THE FIRST CONFORMAL DIRAC EIGENVALUE ON 2-DIMENSIONAL TORI

BERND AMMANN AND EMMANUEL HUMBERT

Abstract. Let M be a compact manifold with a spin structure χ and a Riemannian metric g. Let λ_{g}^{2} be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M, χ) := sup inf \sqrt{λ_{g}^{2}}Vol(M, g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class.

We show that τ(T^{2}, χ) = 2\sqrt{π} if χ is “the” non-trivial spin structure on T^{2}. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2\sqrt{π} at one end of the spin-conformal moduli space.

1 MSC 2000: 53 A 30, 53C27 (Primary) 58 J 50 (Secondary)

Contents

1. Introduction 1 2. The spin-conformal moduli space of T^{2} 3 3. Main results 6 4. Some preliminaries 6 5. Proof of the Main results 8 References 13

1. Introduction

Let (M, g, χ) be a compact spin manifold of dimension n ≥ 2. For any metric g̃ in the conformal class [g] of g, let λ_{1}(D2 ; g̃) be the smallest eigenvalue of the square of the Dirac operator. We define

λ_{min}(M, g, χ) = inf_{g̃∈[g]} \sqrt{λ_{1}(D2 ; g̃)} Vol(M, g̃)^{1/n}.

Several works have been devoted to the study of this conformal invariant and some variants of it [Hij86, Lott86, Bär92, Am03, Am03b, Am03a]. J. Lott [Lott86, Am03] proved that λ_{min}(M, g, χ) = 0 if and only if ker D_{g} ≠ {0}. From [Hij86, Bär92] we deduce λ_{min}(𝕊^{n}) = \frac{n}{2} ωn^{\frac{1}{n}}, where 𝕊^{n} is the sphere with constant sectional curvature 1 and where ω_{n} is its volume. Furthermore, in [Am03, AHM03], we have seen that

λ_{min}(M, g, χ) ≤ λ_{min}(𝕊^{n}) = \frac{n}{2} ωn^{\frac{1}{n}}  (1)

for all Riemannian spin manifolds. Furthermore we define

τ(M, χ) := sup λ_{min}(M, g, χ)

where the supremum runs over all conformal classes on M. Obviously, τ(M, χ) is an invariant of a differentiable manifold with spin structure.

Date: March 2005.

1ammann@iecn.u-nancy.fr, humbert@iecn.u-nancy.fr

We consider it as interesting to determine τ or at least some bounds for τ in as many cases as possible. There are several motivations for studying these invariants λ_{min}(M, g, χ) and τ(M, χ).

Our first motivation is the analogy and the relation to Schoen’s σ-constant, which is defined as

σ(M) := sup inf \frac{∫ Scal_{g} dv_{g}}{Vol(M, g)^{\frac{n−2}{n}}}

where the infimum runs over all metrics in a conformal class g ∈ [g_{0}], and where the supremum runs over all conformal classes.

In the case σ(M) ≥ 0 and n ≥ 3, there is also an alternative definition of the τ-invariant that is analogous to our definition of the τ-invariant. More exactly, in this case

σ(M) := sup inf λ_{1}(L_{g})Vol(M, g)^{2/n},

where λ_{1}(L_{g}) is the first eigenvalue of the conformal Laplacian L_{g} := 4 \frac{n−1}{n−2} Δ_{g} +Scal_{g}. Once again, the infimum runs over all metrics in a conformal class g ∈ [g_{0}], and where the supremum runs over all conformal classes. Many conjectures about the value of the σ-constant exist, but unfortunately it can be calculated only in very few special cases, e.g. σ(S^{n}) = n(n − 1)ω_{n}^{2/n}, σ(S^{n−1} × S^{1}) = n(n − 1)ω_{n}^{2/n}, σ(T^{n}) = 0 and σ(ℝP^{3}) = n(n − 1) (\frac{ω_{n}}{2} )^{2/n}. The reader might consult [BN04] for a very elegant and amazing calculation of σ(ℝP^{3}) and for a good overview over further literature.

For other quotients of the sphere Γ\S^{n}, Γ ⊂ O(n + 1) it is conjectured that

σ(Γ\S^{n}) = n(n − 1) (\frac{ω_{n}}{#Γ})^{2/n}.  (2)

It is not difficult to show that for any metric conformal to the round metric on Γ\S^{n} one has the inequality

λ_{1}(L_{g})Vol(Γ\S^{n}, g)^{2/n} ≥ n(n − 1) (\frac{ω_{n}}{#Γ})^{2/n}. This immediately implies σ(Γ\S^{n}) ≥ n(n − 1) (\frac{ω_{n}}{#Γ})^{2/n}, i.e. the lower bound on σ in (2). However, it is very difficult to obtain the upper bound on σ.

The τ-invariant is not only a formal analogue to Schoen’s σ-constant, but it is also tightly related to it via Hijazi’s inequality [Hij86, Hij91, Hij01]. Hijazi’s inequality implies that if M carries a spin structure χ, then

τ(M, χ)^{2} ≥ \frac{n}{4(n − 1)} σ(M).  (3)

Equality is attained in this inequality if M = S^{n}. Hence, upper bounds for τ(M, χ) may help to determine the σ-constant.

This is one reason for studying the τ-invariant.

Another motivation for studying τ(M, χ) and λ_{min}(M, g, χ) comes from the connection to constant mean curvature surfaces. Let n = 2. If g̃ is a minimizer that attains the infimum in the definition of λ_{min}(M, g, χ), and if Vol(M, g̃) = 1, then any simply connected open subset U of M can be isometrically embedded into ℝ^{3}, (U, g̃) ↪ ℝ^{3}, such that the resulting surface has constant mean curvature λ_{min}(M, g, χ). Vice versa, any constant mean curvature surface gives rise to a stationary point of an associated variational principle. It is shown in [Am03b] that minimizers of λ_{min}(M, g, χ) exist if λ_{min}(M, g, χ) < 2\sqrt{π}.

For the third motivation, let again n ≥ 2 be arbitrary. As indicated above, λ_{min}(M, g, χ) > 0 if and only if ker D_{g} = {0}. Hence, τ(M, χ) > 0 if and only if M carries a metric with ker D = {0}. It follows from the Atiyah-Singer index theorem that any spin manifold M of dimension 4k, k ∈ ℕ with Â(M) ≠ 0 has τ = 0, and the same holds for spin manifolds of dimension 8k + 1 and 8k + 2 with non-vanishing α-genus. C. Bär conjectures [Bär96, BD02] that in all remaining cases one has τ > 0. Using perturbation methods Maier [Mai97] has verified the conjecture in the case n ≤ 4. The conjecture also holds if n ≥ 5 and π_{1}(M) = {e}. Namely, if M is a compact simply connected spin manifold with vanishing α-genus, then building on Gromov-Lawson’s surgery results [GL80] Stolz showed [St92] that M carries a metric g_{+} of positive scalar curvature. Applying the Schrödinger-Lichnerowicz formula we obtain ker D_{g_{+}} = {0}, and hence τ(M, χ) ≥ λ_{min}(M, g_{+}, χ) > 0 for the unique spin structure χ on M. A good reference for this argument is also [BD02], where the interested reader can also find an analogous statement for the case α(M, χ) ≠ 0. The method of Stolz and Bär-Dahl also applies to some other fundamental groups, but the general case still remains open.

In the present article we want to have a closer look at the τ-invariant on surfaces, in particular 2-dimensional tori. The higher dimensional case will be the subject of another publication.

On surfaces the Yamabe operator cannot be defined as above. The Gauss-Bonnet theorem says that the σconstant of a surface does not depend on the metric:

σ(M) = sup inf ∫ 2K_{g} dv_{g} = 4πχ(M).

It was conjectured by Lott [Lott86] and proved by C. Bär [Bär92] that equation (3) also holds in dimension 2. This amounts in showing τ(S^{2}) = 2\sqrt{π}. If M is a compact orientable surface of higher genus, then inequality (3) is trivial.

We will calulate the τ-invariant for the 2-dimensional torus T^{2}. The 2-dimensional torus T^{2} has 4 different spin structures. The diffeomorphism group Diff(T^{2}) acts on the space of spin structures by pullback, and the action has two orbits: one orbit consisting of only one spin structure, the so-called trivial spin structure χ_{tr} and another orbit consisting of three spin structures. The torus T^{2} equipped with the trivial spin structure has non-vanishing α-genus, thus τ = 0. The main result of this article is the following theorem.

Theorem 1.1. Let χ be a non-trivial spin structure on the 2-dimensional torus T^{2}. Then

τ(T^{2}, χ) = 2\sqrt{π} (= λ_{min}(𝕊^{2})) .

More exactly, for a fixed non-trivial spin structure χ we will study λ_{min}(M, g, χ) as a function on the spinconformal moduli space M. We show that it is continuous (Proposition 3.1), and we show that it can be continuously extended to the natural 2-point compactification of M, i.e. the compactification where both ends are compactified by one point each. It will be easy to show that λ_{min}(M, g, χ) → 0 at one of the ends. However, it is much more involved to prove Theorem 3.2 which states that λ_{min}(M, g, χ) → λ_{min}(𝕊^{2}) = 2\sqrt{π} at the other end.

It is evident that Theorem 3.2 implies Theorem 1.1.

For the proof of Theorem 3.2, we have to establish a qualitative lower bound for the eigenvalues. One important ingredient in the proof of Theorem 3.2 is to study a suitable covering of the 2-torus by a cylinder, and to lift a test spinor to this covering. Using a cut-off argument in a way similar to [AB02] we obtain a compactly supported test spinor on the cylinder. After compactifying the cylinder conformally to the sphere S^{2}, we can use Bär’s 2-dimensional version of (3), to prove λ_{min}(M, g, χ) → λ_{min}(𝕊^{2}) = 2\sqrt{π} at the other end.

Theorem 3.2 and Theorem 1.1 should be seen as a spinorial analogue of [Sch91]. In that article, Schoen studies the Yamabe invariant on the moduli space of O(n)-invariant conformal structures on S^{1} × S^{n−1}, n ≥ 3. He shows that at one end of this moduli space, the Yamabe invariant converges to the Yamabe invariant of 𝕊^{n}, and hence σ(S^{1} × S^{n−1}) = σ(𝕊^{n}). Combining this result with the Hijazi inequality and Theorem 1.1, one obtains

Corollary 1.2. Let n ≥ 2. Then

τ(S^{n−1} × S^{1}, χ) = \begin{cases}0 & if n = 2 and if χ is trivial, \\ \frac{n}{2} ω_{n}^{1/n} & otherwise. \end{cases}

The structure of the article is as follows.

In section 2, we define the spin-conformal moduli M space of 2-tori and recall some well known facts. In section 3, we state and explain our results. In sections 4, we recall some preliminaries which will be useful for the proof of Theorem 3.2. In Section 5 the proof is carried out.

Acknowledgement. The authors want to thank the referee for many useful comments.

2. The spin-conformal moduli space of T^{2}

At the beginning of this section we will recall the definition of a spin structure. We will only give it in the case n = 2. For more information and for the case of general dimension we refer to standard text books [Fr00, LM89, Ro88, BGV91]. More details about the 2-dimensional case can be obtained in [AB02] and [Am98, BS92].

Let (M, g) be an oriented surface with a Riemannian metric g. Let P_{SO}(M, g) denote the set of oriented orthonormal frames over M. The base point map P_{SO}(M, g) → M is an S^{1} principal bundle. Let α : S^{1} → S^{1} be the non-trivial double covering, i.e. α(z) = z^{2}. A spin structure on (M, g) is by definition a pair (P, χ) where P is an S^{1} principal bundle over M and where χ : P → P_{SO}(M, g) is a double covering, such that the diagram

\begin{aligned}P × S^{1} & → & P \\ & & & ↘ \\ ↓ χ × α & & ↓ χ & M \\ & & & ↗ \\ P_{SO}(M) × S^{1} & → & P_{SO}(M) \end{aligned}  (4)

commutes (in this diagram the horizontal flashes denote the action of S^{1} on P and P_{SO}(M)). By slightly abusing the notation we will sometimes write χ for the spin structure, assuming that the domain P of χ is implicitly given. Two spin structures (P, χ) and (P̃, χ̃) are isomorphic if there is an S^{1}-equivariant bijection b : P → P̃ such that χ̃ = χ ∘ b.

If g̃ = f^{2}g is a metric conformal to g. Then P_{SO}(M, g̃) → P_{SO}(M, g), (e_{1}, e_{2}) ↦ (fe_{2}, fe_{2}) defines an isomorphism of S^{1} principal bundles. The pullback of a spin structure on (M, g) is a spin structure on (M, g̃).

In a similar way, if (M_{1}, g_{1}) → (M_{2}, g_{2}) is an orientation preserving conformal map, but not necessarily a diffeomorphism, then any spin structure on (M_{2}, g_{2}) pulls back to a spin structure on (M_{1}, g_{1}).

Examples 2.1.

(1) If g_{0} is the standard metric on S^{2}. Then P_{SO}(S^{2}, g_{0}) = SO(3), and the base point map SO(3) → S^{2} is the map that associates to a matrix in SO(3) the first column. The double cover SU(2) → SO(3) defines a spin structure on (S, g_{0}).

(2) Let g̃ be an arbitrary metric on S^{2}. After a possible pullback by a diffeomorphism S^{2} → S^{2} we can write g̃ = f^{2}g_{0}. The pullback of the spin structure given in (1) under the isomorphism P_{SO}(S^{2}, g̃) → P_{SO}(S^{2}, g_{0}) defines a spin structure on (S^{2}, g̃).

(3) Let g_{1} be a flat metric on the torus T^{2}. Then a parallel frame gives rise to a (global) section of P_{SO}(T^{2}, g_{1}) → T^{2}. Hence, this is a trivial S^{1} principal bundle. The trivial fiberwise double covering T^{2} × S^{1} → T^{2} × S^{1}, (p, z) ↦ (p, z^{2}) defines a spin structure on (T^{2}, g_{1}), the so-called trivial spin structure χ_{tr} on (T^{2}, g_{1})

(4) If g̃ is an arbitrary metric on T^{2}. Then we can write g̃ = f^{2}g_{1} where g_{1} is a flat metric. As above, the trivial spin stucture on (T^{2}, g_{1}) defines a spin structure on (T^{2}, g̃). This spin structure is also called the trivial spin structure χ_{tr}.

(5) For (x_{0}, y_{0}) ∈ ℝ^{2} \ {0} we define

Z_{x_{0},y_{0}} = ℝ^{2}/⟨(x_{0}, y_{0})⟩

where ⟨(x_{0}, y_{0})⟩ is the subgroup of ℝ^{2} spanned by (x_{0}, y_{0}). We will assume that it carries the metric induced by the euclidean metric g_{eucl} on ℝ^{2}. Then P_{SO}(Z_{x_{0},y_{0}}) is a trivial bundle, and a natural trivialization is obtained by a parallel frame. The map Z_{x_{0},y_{0}} × S^{1} → Z_{x_{0},y_{0}} × S^{1}, (p, z) ↦ (p, z^{2}) defines a spin structure on Z_{x_{0},y_{0}}, the trivial spin structure on Z_{x_{0},y_{0}}.

Assume that χ : P → P_{SO}(M, g) is a spin structure on a surface, and assume that β : π_{1}(M) → {−1, +1} is a group homomorphism. Then there is a {−1, +1} principal bundle B_{β} → M with holonomy β. Let P_{β} be the quotient of P × B_{β} by the diagonal action of {−1, +1}. Then P_{β} together with the induced map χ_{β} : P_{β} → P_{SO}(M, g) is also a spin structure on (M, g). Conversely, if (P̃, χ̃) is another spin structure, then one can show that there is a unique β : π_{1}(M) → {−1, +1} such that (P̃, χ̃) and (P_{β}, χ_{β}) are isomorphic. Thus, we see that the space of spin structures is an affine space over the {−1, +1}-vector space Hom(π_{1}(M), {−1, +1}) = H^{1}(M, {−1, +1}).

Examples 2.2.

(1) Any compact oriented surface M carries a spin structure. If k denotes the genus of M, then there are 4^{k} homomorphisms π_{1}(M) → {−1, +1}, hence there are 4^{k} isomorphism classes of spin structures. In particular, the spin structure on S^{2} is unique.

(2) Because of π_{1}(Z_{x_{0},y_{0}}) = ℤ, there are exactly two spin structures on Z_{x_{0},y_{0}}, the trivial one and another one called the non-trivial spin structure.

From now on, let M = T^{2} = ℝ^{2}/Γ where Γ is a lattice in ℝ^{2}. The trivial spin structure defined above can be used to identify Hom(π_{1}(M), {−1, +1}) with the set of isomorphism classes of spin structures. By slightly

−0.5 ^{0.5 1.0}  0.5 □ x

y

M_{1}

Figure 1. The spin conformal moduli space is M = M_{1}/ ∼, where ∼ means identifying (x, y) ∈ ∂M_{1} with (−x, y).

abusing the language we will always write χ for the spin structure (P, χ) and also for the homomorphism π_{1}(M) → {−1, +1}.

The following lemma summarizes some well-known equivalent characterizations of triviality of χ (see e.g. [LM89], [Mil63], [Am98], [Fr84]).

Lemma 2.3. With the above notations, the following statements are equivalent

(1) The spin structure is trivial (in the above sense);

(2) χ(γ) = 1 for all γ ∈ Γ;

(3) The spin structure is invariant under the natural action of the diffeomorphism group Diff(T^{2});

(4) (T^{2}, χ) is the non-trivial element in the 2-dimensional spin-cobordism group;

(5) The α-genus of (T^{2}, χ) is the non-trivial element in ℤ/2ℤ;

(6) The Dirac operator has a non-trivial kernel;

(7) The kernel of the Dirac operator has complex dimension 2.

In particular, we easily see

τ(T^{2}, χ_{tr}) = 0.

From now on, in the rest of this article, we assume that χ is not the trivial spin structure, i.e. χ(γ) = −1 for some γ ∈ Γ.

Definition 2.4. Two 2-dimensional tori with Riemannian metrics, orientations and spin structures are said to be spin-conformal if there is a conformal map between them preserving the orientation and the spin structure. “Being spin-conformal” is obviously an equivalence relation. The spin-conformal moduli space M of T^{2} with the non-trivial spin structure is defined to be the set of these equivalence classes. Furthermore we define

M_{1} := \{(\begin{matrix}x \\ y \end{matrix}) | |x| ≤ \frac{1}{2}, y^{2} + (|x| − \frac{1}{2})^{2} ≥ \frac{1}{4}, y > 0\}

(see also Fig. 1).

Lemma 2.5. Let g be a Riemannian metric on T^{2} = ℝ^{2}/ℤ^{2}, and let χ : ℤ^{2} → {−1, +1} be a non-trivial spin structure. Then there is a lattice Γ ⊂ ℝ^{2}, a spin structure χ′ : Γ → {−1, +1}, such that

(1) Γ is generated by (\begin{matrix}1 \\ 0 \end{matrix}) and (\begin{matrix}x \\ y \end{matrix}) with (\begin{matrix}x \\ y \end{matrix}) ∈ M_{1}

(2) (T^{2}, g, χ) is spin-conformal to (ℝ^{2}/Γ, g_{eucl}, χ′)

(3) χ′((\begin{matrix}1 \\ 0 \end{matrix})) = +1 and χ′((\begin{matrix}x \\ y \end{matrix})) = −1.

Proof. Because of the uniformization theorem we can assume without loss of generality that g is a flat metric. The lemma then follows from elementary algebraic arguments. □

Note that x and y are uniquely determined if (\begin{matrix}x \\ y \end{matrix}) is in the interior of M_{1}, i.e. if |x| < 1/2 and y^{2}+(|x|−1/2)^{2} >

1/4. If (\begin{matrix}x \\ y \end{matrix}) is on the boundary of M_{1}, then y and |x| are determined, but not the sign of x. Hence, after

gluing (\begin{matrix}x \\ y \end{matrix}) ∈ ∂M_{1} with (\begin{matrix}−x \\ y \end{matrix}) we obtain the spin-conformal moduli space M.

Notation. Let (x_{0}, y_{0}) ∈ M_{1}. The lattice generated by by (\begin{matrix}1 \\ 0 \end{matrix}) and (\begin{matrix}x \\ y \end{matrix}) is noted as Γ_{x_{0},y_{0}}. Furthermore, we

write T_{x_{0},y_{0}} for the 2-dimensional torus ℝ^{2}/Γ_{x_{0},y_{0}} equipped with the euclidean metric.

The quantity λ_{min}(T^{2}, g, σ) is a spin-conformal invariant, hence λ_{min} can be viewed as a function on M or on M_{1}.

3. Main results

In this article, we study λ_{min} as a function on the spin-conformal moduli space with the non-trivial spin structure. This function takes values in [0, λ_{min}(𝕊^{2})] because of (1). As the spin structure is non-trivial, Lott’s results states that 0 is not attained. As a preliminary result we will prove that this function is continuous.

Proposition 3.1. The function

λ_{min} : | \begin{matrix}M_{1} & → & ]0, λ_{min}(𝕊^{2})] \\ (x_{0}, y_{0}) & ↦ & λ_{min}^{x_{0},y_{0}} \end{matrix}

is continuous on M_{1}.

The spin-conformal moduli space M (resp. M_{1}) has two ends. We will compactify each end by adding one point. The point added at the end y → ∞ will be denoted by ∞ and the point added at the end y → 0 is denoted by (0, 0).

Theorem 3.2. The function

λ_{min} : | \begin{matrix}M_{1} & → & ]0, λ_{min}(𝕊^{2})] \\ (x_{0}, y_{0}) & ↦ & λ_{min}^{x_{0},y_{0}} \end{matrix}

extends continuously to M_{1} ∪ {(0, 0), ∞} by setting λ_{min}^{0,0} = λ_{min}(𝕊^{2}) and λ_{min}^{∞} = 0.

The continuous extension at ∞ is is easy to see. The first eigenvalue of the Dirac operator on (T_{x_{0},y_{0}}, g_{eucl}, χ_{x_{0},y_{0}}), is π/y_{0}, the area is y_{0}, hence

λ_{min}^{x_{0},y_{0}} ≤ π/\sqrt{y_{0}} → 0 for y_{0} → ∞.

However, the limit (x_{0}, y_{0}) → (0, 0) is much more difficult to obtain.

Clearly, Theorem 3.2 implies Theorem 1.1.

4. Some preliminaries

Variational characterization of λ_{min}. Let (M, g, χ) be a compact spin manifold of dimension n ≥ 2 with ker D_{g} = {0}. For ψ ∈ Γ(ΣM), we define

J_{g}(ψ) = \frac{( ∫_{M} |Dψ|^{\frac{2n}{n+1}} dv_{g})^{\frac{n+1}{n}}}{| ∫_{M}⟨Dψ, ψ⟩ dv_{g}|} .

Lott [Lott86] proved that

λ_{min}(M, [g], χ) = inf_{ψ} J_{g}(ψ)  (5)

where the infimum is taken over the set of smooth spinor fields for which

(∫_{M}⟨Dψ, ψ⟩ dv_{g}) ≠ 0.

The functional J_{g} for the torus T_{x_{0},y_{0}} is noted as J^{x_{0},y_{0}}.

Remark 4.1. The exponents in J_{g} are chosen such that J_{g} is conformally invariant. More exactly, if g and g̃ are conformal, then the spinor bundles of (M, g, χ) and (M, g̃, χ) can be identified in such a way that J_{g}(ψ) = Jg̃(ψ).

Cylinders and doubly pointed spheres.

Let Z_{x_{0},y_{0}} be defined as in Examples 2.1 (5).

Lemma 4.2 (Mercator, around 1569). Let N, S ∈ 𝕊^{2} be respectively the North pole and the South pole of 𝕊^{2}. Then there is a conformal diffeomorphism F_{x_{0},y_{0}} from (Z_{x_{0},y_{0}}, g_{eucl}) to (𝕊^{2} \ {N, S}).

Proof. In the case (x_{0}, y_{0}) = (0, 2π) we see that the application

F_{0,2π} : (\begin{matrix}x \\ y \end{matrix}) ↦ ( \frac{sin y}{cosh x} ; \frac{cos y}{cosh x} ; tanh x)

is conformal and defines a conformal bijection Z_{0,2π} → S^{2} \ {N, S}. The general case follows by composing with a linear conformal map Z_{x_{0},y_{0}} → Z_{0,2π}. □

The map F induces a map between the frame bundles.

F̃_{x_{0},y_{0}} : P_{SO}(Z_{x_{0},y_{0}}) → P_{SO}(𝕊^{2})
F̃_{x_{0},y_{0}}((p, X, Y)) := (F_{x_{0},y_{0}}(p), \frac{dF_{x_{0},y_{0}}(X)}{|dF_{x_{0},y_{0}}(X)|}, \frac{dF_{x_{0},y_{0}}(Y)}{|dF_{x_{0},y_{0}}(Y)})
X, Y ∈ T_{p}Z_{x_{0},y_{0}} are orthonormal and oriented

The unique spin structure on 𝕊^{2} pulls back to a spin structure on Z_{x_{0},y_{0}}, that we will denote as χ_{x_{0},y_{0}}.

Lemma 4.3. The spin structure χ_{x_{0},y_{0}} is the non-trivial spin structure on Z_{x_{0},y_{0}}.

Proof. We will show the lemma for the case (x_{0}, y_{0}) = (0, 2π). As before, the general case then follows by composing with a linear map Z_{x_{0},y_{0}} → Z_{0,2π}.

We define the loop γ : [0, 2π] → Z_{0,2π}, γ(t) := (0, t) and the parallel section

α : t ↦ (\frac{∂}{∂x}|_{γ(t)}, \frac{∂}{∂y} |_{γ(t)})

of P_{SO}(Z_{0,2π}) along γ. The spin structure (P, χ_{0,2π}) on Z_{0,2π} is trivial if and only if there is a section α̃ of P along γ such that χ_{0,2π} ∘ α̃ = α and α̃(0) = α̃(2π).

The composition F̃_{0,2π} ∘ α is a section of P_{SO}(𝕊^{2}) = SO(3) along F_{0,2π} ∘ γ. One checks that

F̃_{0,2π} ∘ α(t) = (\frac{∂F_{0,2π}}{∂x} |_{(0,t)}, \frac{∂F_{0,2π}}{∂y} |_{(0,t)}, F (0, t)) = (\begin{matrix}0 & cos y & sin y \\ 0 & − sin y & cos y \\ 1 & 0 & 0 \end{matrix})

We lift this loop to a path α̂ in SU(2), then one easily sees that α̂(0) = −α̂(2π). As χ_{0,2π} is defined as the pullback of the spin structure on S^{2}, we see that any lift α̃ of α also satisfies α̃(0) ≠ α̃(2π). Hence, we have proved non-triviality of χ_{0,2π}. □

Corollary 4.4. Let Z_{x_{0},y_{0}} carry its non-trivial spin structure. Then,

\frac{(∫_{Z_{x_{0},y_{0}}} |Dψ|^{\frac{4}{3}} dx)^{\frac{3}{2}}}{|∫_{Z_{x_{0},y_{0}}} ⟨ψ, Dψ⟩dx|} ≥ λ_{min}(𝕊^{2})

for any compactly supported spinor ψ ∈ Γ(ΣZ_{x_{0},y_{0}}) such that ∫ ⟨ψ, Dψ⟩ ≠ 0.

Let f : Z_{x_{0},y_{0}} →]0, +∞[ be such that F_{x_{0},y_{0}}^{∗}g_{0} = f^{2}g_{eucl}. It is well known (see for example [Hit74, Hij86] ) that F_{x_{0},y_{0}} induces a pointwise isometry

| \begin{matrix}Σ(T_{x_{0},y_{0}}, g_{eucl}) & → & Σ(𝕊^{2} \ \{N, S\}, g_{0}) \\ ψ & ↦ & ψ̄, \end{matrix}

such that

D̄f^{− \frac{1}{2}} ψ̄ = f^{− \frac{3}{2}} \overline{Dψ}.

where D̄ denotes the Dirac operator on 𝕊^{2}. Moreover, ψ̄ is smooth on 𝕊^{2} since ψ ≡ 0 in a neighborhood of N and S. It is well known that the functional J defined at the beginning of section 4 is conformally invariant. This implies that

\frac{(∫_{Z_{x_{0},y_{0}}} |Dψ|^{\frac{4}{3}} dx)^{\frac{3}{2}}}{| ∫_{Z_{x_{0},y_{0}}} ⟨ψ, Dψ⟩ dx|} = \frac{(∫_{𝕊^{2}} |D̄(f^{− \frac{1}{2}} ψ̄)|^{\frac{4}{3}} dv_{g_{0}})^{\frac{3}{2}}}{| ∫_{𝕊^{2}} ⟨f^{− \frac{1}{2}} ψ̄, D̄(f^{− \frac{1}{2}} ψ̄)⟩ dv_{g_{0}}|} ≥ λ_{min}(𝕊^{2}).

5. Proof of the Main results

For the proof we will need the following well known elliptic estimates. These estimates are a consequence of techniques explained for example in [Ta81], see also [Aub98]. However, in our special situation a proof is much easier. Hence, for the convenience of the reader we will include an elementary proof here.

Lemma 5.1 (Elliptic estimates). Let (x_{0}, y_{0}) ∈ M_{1}, and note T^{2} for T_{x_{0},y_{0}}. There exists C > 0 depending only on x_{0} and y_{0} such that

∫_{T^{2}} |Dψ|^{\frac{4}{3}} dv_{g} ≥ C ∫_{T^{2}} |∇ψ|^{\frac{4}{3}} dv_{g}  (6)

and

(∫_{T^{2}} |ψ|^{4}dv_{g})^{\frac{1}{3}} ≤ C ∫_{T^{2}} |∇ψ|^{\frac{4}{3}} dv_{g}  (7)

for any smooth spinor ψ.

Proof. Let q = \frac{4}{3}. Assume that (6) is false. Then, for all ε > 0, we can find a smooth spinor ψ_{ε} ∈ Γ(Σ(T^{2})) such that

∫_{T^{2}} |Dψ_{ε}|^{q} dv_{g} ≤ ε  and  ∫_{T^{2}} |∇ψ_{ε}|^{q} dv_{g} = 1.  (8)

Now, assume that

lim_{ε→0} (∫_{T} |ψ_{ε}|^{q} dv_{g})^{\frac{1}{q}} = +∞.

Then, we set

ψ′_{ε} = \frac{ψ_{ε}}{(∫_{T^{2}} |ψ_{ε}|^{q}dv_{g})^{\frac{1}{q}}} .

The sequence (ψ′_{ε}) is bounded in W^{1,q}(T^{2}) and since W^{1,q}(T^{2}) is reflexive, we can find ψ′_{0} ∈ W^{1,q}(T^{2}) such that there is sequence ε_{i} → 0, with lim_{i→∞} ψ′_{ε_{i}} = ψ′_{0} weakly in W^{1,q}(T^{2}). Then, we would have

∫_{T^{2}} |∇ψ′_{0}|^{q}dv_{g} ≤ lim inf_{ε} ∫_{T^{2}} |∇ψ′_{ε}|^{q}dv_{g} = 0

We would get that ψ′_{0} is parallel which cannot occur since the structure on T^{2} is not trivial. This proves that (ψ_{ε}) is bounded in L^{q}(T^{2}) and hence, by (8) in W^{1,q}(T^{2}). Again by reflexivity of W^{1,q}(T^{2}), we get the existence of a spinor ψ_{0}, weak limit of a subsequence ψ_{ε_{i}} in W^{1,q}(T^{2}). By weak convergence of Dψ_{ε_{i}} to Dψ_{0} in L^{q}(T^{2}), we have

∫_{T^{2}} |Dψ_{0}|^{q}dv_{g} ≤ lim inf_{i} ∫_{T^{2}} |Dψ_{ε_{i}}|^{q}dv_{g} = 0.

This is impossible since the Dirac operator on T^{2} has a trivial kernel. This proves (6). As one can check, relation (7) can be proved with the same type of arguments. □

Proof of Proposition 3.1. The proposition states that λ_{min} is continuous on M_{1}. Let (x_{k}, y_{k})_{k} ∈ M_{1} be a sequence tending to (x_{0}, y_{0}) ∈ M_{1}. We identify T^{2} with ℝ^{2}/ℤ^{2}. The conformal structures corresponding to (x_{k}, y_{k}) and (x_{0}, y_{0}) are represented by flat metrics g_{x_{k},y_{k}} and g_{x_{0},y_{0}} on ℝ^{2}/ℤ^{2}, that are invariant under translations, and such that g_{x_{k},y_{k}} → g_{x_{0},y_{0}} in the C^{∞}-topology .

Let ε > 0 be small and let ψ_{0} and (ψ_{k})_{k} be smooth spinors such that

J_{x_{0},y_{0}}(ψ_{0}) ≤ λ_{min}^{x_{0},y_{0}} + ε and J_{x_{k},y_{k}}(ψ_{k}) ≤ λ_{min}^{x_{k},y_{k}} + ε.

At first, since (g_{x_{k},y_{k}})_{k} tends to g_{x_{0},y_{0}}, it is easy to see that

lim_{k} J_{x_{k},y_{k}}(ψ_{0}) = J_{x_{0},y_{0}}(ψ_{0})

and hence lim sup_{k} λ_{min}^{x_{k},y_{k}} ≤ λ_{min}^{x_{0},y_{0}} + ε for the given ε > 0 that we can choose as small as we want. Thus

lim sup_{k} λ_{min}^{x_{k},y_{k}} ≤ λ_{min}^{x_{0},y_{0}}.

Now, let us prove that

lim sup_{k} J_{x_{0},y_{0}}(ψ_{k}) ≤ lim inf_{k} J_{x_{k},y_{k}}(ψ_{k})  (9)

We let (v, w) be a orthormal basis for g_{x_{0},y_{0}} and (v_{k}, w_{k})_{k}, orthonormal basis for g_{x_{k},y_{k}} which tends to (v, w). One can write for all k, v_{k} = a_{k}v+b_{k}w and w_{k} = c_{k}v+d_{k}w with lim_{k} a_{k} = lim_{k} d_{k} = 1 and lim_{k} b_{k} = lim_{k} c_{k} = 0. We have

(∫_{T^{2}} |D_{x_{k},y_{k}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{k},y_{k}}} )^{\frac{3}{4}} = (∫_{T^{2}} |v_{k} · ∇_{v_{k}}ψ_{k} + w_{k} · ∇_{w_{k}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{k},y_{k}}} )^{\frac{3}{4}}

= (∫_{T^{2}} |D_{x_{0},y_{0}}ψ_{k} + θ_{k}|^{\frac{4}{3}} dv_{g_{x_{k},y_{k}}} )^{\frac{3}{4}} .

with

|θ_{k}| = |(a_{k}^{2} + c_{k}^{2} − 1)v · ∇_{v}ψ_{k} + (b_{k}^{2} + d_{k}^{2} − 1)w · ∇_{w}ψ_{k} + (a_{k}b_{k} + c_{k}d_{k})⟨w · ∇_{v}ψ_{k} + v · ∇_{w}ψ_{k}⟩| ≤ α_{k}|∇ψ_{k}|

where (α_{k})_{k} is a sequence of positive numbers which tends to 0. Note that because of the translation invariance of the metrics, the Levi-Civita connection does not depenpend on k. Since lim_{k} g_{x_{k},y_{x}} = g_{x_{0},y_{0}}, one gets that

(∫_{T^{2}} |D_{x_{k},y_{k}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{k},y_{k}}} )^{\frac{3}{4}} ≥ (∫_{T^{2}} |D_{x_{0},y_{0}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} )^{\frac{3}{4}} − α′_{k}(∫_{T^{2}} |∇ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} )^{\frac{3}{4}}

where lim_{k} α′_{k} = 0. Together with Lemma 5.1, we get that

(1 − Cα′_{k})(∫_{T^{2}} |D_{x_{0},y_{0}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} )^{\frac{3}{4}} ≤ (∫_{T^{2}} |D_{x_{k},y_{k}}ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{k},y_{k}}} )^{\frac{3}{4}}  (10)

where C is a positive constant independent of k. Now, in the same way, we can write

∫_{T^{2}} ⟨ψ_{k}, D_{x_{0},y_{0}}ψ_{k}⟩ dv_{g_{x_{0},y_{0}}} ≥ ∫_{T^{2}} ⟨ψ_{k}, D_{x_{k},y_{k}}ψ_{k}⟩ dv_{g_{x_{k},y_{k}}} − β_{k} ∫_{T^{2}} |ψ_{k}||∇ψ_{k}|dv_{g_{x_{0},y_{0}}}

where lim_{k} β_{k} = 0. Using Hölder inequality, we have

∫_{T^{2}} |ψ_{k}||∇ψ_{k}|dv_{g_{x_{0},y_{0}}} ≤ (∫_{T^{2}} |ψ_{k}|^{4}dv_{g_{x_{0},y_{0}}} )^{\frac{1}{4}} (∫_{T^{2}} |∇ψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} )^{\frac{3}{4}} .

Using (6) and (7), this gives

∫_{T^{2}} |ψ_{k}||∇ψ_{k}|dv_{g_{x_{0},y_{0}}} ≤ C (∫_{T^{2}} |Dψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} )^{3/2} .

We obtain

∫_{T^{2}} ⟨ψ_{k}, D_{x_{0},y_{0}}ψ_{k}⟩ dv_{g_{x_{0},y_{0}}} ≥ ∫_{T^{2}} ⟨ψ_{k}, D_{x_{k},y_{k}}ψ_{k}⟩ dv_{g_{x_{k},y_{k}}} − β_{k} (∫_{T^{2}} |Dψ_{k}|^{\frac{4}{3}} dv_{g_{x_{0},y_{0}}} .)^{3/2}

Together with (10), we get (9). This immediatly implies that

lim inf_{k} λ_{min}^{x_{k},y_{k}} ≥ λ_{min}^{x_{0},y_{0}}

and ends the proof of the proposition. □

Proof of Theorem 3.2. Any calculation in this proof will be carried out in Riemannian normal coordinates with respect to a flat metric. In the following, (e_{1}, e_{2}) will denote the canonical basis of ℝ^{2}.

In order to prove lim_{(x_{0},y_{0})→(0,0)} λ_{min}^{x_{0},y_{0}} = λ_{min}(𝕊^{2}) we will show that there is no sequence (x_{k}, y_{k}) → (0, 0) such that lim_{(x_{k},y_{k})→(0,0)} λ_{min}^{x_{k},y_{k}} < λ_{min}(𝕊^{2}). We may assume that λ_{min}^{x_{k},y_{k}} < λ_{min}(𝕊^{2}) for all k. Note that the spectrum of D is symmetric in dimension 2. By [Am03a], we then can find a sequence of spinors ψ_{k} of class C^{1} such that on T_{x_{k},y_{k}}

Dψ_{k} = λ_{min}^{x_{k},y_{k}}|ψ_{k}|^{2}ψ_{k}  (11)

and such that

∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx = 1.  (12)

Moreover, we have

J_{x_{k},y_{k}}(ψ_{k}) = λ_{min}^{x_{k},y_{k}}.  (13)

Sometimes we will identify ψ_{k} with its pullback to ℝ^{2}. In this picture ψ_{k} is a doubly periodic spinor on ℝ^{2}.

Step 1. There exists C > 0 such that for all k, we have λ_{min}^{x_{k},y_{k}} ≥ Cy_{k}^{1/2}.

Here and in the sequel, C will always denote a positive constant which does not depend on k.

For the proof of the first step, we let Ω = {(x, y) ∈ M_{1} | 1/2 ≤ y ≤ 3/2}. Since Ω is compact and since λ_{min} is continuous and positive, there exists C > 0 such that for all

λ_{min} ≥ C on Ω.  (14)

Now, assume that

lim_{k} \frac{λ_{min}^{x_{k},y_{k}}}{yk^{\frac{1}{2}}} = 0.

We can find a sequence (N_{k})_{k} which tends to +∞ such that (3^{N_{k}}x_{k}, 3^{N_{k}}y_{k}) ∈ Ω. Note that the locally isometric covering T_{px_{k},py_{k}} → T_{x_{k},y_{k}}, p ∈ ℕ, preserves the spin structures if and only if p is odd. Let ψ̃_{k} be the pullback of ψ_{k} with respect to covering T_{3^{N_{k}} x_{k},3^{N_{k}} y_{k}} → T_{x_{k},y_{k}}. We now have

∫_{T_{3^{N_{k}} x_{k},3^{N_{k}} y_{k}}} |Dψ_{k}|^{\frac{4}{3}} dx = 3^{N_{k}} ∫_{T_{x_{k},y_{k}}} |Dψ_{k}|^{\frac{4}{3}} dx

and

∫_{T_{3^{N_{k}} x_{k},3^{N_{k}} y_{k}}} ⟨ψ_{k}, Dψ_{k}⟩ dx = 3^{N_{k}} ∫_{T_{x_{k},y_{k}}} ⟨ψ_{k}, Dψ_{k}⟩ dx.

We then get by (14) that

C ≤ λ_{min}^{3^{N_{k}} x^{k},3^{N_{k}} y_{k}} ≤ J_{3^{N_{k}} (x_{k},y_{k})}(ψ_{k}) = 3^{\frac{N_{k}}{2}} λ_{min}^{x_{k},y_{k}} ≤ Cy_{k}^{−1/2}λ_{min}^{x_{k},y_{k}}.

Step 2. There exists C > 0 such that for all k, we have λ_{min}^{x_{k},y_{k}} ≥ C.

Let η : ℝ → [0, 1] be a cut-off function defined on ℝ which is equal to 0 on ℝ \ [−1, 2] and which is equal to 1 on [0, 1]. We may assume that η is smooth. Let v_{k} = (x_{k}, y_{k}). Since (e_{1}, v_{k}) is a basis of ℝ^{2}, we can define η_{k} : ℝ^{2} → [0, 1] by

η_{k}(tv_{k} + se_{1}) = η(s)

Since v_{k} is asymptotically orthogonal to e_{1}, we can find C > 0 independent of k such that

|∇η_{k}| ≤ C  (15)

THE FIRST CONFORMAL DIRAC EIGENVALUE ON 2-DIMENSIONAL TORI 11

Moreover, by corollary 4.4, we have

\frac{(∫_{Z_{x_{k},y_{k}}} |Dη_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{2}}}{| ∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, Dη_{k}ψ_{k}⟩dx|} ≥ λ_{min}(𝕊^{2}).  (16)

Now, we write that

(∫_{Z_{x_{k},y_{k}}} |Dη_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} = (∫_{Z_{x_{k},y_{k}}} |∇η_{k} · ψ_{k} + η_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}}
≤ (∫_{Z_{x_{k},y_{k}}} |∇η_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} + (∫_{Z_{x_{k},y_{k}}} |η_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} .

By (15) and Hölder inequality, we have

(∫_{Z_{x_{k},y_{k}}} |∇η_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ C(∫_{Z_{x_{k},y_{k}} ∩Supp(∇η_{k})} |ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}}
≤ C(∫_{Z_{x_{k},y_{k}} ∩Supp(∇η_{k})} |ψ_{k}|^{4}dx)^{\frac{1}{4}} Vol(Z_{x_{k},y_{k}} ∩ Supp(∇η_{k}))^{\frac{1}{2}} .

We then have

Vol(Z_{x_{k},y_{k}} ∩ Supp(∇η_{k})) ≤ 3y_{k}.

By (12) and step 1, this gives that

(∫_{Z_{x_{k},y_{k}}} |∇η_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ Cyk^{\frac{1}{2}} ≤ Cλ_{min}^{x_{k},y_{k}}.

With the same argument and using relations (11) and (12), it follows that

(∫_{Z_{x_{k},y_{k}}} |η_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ 3^{\frac{3}{4}} λ_{min}^{x_{k},y_{k}}(∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx)^{\frac{3}{4}} ≤ Cλ_{min}^{x_{k},y_{k}}.

Finally, we get that

(∫_{Z_{x_{k},y_{k}}} |Dη_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{2}} ≤ C(λ_{min}^{x_{k},y_{k}})^{2}.  (17)

We now write that

∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, Dη_{k}ψ_{k}⟩ dx = ∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, ∇η_{k} · ψ_{k} + η_{k}Dψ_{k}⟩ dx.

Moreover, the left hand side of this equality is real since D is an autoadjoint operator. Since

∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, ∇η_{k} · ψ_{k}⟩ dx ∈ iℝ.

Together with equation (11), this implies that

∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, Dη_{k}ψ_{k}⟩dx = ∫_{Z_{x_{k},y_{k}}} η_{k}^{2}λ_{min}^{x_{k},y_{k}}|ψ_{k}|^{4}dx.

Using (12), we obtain that

∫_{Z_{x_{k},y_{k}}} ⟨η_{k}ψ_{k}, Dη_{k}ψ_{k}⟩ dx ≥ λ_{min}^{x_{k},y_{k}} ∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx = λ_{min}^{x_{k},y_{k}}.  (18)

Finally, plugging (17) and (18) in (16), we obtain that λ_{min}(𝕊^{2}) ≤ Cλ_{min}^{x_{k},y_{k}}. This proves the step. Step 3. The function λ_{min} can be extended continuously to M_{1} ∪ {(0, 0)} by setting λ_{min}^{0,0} = λ_{min}(𝕊^{2}).

In other words, we show that lim_{k} λ_{min}^{x_{k},y_{k}} = λ_{min}(𝕊^{2}). The method is quite similar than the one of previous step. Let ζ_{k} : ℝ → [0, 1] be a smooth cut-off function defined on ℝ which is equal to 0 on ℝ \ [−y_{k}, 1 + y_{k}], which is equal to 1 on [0, 1] and which satisfies |∇ζ_{k}| ≤ \frac{2}{y_{k}}. As in the last step, we can define γ_{k} : ℝ^{2} → [0, 1] by

γ_{k}(tv_{k} + se_{1}) = ζ_{k}(s).

Since v_{k} is asymptotically orthogonal to e_{1}, we can find C > 0 independent of k such that

|γ_{k}| ≤ \frac{C}{y_{k}}  (19)

As in step 2, we have

\frac{(∫_{Z_{x_{k},y_{k}}} |Dγ_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{2}}}{| ∫_{Z_{x_{k},y_{k}}} ⟨γ_{k}ψ_{k}, Dγ_{k}ψ_{k}⟩ dx|} ≥ λ_{min}(𝕊^{2}).  (20)

We first prove that we can assume that

∫_{Z_{x_{k},y_{k}} ∩Supp(∇γ_{k})} |ψ_{k}|^{4}dx ≤ Cy_{k}.  (21)

We let n_{k} = [(2y_{k})^{−1}] be the integer part of 2y_{k}^{−1}. For all l ∈ [0, n_{k} − 1], we define

A_{k,l} = \{te_{1} + sv_{k}|s ∈ [0, 1[ and t ∈ [\frac{l − \frac{1}{2}}{n_{k}} , \frac{l + \frac{1}{2}}{n_{k}} ] \}.

The family of sets (A_{k,l})_{l∈[0,n_{k}−1]} is a partition of T′_{x_{k},y_{k}} which is the image of T_{x_{k},y_{k}} by the translation of vector −\frac{1}{2n_{k}} e_{1}. By periodicity, (A_{k,l})_{l∈[0,n_{k}−1]} can be seen as a partition of T_{x_{k},y_{k}}. Consequently, we can write that

1 = ∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx = ∑_{l=0}^{n_{k}−1} ∫_{A_{k,l}} |ψ_{k}|^{4}dx.

Hence, there exists l_{0} ∈ [0, n_{k} − 1] such that

∫_{A_{k,l_{0}}} |ψ_{k}|^{4}dx = min_{l∈[0,n_{k}−1]} ∑_{l=0}^{n_{k}−1} ∫_{A_{k,l}} |ψ_{k}|^{4}dx ≤ \frac{1}{n_{k}} .

Obviously, without loss of generality, we can replace ψ_{k} by ψ_{k} ∘ t_{0} where t_{0} is the translation of vector −l_{0}e_{1}. In this way, we can assume that l_{0} = 0. By periodicity, Supp(∇γ_{k}) ⊂ A_{k,0}. Hence,

∫_{Z_{x_{k},y_{k}} ∩Supp(∇γ_{k})} |ψ_{k}|^{4}dx ≤ \frac{1}{n_{k}} .

Since n_{k} ∼ \frac{2}{y_{k}}, equation (21) follows.

Now, we proceed as in step 2. We write that

(∫_{Z_{x_{k},y_{k}}} |Dγ_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} = (∫_{Z_{x_{k},y_{k}}} |∇γ_{k} · ψ_{k} + γ_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}}

≤ (∫_{Z_{x_{k},y_{k}}} |∇γ_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} + (∫_{Z_{x_{k},y_{k}}} |γ_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} .

It follows from (19) and the Hölder inequality that

(∫_{Z_{x_{k},y_{k}}} |∇γ_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ \frac{C}{y_{k}} (∫_{Z_{x_{k},y_{k}} ∩Supp(∇γ_{k})} |ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}}

≤ \frac{C}{y_{k}} (∫_{Z_{x_{k},y_{k}} ∩Supp(∇γ_{k})} |ψ_{k}|^{4}dx)^{\frac{1}{4}} (Vol(Z_{x_{k},y_{k}} ∩ Supp(∇γ_{k})))^{\frac{1}{2}} .

Clearly, we have

Vol(Z_{x_{k},y_{k}} ∩ Supp(∇γ_{k})) ≤ Cy_{k}^{2}.

By (21), we obtain

(∫_{Z_{x_{k},y_{k}}} |∇γ_{k} · ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ Cy_{k}^{−1+ \frac{1}{4} +1} ≤ Cyk^{\frac{1}{4}} = o(1).

For the other term, we write, using (11)

(∫_{Z_{x_{k},y_{k}}} |γ_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} = λ_{min}^{x_{k},y_{k}}(∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx + ∫_{Z_{x_{k},y_{k}} ∩\{0<γ_{k}<1\}} |ψ_{k}|^{4}dx)^{\frac{3}{4}} .

Clearly, we can construct γ_{k} such that {0 < γ_{k} < 1} ⊂ Supp(∇γ_{k}). It then follows from (21) that

(∫_{Z_{x_{k},y_{k}}} |γ_{k}Dψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{4}} ≤ λ_{min}^{x_{k},y_{k}} + o(1).

Finally, we obtain

(∫_{Z_{x_{k},y_{k}}} |Dγ_{k}ψ_{k}|^{\frac{4}{3}} dx)^{\frac{3}{2}} ≤ (λ_{min}^{x_{k},y_{k}})^{2} + o(1).  (22)

Now, as in step 2, we write that

∫_{Z_{x_{k},y_{k}}} ⟨γ_{k}ψ_{k}, Dγ_{k}ψ_{k}⟩ dx = ∫_{Z_{x_{k},y_{k}}} γ_{k}^{2}λ_{min}^{x_{k},y_{k}}|ψ_{k}|^{4}dx.

Using (12), we obtain that

∫_{Z_{x_{k},y_{k}}} ⟨γ_{k}ψ_{k}, Dγ_{k}ψ_{k}⟩ dx ≥ λ_{min}^{x_{k},y_{k}} ∫_{T_{x_{k},y_{k}}} |ψ_{k}|^{4}dx = λ_{min}^{x_{k},y_{k}}.  (23)

Plugging (22) and (23) in (20), we obtain that

λ_{min}(𝕊^{2}) ≤ \frac{(λ_{min}^{x_{k},y_{k}})^{2} + o(1)}{λ_{min}^{x_{k},y_{k}}}

which implies that either λ_{min}^{x_{k},y_{k}} → 0 or λ_{min}^{x_{k},y_{k}} → λ_{min}(𝕊^{2}). Hence, step 2 yields the statement of the theorem.

□

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Authors’ address:

Bernd Ammann and Emmanuel Humbert Institut Élie Cartan BP 239 Université Henri Poincaré, Nancy 54506 Vandoeuvre-lès -Nancy Cedex France

E-Mail:

ammann at iecn.u-nancy.fr and humbert at iecn.u-nancy.fr