head, when you are a doctor or a lawyer, that are the good jobs you can have, so he wanted us to study high.[27, female, HBO, nursing] Immigrant parents develop a strongly instrumental view on education where extrinsic rewards have the utmost importance when choosing a field. Parents are driven by the potential earnings and the high prestige of the profession when they encourage their children to pursue a medical career without consulting other options.

  is simply compatible with the futures they imagined having.I got discouraged by a person who tells me that [as a doctor] you basically don't have a social life any more. He said you can get paged up for an emergency when you're just playing with your children; I thought no, that's not for me. I value the time I have with my family.[22, male, HBO, economics]

  If you have higher educational level, you are just thinking in a different way, because you are trying to follow your interest, you are challenging yourself -you are not looking forward to have a manager who tells you to do this and that, you know you are not looking for a job which is from nine-to-five.

  security which pulls students towards this choice. Offering a seemingly perfect balance of work and family life, in line with the notion of nine-tofive, is what fits so well with the imagined futures of Turkish young people in the Dutch society. Jobs 'in suits' in business, government or the banking sector provide a strong contrast to parental employment experiences, as having to work long hours and doing dirty and heavy physical work for a minimal wage is clearly not how the children of immigrants imagine their futures. The futures of second generation Turks are not aimed to centre around their working life but around their families, and economics is the field with the right attributes to enable their dreams to become a reality within the Dutch society.

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	MODOOD, TARIQ 2006 'Ethnicity, Muslims and higher education entry in Britain',

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  MODOOD, TARIQ and ACLAND, TONY 1998 Race and Higher Education, London:

	Policy Studies Institute
	PÁSZTOR, ADÉL 2009 'Different settings, different choosers? Applying Ball's
	framework on the case of second generation Turks', International Studies in Sociology
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	y

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  'Choices of Degree or Degrees of Choice? Class, 'Race' and the Higher Education Choice Process', Sociology, vol. 35, no. 4, pp. 855-74 REAY, DIANE, DAVID, MIRIAM and BALL, STEPHEN J. 2005 Degrees of Choice. Social class, race and gender in higher education. Stoke on Trent: Trentham Books TESSER, PAUL and DRONKERS, JAAP 2007 'Equal Opportunities or Social Closure in the Netherlands', in Anthony Heath and Sin Yi Cheung (eds), Unequal Chances: Ethnic Minorities in Western Labour Markets, Oxford: Oxford University Press WOLFF, RICK and PÁSZTOR, ADÉL 2010 'Equal Opportunity in a Public System. Experiences of Ethnic Minority Students in Dutch Higher Education', in Fred Lazin, Matt Evans and N. Jayaram (eds), Higher Education and Equality of Opportunity: Cross-National Perspectives, Lanham, MD: Lexington Books pp. 271-89 ZHOU, MIN and BANKSTON, CARL. L. 1998 Growing up American. How Vietnamese children adapt to life in the US, New York: Russel Sage Foundation

	Ethnic origin	HE entry (%)	Females (%)	University (%)
	Native Dutch	54	52	26
	Turkey	29	53	16
	Morocco	33	56	13
	Suriname	42	59	17
	Netherlands Antilles and Aruba F Western-migrants (EU) o r Average	36 54 51	60 55 53	21 32 21
	P
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				y

Table 1 .

 1 First time entry into higher education by ethnic origin; 2005 (age cohort: 18-20) (adapted from Jennissen 2007) ADÉL PÁSZTOR is a lecturer in the Department of Social Sciences at Northumbria University. ADDRESS: Department of Social Sciences, Northumbria University, Sandyford Road, Newcastle upon Tyne, NE1 8ST, UK. Email: adel.pasztor@northumbria.ac.uk Notes 1 Similarly to the former polytechnics in the UK the Dutch HBOs provide higher professional education (in Dutch: hoger beroepsonderwijs). As of 2001 there were 56 government funded HBOs and 13 universities, and 61 private HBOs and 8 private universities in the Dutch higher education system (Marginson et al. 2008). 2 VWO (voorbereidend wetenschappelijk onderwijs) is the most selective secondary school track providing pre-university education. HAVO (hoger algemeen voortgezet onderwijs) is a less selective secondary school type offering higher general secondary education (with direct access to HBO but not to university). The MBO (middelbaar beroepsonderwijs) provides vocational training for those coming from the least selective, vocationally oriented secondary school tracks such as the MAVO (middelbaar algemeen voortgezet onderwijs) or the VBO (voorbereidend beroepsonderwijs). The last two types were recently merged into one stream called VMBO (voorbereidend middelbaar beroepsonderwijs). The MBO certificate allows students to access the HBO.

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