Spectral estimates on 2-tori

Bernd Ammann^{∗}

April 2001

Abstract

We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particular we give a lower bound for the first eigenvalue of the Dirac operator for nontrivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so far that uses information about the spin structure.

As a corollary we obtain lower bounds for the Willmore functional of a torus embedded into S^{3}.

In the final section we compare Dirac spectra for two different spin structures on an arbitrary Riemannian spin manifold.

Keywords: Dirac operator, Laplace operator, spectrum, conformal metrics, two-dimensional torus, spin structures, Willmore functional

Mathematics Classification 2000: Primary: 53C27, Secondary: 58J50 53C80

1 Introduction

The Dirac operator is an elliptic differential operator of order one playing an important role both in modern physics and in mathematics. In physics, particles with non-integer spin, so-called fermions, are described by the Dirac equation. Let us assume that the space-time M is stationary, M = ℝ × N and that the spatial component N is compact and admits a spin structure. Then stationary fermions have a wave function of the form

Ψ(t, x) = e^{iEt}Ψ_{0}(x) t ∈ ℝ, x ∈ N

where Ψ_{0} is an eigenspinor of D_{N}^{2}, the square of the Dirac operator on N, that belongs to the eigenvalue λ . The energy E and the eigenvalue λ are related via the formula

E^{2} = λ + m^{2}

with m being the rest mass of the particle. Knowing the spectrum therefore means knowing possible energies. The first eigenvalue is of particular interest as it characterizes the energy of the state of lowest energy — the vacuum. On an arbitrary Riemannian manifold, exact calculation of the spectrum is impossible, thus one tries try to find bounds for the eigenvalues.

Bounding eigenvalues of the Dirac operator on a compact Riemannian manifold N is also an important tool in differential geometry and topology. If N is spin and carries a metric whose scalar curvature is greater than or equal to s_{0} > 0 at every point, then with the help of the Schrödinger-Lichnerowicz formula it is easy to prove that the first eigenvalue λ_{1} of D^{2} is bounded from below by s_{0}/4. On the other hand Atiyah-Singer index theorem tells us that positivity of the first eigenvalue of D^{2} on a compact Riemannian manifold N implies that the Â-genus vanishes. Therefore any compact spin manifold admitting a positive scalar curvature metric has vanishing Â-genus.

∗Partially supported by The European Contract Human Potential Programme, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118

Lower bounds for Dirac eigenvalues can also be applied to problems in classical differential geometry. For any immersion F : N → ℝ^{n} of a compact manifold N, Christian Bär [Bär98b] proved

∫_{N} |H|^{2} ≥ μ_{1}area(N).  (1)

Here N carries the induced metric, μ_{1} is the first eigenvalue of the square of a twisted Dirac operator and H is the mean curvature vector field of F (N) ⊂ ℝ^{n}. If N is the 2-dimensional torus T^{2}, then the left hand side of (1) is the so-called Willmore functional. The Willmore conjecture states

∫_{T^{2}} |H|^{2} ≥ 2π^{2}

for any immersion F : T^{2} → ℝ^{n}. This conjecture first appeared in [Wil65] for the case n = 3. In the meantime the conjecture has been verified for several classes of immersions, for example for immersions with rotational symmetry [LS84] or for non-injective immersions [LY82]. Nevertheless the conjecture remains open until now. For further information on this conjecture the reader may read the introductions of [Top98b] or [Amm00].

Now assume for simplicity that F is an embedding and F (T^{2}) ⊂ S^{3} ⊂ ℝ^{4}. In this case, the twisting bundle is trivial, and μ_{1} is the first eigenvalue of the square of the classical Dirac operator associated to a non-trivial spin structure. Our goal is to use inequality (1) in order to derive lower bounds for the Willmore functional. If the induced metric on T^{2} is flat, the spectrum of D has been explicitely calculated [Fri84] and we obtain a lower bound for ∫_{T^{2}} |H|^{2}.

Obtaining lower eigenvalue estimates for non-flat tori is much harder. John Lott [Lot86, Proposition 1] proved the existence of a constant CLott > 0 depending on the spin-conformal type of the torus such that

μ_{1}area ≥ CLott.  (2)

Unfortunately, Lott’s article does not give an explicit value and it seems hard to express such a constant CLott in terms of meaningful geometric data. Lott’s estimate uses the L^{p}-boundedness of zero order pseudo-differential operators and Sobolev embedding theorems, hence corresponding constants are hard to interpret without using explicit coordinates.

The starting point of the author’s PhD thesis [Amm98] and of the present article is to find an explicit lower bound for μ_{1} that uses information about the spin structure. All explicit lower estimates known before did not use any information about the spin structure.

For general compact Riemannian manifolds the problem of finding such estimates is rather difficult. It is not clear at all what kind of data from the spin structure could be used in order to get an additional term in a lower eigenvalue estimate. Take for example a compact manifold with non-vanishing Â-genus. It has μ_{1} = 0 for any spin structure, thus the contribution of the spin structure in the estimate has to vanish.

As the general case is hard to handle, most of the article will specialize to the 2-dimensional torus T^{2}. By the uniformization theorem any 2-dimensional torus is conformally equivalent to a flat torus. We use this fact in order to control the geometry. An important, but also very technical step for this is the estimate of the oscillation of the conformal factor (Section 9). Although our main goal was to find lower estimates for the Dirac eigenvalues, it turns out that this method gives upper and lower bounds for all eigenvalues both of the Laplace operator and the Dirac operator and for any spin structure. We prove different versions of the estimates. Theorem 2.2 for example states for the first eigenvalue μ_{1} of the square of the Dirac operator

μ_{1}area ≥ CAmmann · κ  (3)

where CAmmann > 0 is an explicit constant depending on the spin-conformal class and κ ≤ 1 is a curvature expression that satisfies κ = 1 if the metric is flat. This estimate is sharp for any flat metric.

In view of Lott’s result (2), it is tempting to conjecture that we can drop the curvature term, i. e. μ_{i}area ≥ CAmmann. This is false however: we can prove by example given at the end of section 12 that for many spinconformal structures the optimal constant in Lott’s estimate is not attained by a flat torus.

In section 12 we will prove some lower bounds for the Willmore functional that are strongly related to our lower estimates of the Dirac eigenvalues. In particular we prove for embeddings T^{2} → S^{3} that under a curvature condition the Willmore functional converges to ∞ if the spin-conformal type of the embedding converges to one end of the spin-conformal moduli space (Corollary 12.5).

The results in this paper about the Willmore conjecture are strongly related to another preprint of the author [Amm00]. The results of the present article are stronger near one of the ends of the spin-conformal moduli space but they have other drawbacks. Namely, they do not generalize easily to higher codimensions and they impose a restriction on the spin-conformal class.

The structure of the article is as follows: In section 2 we will state our spectral estimates on 2-tori. Sections 3 to 11 provide proofs of the statements in section 2. We then apply Theorem 9.1 once again and derive an application to the Willmore functional that is related to our lower eigenvalue estimates.

Finally in section 13 we will prove a result for arbitrary spin manifolds M. Let M carry two different spin structures ϑ and ϑ′. The difference of these spin structures χ := ϑ−ϑ′ is an element in H^{1}(M, ℤ_{2}) = Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}). Assume that χ vanishes on the torsion part of H_{1}(M, ℤ). We will define a norm ∥χ∥_{L^{∞}}, the stable norm of χ. We prove that the eigenvalues (ρ_{i})_{i∈ℤ} of the Dirac operator corresponding to ϑ and the eigenvalues (ρ′_{i})_{i∈ℤ} corresponding to ϑ′ can be numbered so that

|ρ_{i} − ρ′_{i}| ≤ 2π ∥ϑ − ϑ′∥_{L^{∞}} .

If the spectrum is known for ϑ and if |ρ_{i}| > 2π ∥ϑ − ϑ′∥_{L^{∞}} for any i ∈ ℤ, then this yields a lower bound for any ρ′_{i}.

At the end of the introduction we want to mention some other publications that treat the interplay between spin structures and the spectrum of the Dirac operator. However, they do not derive explicit eigenvalue bounds for generic metrics. We will restrict to the most recent ones. For further references and a good overview of the subject we refer to [Bär00].

Dahl [Dah99] shows that the difference of the eta-invariants corresponding to two different spin structures is an integer, if the difference of the spin structures viewed as an element in Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}) vanishes on the torsion part. Bär [Bär98a] calculated the essential spectrum of hyperbolic 2- and 3-manifolds of finite volume. In these examples, the essential spectrum depends on the spin structure at the cusps. Pfäffle [Pfä99] calculated the spectrum and the η-invariants of flat Bieberbach manifolds. These spectra also depend on the spin structure.

Several results in the present article already appeared in the author’s PhD thesis [Amm98].

2 Main results

In this section we summarize our results about the spectra of Dirac and Laplace operators on 2-tori.

The spectrum of the Dirac operator depends on the spin structure. At first, we recall some important facts about spin structures and introduce some notation. Spin structures will be discussed in more detail in section 4.

Let M be a compact orientable manifold with vanishing second Stiefel-Whitney class w_{2}(T M) = 0. Such manifolds admit a spin structure. However, the spin structure is not unique in general. The group H^{1}(M, ℤ_{2}) acts freely and transitively on the set of spin structures 𝔖𝔭𝔦𝔫(M), i. e. 𝔖𝔭𝔦𝔫(M) is an affine space associated to the vector space H^{1}(M, ℤ_{2}). After fixing a spin structure and a Riemannian metric on M we can define the spinor bundle ΣM → M and a Dirac operator D : Γ(ΣM) → Γ(ΣM).

We are mainly interested in the case M = T^{2}. The 2-dimensional torus T^{2} is spin. Because of # 𝔖𝔭𝔦𝔫(T^{2}) = #H^{1}(T^{2}, ℤ_{2}) = 4 there are 4 spin structures on T^{2}. There is exactly one spin structure in 𝔖𝔭𝔦𝔫(T^{2}) for which 0 lies in the spectrum of D, regardless of the underlying metric g. This spin structure will be called trivial (see section 4 for other characterizations). We will identify the trivial spin structure with 0 ∈ H^{1}(T^{2}, ℤ_{2}). This identification yields an identification of the affine space 𝔖𝔭𝔦𝔫(T^{2}) with H^{1}(T^{2}, ℤ_{2}). On the other hand, we will identify H^{1}(T^{2}, ℤ_{2}) with Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2}). Hence spin structures on T^{2} are in a canonical one-to-one relation to such homomorphisms. Frequently, we will use the term “spin homomorphism” instead of “spin structure” in order to indicate that we regard the spin structure as an element in Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2}).

If the torus T^{2} is endowed with a flat metric, it is very helpful to write the torus as ℝ^{2}/Γ with a lattice Γ ≅ H_{1}(T^{2}, ℤ). We always assume that ℝ^{2}/Γ carries the metric induced by the Euclidean metric on ℝ^{2}. Let Γ^{∗} be the lattice dual to Γ. Elements χ ∈ Hom_{ℤ}(Γ, ℤ_{2}) are represented by vectors α ∈ (1/2)Γ^{∗} with the property

χ(x) = (−1)^{2α(x)} ∀x ∈ Γ.

Note that χ determines α only up to elements in Γ^{∗}.

We define the function S : [0, 4π[ × [0, ∞[×]1, ∞[ ×]0, ∞] → ]0, ∞] by

S(K, K′, p, V) := \frac{p}{p − 1} [\frac{K′}{4π} + \frac{1}{2}| log (1 − \frac{K}{4π} ) | + \frac{K}{8π − 2K} log (\frac{2K′}{K} )] + \frac{KV}{8}

for K > 0 and S(0, K′, p, V) := \frac{p}{p−1} \frac{K′}{4π}.

Let Area_{g} be the area of (T^{2}, g).

THEOREM 2.1. Let (T^{2}, g) be a Riemannian 2-torus with spin homomorphism χ. Choose a lattice Γ in ℝ^{2} with vol(ℝ^{2}/Γ) = 1 together with a conformal map A : ℝ^{2}/Γ → (T^{2}, g). Assume that A^{∗}(χ) is represented by α ∈ (1/2)Γ^{∗}. Let 0 ≤ ℓ_{0} ≤ ℓ_{1} ≤ ℓ_{2} ≤ … be the sequence of lengths of Γ^{∗} + α (with multiplicities), and let (μ_{i} | i = 1, 2, …) be the spectrum of D^{2} on (T^{2}, g, χ).

Then

e^{−2 osc u} 4π^{2} ℓ[\frac{i−1}{2}]^{2} ≤ μ_{i} Area_{g} ≤ e^{2 osc u} 4π^{2} ℓ[\frac{i−1}{2}]^{2}.

If ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π, then

osc u ≤ S (∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, σ_{1}(T^{2}, g)^{−2})  (4)

with σ_{1}(T^{2}, g) := inf {length(β) | β ∈ Γ − {0}}.

The number σ_{1}(T^{2}, g) is a conformal invariant of (T^{2}, g) which will be called cosystole.

The most difficult step in the proof of this theorem is to find the estimate (4). This step will be performed in Theorem 9.1. For proving the above theorem, we will use the explicit formula for the spectra of flat tori (Proposition 7.2, [Fri84]). Another important tool for the proof is the following proposition.

PROPOSITION 5.2. Let M be a compact manifold with two conformal metrics g̃ and g = e^{2u}g̃. Let D and D̃ be the corresponding Dirac operators with respect to a common spin structure. We denote the eigenvalues of D^{2} by μ_{1} ≤ μ_{2} ≤ … and the ones of D̃^{2} by μ̃_{1} ≤ μ̃_{2} ≤ . . ..

Then

μ_{i} min_{m∈M} e^{2u(m)} ≤ μ̃_{i} ≤ μ_{i} max_{m∈M} e^{2u(m)} ∀i = 1, 2, ….

This proposition is based on Hitchin’s transformation formula for spinors [Hit74] (see section 5 for a proof).

In section 6 we will define a norm on H^{1}(T^{2}, ℤ_{2}), the L^{2}-norm. This norm allows us to derive explicit lower bounds for the first eigenvalue of D^{2} on T^{2}. This lower bound is non-trivial if the spin structure is non-trivial. The cosystole σ_{1}(T^{2}, g) can also be expressed in terms of the L^{2}-norm

σ_{1}(T^{2}, g) := inf {∥α∥_{L^{2}} | α ∈ H^{1}(T^{2}, ℤ_{2}), α ≠ 0} .

(see section 6, in particular Proposition 6.1 (a)).

THEOREM 2.2. Let (T^{2}, g) be a Riemannian 2-torus with spin homomorphism χ. Assume that ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π. Then the first eigenvalue μ_{1} of D^{2} satisfies

μ_{1}Area_{g} ≥ \frac{4π^{2} ∥χ∥_{L^{2}}^{2}}{exp (2S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, σ_{1}(T^{2}, g)^{−2})},

The equality is attained if and only if g is flat.

From this theorem we will obtain two corollaries estimating μ_{1} in terms of the systole sys_{1}, the spinning systole spin-sys_{1} and the non-spinning systole nonspin-sys_{1}.

sys_{1}(T^{2}, g) := inf {length(γ) | γ is a non-contractible loop.}
spin-sys_{1}(T^{2}, g, χ) := inf {length(γ) | γ is a loop with χ([γ]) = −1.}
nonspin-sys_{1}(T^{2}, g, χ) := inf {length(γ) | γ is a non-contractible loop with χ([γ]) = 1
and [γ] is a primitive element in H_{1}(T^{2}, ℤ).}

An element α ∈ H_{1}(T^{2}, ℤ) is called primitive if there are no k ∈ ℕ, k ≥ 2, β ∈ H_{1}(T^{2}, ℤ) with α = k · β.

COROLLARY 2.3. Let (T^{2}, g) be a Riemannian 2-torus with non-trivial spin homomorphism χ. Assume that ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π. Then the first eigenvalue μ_{1} of D^{2} satisfies

μ_{1}Area_{g}^{2} ≥ \frac{π^{2} nonspin-sys_{1}(T^{2}, g, χ)^{2}}{exp (2S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, \frac{Area_{g}}{sys_{1}(T^{2},g)^{2}} )}.

The equality is attained if and only if g is flat.

COROLLARY 2.4. Let (T^{2}, g) be a Riemannian 2-torus with non-trivial spin homomorphism χ. Assume that ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π. Then the first eigenvalue μ_{1} of D^{2} satisfies

μ_{1}spin-sys_{1}(T^{2}, g, χ)^{2} ≥ \frac{π^{2}}{exp (4S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, \frac{Area_{g}}{sys_{1}(T^{2},g)^{2}} )}.

The equality is attained if and only if

(a) g is flat, i. e. (T^{2}, g) is isometric to ℝ^{2}/Γ for a suitable lattice Γ, and

(b) there are generators γ_{1}, γ_{2} for Γ statisfying γ_{1} ⊥ γ_{2}, χ(γ_{1}) = 1 and χ(γ_{2}) = −1.

Using Proposition 6.1 and the inequalities from section 10 the two corollaries immediately follow from Theorem 2.2.

We now turn to the Laplace operator and to the Dirac operator associated to a trivial spin structure. We recall a well-known proposition that is the analogue of Proposition 5.2 for the Laplacian on surfaces (section 5).

PROPOSITION 5.1. Let M be a compact 2-dimensional manifold with two conformal metrics g̃ and g = e^{2u}g̃. The eigenvalues of the Laplacian on functions corresponding to g and g̃ will be denoted as 0 = λ_{0} < λ_{1} ≤ λ_{2} … and 0 = λ̃_{0} < λ̃_{1} ≤ λ̃_{2} … respectively.

Then

λ_{i} min_{m∈M} e^{2u(m)} ≤ λ̃_{i} ≤ λ_{i} max_{m∈M} e^{2u(m)} ∀i = 1, 2, ….

Together with Proposition 7.1 and Theorem 9.1 we obtain

THEOREM 2.5. Let (T^{2}, g) be a torus conformally equivalent to ℝ^{2}/Γ, vol(ℝ^{2}/Γ) = 1. Let Γ^{∗} be the lattice dual to Γ. Let 0 ≤ ℓ_{0} ≤ ℓ_{1} ≤ ℓ_{2} ≤ … be the sequence of lengths of Γ^{∗}, and let (λ_{i} | i = 0, 1, 2, …) be the spectrum of the Laplacian on functions on (T^{2}, g), then

e^{−2 osc u} 4π^{2} ℓ_{i}^{2} ≤ λ_{i} Area_{g} ≤ e^{2 osc u} 4π^{2} ℓ_{i}^{2}.

If ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π, then

osc u ≤ S (∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, σ_{1}(T^{2}, g)^{−2}) .

Note that this theorem also provides bounds for the Laplacian on forms: By Poincaré duality the spectrum on 2-forms is the same as the spectrum on functions, and the Laplacian on 1-forms also has the same non-zero eigenvalues, but each with multiplicity two.

The theorem implies, in particular, a lower bound on the first positive eigenvalue.

THEOREM 2.6. Let (T^{2}, g) be a Riemannian 2-torus. Assume that ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π. Then the first positive eigenvalue λ_{1} of the Laplacian on functions satisfies

λ_{1}Area_{g} ≥ \frac{4 π^{2} σ_{1}(T^{2}, g)^{2}}{exp (2S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, σ_{1}(T^{2}, g)^{−2})}.

The equality is attained if and only if g is flat.

COROLLARY 2.7. Let (T^{2}, g) be a Riemannian 2-torus. Assume that ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π. Then the first positive eigenvalue λ_{1} of the Laplacian on functions satisfies

λ_{1}Area_{g}^{2} ≥ \frac{4 π^{2} sys_{1}(T^{2}, g)^{2}}{exp (2S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, \frac{Area_{g}}{sys_{1}(T^{2},g)^{2}} )}.

The equality is attained if and only if g is flat.

Remark. Theorem 2.6 and Corollary 2.7 also hold for the first positve eigenvalue of D^{2}, if the spin structure is trivial. Theorem 2.5 holds for the spectrum of D^{2}, if we double the multiplicities.

The structure of the paper is as follows: In the following sections (sections 3–11) we will prove our main results. In section 12, we will apply the inequalities in Proposition 9.1 in order to obtain a lower bound on the Willmore functional. Finally, in section 13 we assume that a manifold of abitrary dimension n ≥ 2 carries two spin structures. We derive an upper bound for the spectra of the corresponding Dirac operators.

3 Overview

We want to obtain upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on a Riemannian 2-torus (T^{2}, g).

The Clifford action of the volume element on spinors anticommutes with the Dirac operator D. Thus, the spectrum of D is symmetric and is uniquely determined by the spectrum of its square D^{2}. Therefore we will study the spectrum of D^{2} instead of the spectrum of D. In the literature D^{2} is often called the Dirac Laplacian.

In order to prove bounds on eigenvalues we use the uniformization theorem which tells us that we can write g as g = e^{2u}g_{0} with a real-valued function u and a flat metric g_{0}. For flat tori the spectrum of the Laplacian and the Dirac operator is known: the spectra can be calculated in terms of the dual lattice corresponding to (T^{2}, g_{0}).

We obtain bounds through the following steps.

(a) Comparison of the spectrum of (T^{2}, g) and the spectrum of (T^{2}, g_{0}) (Propositions 5.1 and 5.2).

(b) Introduction of certain spin-conformal invariants that contain information about the dual lattice corresponding to (T^{2}, g_{0}) (section 6).

(c) The knowledge of spectra of flat tori (section 7).

(d) A bound on osc u = max u − min u (section 9).

(e) Derivation, in section 10, of certain inequalities that are in a sense inverse to the inequalities in Proposition 6.1 and contain a curvature term.

In section 11, we combine the inequalities and derive the main results.

4 Spin structures

The eigenvalues of D depend on the spin structures and we want to find estimates depending on the spin structure. In this section we recall some important facts about spin structures. Good references about spin structures are [LM89], [BG92] and [Swi93, section II]. We will define spin structures without fixing a Riemannian metric. This definition will allow us to identify spin structures on diffeomorphic but non isometric manifolds (see Proposition 5.2).

Let M be an oriented manifold of dimension n ≥ 2. The bundle GL^{+}(M) of oriented bases over M is a principal GL^{+}(n, ℝ)-bundle. The fundamental group of GL^{+}(n, ℝ) is ℤ for n = 2 and ℤ_{2} for n ≥ 3. Therefore GL^{+}(n, ℝ) has a unique connected double covering Θ : G̃L^{+}(n, ℝ) → GL^{+}(n, ℝ).

Definition. A spin structure on M is a pair (G̃L^{+}(M), ϑ) where G̃L^{+}(M) is a principal G̃L^{+}(n, ℝ)-bundle over M and ϑ is a double covering G̃L^{+}(M) → GL^{+}(M) such that

G̃L^{+}(M) × G̃L^{+}(n, ℝ) → G̃L^{+}(M)
↘
↓ ϑ × Θ ↓ ϑ M
↗
GL^{+}(M) × GL^{+}(n, ℝ) → GL^{+}(M)

(5)

commutes. The horizontal arrows are given by the group action.

There is a spin structure on M if and only if the second Stiefel-Whitney class w_{2}(T M) vanishes. Such manifolds are called spin. From now on we assume that M is spin.

Two spin structures (G̃L^{+}(M), ϑ) and (G̃L_{1}^{+}(M), ϑ_{1}) are identified if there is a fiber preserving isomorphism of principal G̃L^{+}(n, ℝ)-bundles α : G̃L^{+}(M) → G̃L_{1}^{+}(M) with ϑ = ϑ_{1} ∘ α.

The set of all spin structures (G̃L^{+}(M), ϑ) over M will be denoted by 𝔖𝔭𝔦𝔫(M). The set 𝔖𝔭𝔦𝔫(M) has the structure of an affine space associated to the vector space H^{1}(M, ℤ_{2}), i. e. H^{1}(M, ℤ_{2}) acts freely and transitively on 𝔖𝔭𝔦𝔫(M). We will describe this action: Elements in H^{1}(M, ℤ_{2}) can be viewed as principal ℤ_{2}-bundles over M [LM89, Appendix A]. Let π : P_{χ} → M be the ℤ_{2}-bundle defined by χ ∈ H^{1}(M, ℤ_{2}). Let (G̃L^{+}(M), ϑ) be a

spin structure. The group ℤ_{2} acts by deck-transformations both on G̃L^{+}(M) and P_{χ}. We define

G̃L_{1}^{+}(M) := (G̃L^{+}(M) ×_{M} P_{χ})/ℤ_{2}

where ℤ_{2} acts diagonally on the fiberwise product of the bundles. The map

ϑ ×_{M} π : G̃L^{+}(M) ×_{M} P_{χ} → GL^{+}(M) (A, α) ↦ ϑ(A)

is invariant under the ℤ_{2}-action and therefore defines a map ϑ_{1} : G̃L_{1}^{+}(M) → GL^{+}(M) compatible with (5). The action of χ maps (G̃L^{+}(M), ϑ) to the spin structure (G̃L_{1}^{+}(M), ϑ_{1}). This action is free and transitive [LM89, II§1].

Now we fix a Riemannian metric g on M. This reduces our structure group from GL^{+}(n, ℝ) to SO(n). The bundle of positively oriented orthonormal bases SO(M, g) is a principal SO(n)-bundle. The spin group is defined by Spin(n) := Θ^{−1}(SO(n)) and is the unique connected double covering of SO(n). A metric spin structure is a pair (Spin(M, g), ϑ) where Spin(M, g) is a principal Spin(n)-bundle over M and ϑ is a double covering Spin(M, g) → SO(M, g) satisfying a compatibility condition analogous to (5). For any spin structure (G̃L^{+}(M), ϑ) we obtain a metric spin structure (Spin(M, g), ϑ′) by restriction:

Spin(M, g) := ϑ^{−1}(SO(M, g)) ϑ′ := ϑ|_{Spin(M,g)}.

Via this restriction map, the set of metric spin structures is in a natural one-to-one correspondance to 𝔖𝔭𝔦𝔫(M) [Swi93].

Metric spin structures are used to define spinors and the Dirac operator. Let γ_{n} : Spin(n) → SU(Σ_{n}) be the complex spinor representation of Spin(n). This is a complex representation of dimension 2^{[n/2]}. It is irreducible for n odd. For n even, it consists of two irreducible components γ_{n}^{+} and γ_{n}^{−}, γ_{n}^{±} : Spin(n) → SU(Σ_{n}^{±}). The representation γ_{n} is not a pullback from a representation of SO(n). The associated vector bundle ΣM := Spin(M) ×_{γ_{n}} Σ_{n} is called spinor bundle and its sections are spinor fields. The Dirac operator (see [LM89] for a definition) is an elliptic operator acting on the space of smooth spinor fields.

Large parts of this article will deal with the case M = T^{2}. In this case many of our definitions simplify. Let f : ℝ^{2} → T^{2} be a smooth covering map with deck transformation group ℤ^{2} acting by translations. Then

τ_{f} : T^{2} × GL^{+}(2) → GL^{+}(T^{2})
(f(p), A) ↦ (∂_{x}f(p), ∂_{y}f(p)) · A

yields a trivialization of GL^{+}(T^{2}).

Definition. The trivial spin structure on T^{2} (with respect to f) is the one given by σ_{f} := (G̃L^{+}(T^{2}), ϑ) with

G̃L^{+}(T^{2}) := T^{2} × G̃L^{+}(2) ϑ := τ_{f} ∘ (id ×Θ).

Consider the bijection

ι_{f} : H^{1}(T^{2}, ℤ_{2}) → 𝔖𝔭𝔦𝔫(T^{2}), χ ↦ χ + σ_{f}.

The following proposition shows that ι_{f} does not depend on the choice of f. This will allow us to identify H^{1}(T^{2}, ℤ_{2}) and 𝔖𝔭𝔦𝔫(T^{2}) via ι_{f}.

PROPOSITION 4.1. Let (G̃L^{+}(T^{2}), ϑ) be a spin structure on T^{2}. Let χ_{f} be the element in H^{1}(T^{2}, ℤ_{2}) =

Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}) with ι_{f}(χ_{f}) = (G̃L^{+}(T^{2}), ϑ). Fix a complex structure J on T T^{2}.

Then for any non-contractible smooth embedding c : S^{1} → T^{2} the following conditions are equivalent

(1) χ_{f}([c]) = 1.

(2) (ċ, J(ċ)) : S^{1} → GL^{+}(T^{2}) lifts to G̃L^{+}(T^{2}) via ϑ.

Characterization (2) is independent from the choice of f, characterization (1) is independent from the choice of J. Therefore ι_{f} depends neither on f nor J. The above proposition is an immediate consequence of the following lemma.

LEMMA 4.2. Let c : S^{1} → ℝ^{2}/ℤ^{2} be a non-contractible smooth embedding. Choose a lift C : ℝ → ℝ^{2} = ℂ, i. e. C(t)/ℤ^{2} = c(e^{2πit}) . Then

(1) the homology class [c] ∈ H_{1}(ℝ^{2}/ℤ^{2}, ℤ) is primitive, i. e. not a multiple of another element in H_{1}(ℝ^{2}/ℤ^{2}, ℤ).

(2) the map

v(c) : S^{1} → S^{1}, e^{2πit} ↦ \frac{Ċ(t)}{|Ċ(t)|}

has degree 0.

Proof. The curve c can be lifted to the cylinder Z := ℝ^{2}/⟨[c]⟩. The lift will be denoted by c^{Z}. It is a simple closed curve generating π_{1}(Z). By Jordan’s theorem about simple closed curves in ℝ^{2} we know that this curve divides Z into two connected components Z^{+} and Z^{−}. Each of the components contains one end of the cylinder.

Let us assume that [c] is not primitive, i. e. [c] = k · a with k ∈ ℕ, k ≥ 2 and a ∈ H_{1}(M, ℤ) primitive. The action of a on Z maps Z^{+} to Z^{+} and Z^{−} to Z^{−}. Hence the image of c^{Z} is mapped to itself. This contradicts k ≥ 2. Thus we have proven (1).

Now let c_{1} : S^{1} → T^{2} be another embedding, homotopic to c. A suitable lift c_{1}^{Z} of c_{1} divides Z^{+} into a bounded and an unbounded part. The bounded part has c^{Z} and c_{1}^{Z} as boundaries and has Euler characteristic 0. Therefore the Gauss-Bonnet theorem for the Euclidean metric on Z yields v(c) = v(c_{1}). Thus the lemma only has to be checked for one representative in each primitive class. As this is trivial, (2) follows. □

From now on we will identify 𝔖𝔭𝔦𝔫(T^{2}) with H^{1}(T^{2}, ℤ_{2}) and Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2}). Frequently, we will use the term “spin homomorphism” instead of “spin structure” in order to indicate that we regard the spin structure as an element in Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2}).

From Proposition 5.2 below it is clear that the trivial spin structure is the only spin structure such that 0 is in the spectrum of the Dirac operator D. Therefore our definition of “trivial spin structure” coincides with the definition in section 2.

Remark. On oriented surfaces there is an alternative approach to defining spin structures. We fix a conformal structure on M. Therefore T M is a complex line bundle. A line bundle spin structure is a pair (Σ^{+}M, ϑ) of a complex line bundle Σ^{+}M and a map ϑ : Σ^{+}M → T M satisfying

ϑ(z · q) = z^{2} · ϑ(q), ∀q ∈ Σ^{+}M, z ∈ ℂ.

It is not hard to show that there is a natural bijection from the set of line bundle spin structures to the set of spin structures. For M = T^{2}, the trivial spin structure is characterized by the fact that for any non-contractible embedding S^{1} → T^{2} the tangent vector field ċ : S^{1} → T T^{2} lifts to Σ^{+}T^{2}. The line bundle spin structure definition is used by [KS97] for example. The Arf invariant [KS97] can also be used to distinguish the trivial spin structure from the non-trivial ones. The Arf invariant is equal to −1 for the trivial spin structure, and equal to 1 for all others.

5 Comparing spectra of conformal manifolds

In this section we will compare Dirac and Laplace eigenvalues on 2-tori. We recall a proof of a well-known proposition (see e.g. [Dod82, Proposition 3.3] for a more general version).

PROPOSITION 5.1. Let M be a compact 2-dimensional manifold with two conformal metrics g̃ and g = e^{2u}g̃. The eigenvalues of the Laplacian on functions corresponding to g and g̃ will be denoted as 0 = λ_{0} < λ_{1} ≤ λ_{2} … and 0 = λ̃_{0} < λ̃_{1} ≤ λ̃_{2} … respectively.

Then

λ_{i} min_{m∈M} e^{2u(m)} ≤ λ̃_{i} ≤ λ_{i} max_{m∈M} e^{2u(m)} ∀i = 1, 2, ….

Proof. Let f_{0}, …, f_{i} be eigenfunctions of Δ_{g} to the eigenvalues λ_{0}, …, λ_{i}. Let U_{i} be the subspace of V := C^{∞}(T^{2}) generated by f_{0}, …, f_{i}. We are bounding λ̃_{i} by the Rayleigh quotient:

λ̃_{i} ≤ max_{f∈U_{i}−\{0\}} \frac{(Δg̃f, f)g̃}{(f, f)g̃} .

We obtain for the numerator and the denominator:

(Δg̃f, f)g̃ = ∫ (Δg̃f)f̄ dvolg̃ = ∫ (Δ_{g}f)f̄ dvol_{g}
= (Δ_{g}f, f)_{g} ≤ λ_{i}(f, f)_{g}

(f, f)g̃ = ∫ ff̄dvolg̃ = ∫ ff̄e^{−2u} dvol_{g} ≥ e^{−2 max u} (f, f)_{g}.

Therefore we obtain

λ̃_{i} ≤ λ_{i} e^{2 max u}.

The other inequality can be proven in a completely analogous way. □

There is a similar proposition for the Dirac operator.

PROPOSITION 5.2. Let M be a compact manifold with two conformal metrics g̃ and g = e^{2u}g̃. Let D and D̃ be the corresponding Dirac operators with respect to a common spin structure. We denote the eigenvalues of D^{2} by μ_{1} ≤ μ_{2} ≤ … and the ones of D̃^{2} by μ̃_{1} ≤ μ̃_{2} ≤ . . ..

Then

μ_{i} min_{m∈M} e^{2u(m)} ≤ μ̃_{i} ≤ μ_{i} max_{m∈M} e^{2u(m)} ∀i = 1, 2, …

Proof. Let n := dim M. We have

dvol_{g} = e^{nu}dvolg̃.

There is an isomorphism of vector bundles [Hit74], [Bau81, Satz 3.14] or [Hij86, 4.3.1]

ΣM → Σ̃M
Ψ ↦ Ψ̃

over the identity id : M → M satisfying

D̃(Ψ̃) = e^{u}D̃Ψ

and

|Ψ̃| = e^{\frac{n−1}{2} u}|Ψ|.

Let (Ψ_{i} | i = 1, 2, …) be an orthonormal basis of the sections of ΣM with Ψ_{i} being an eigenspinor of D^{2} to the eigenvalue μ_{i} with respect to the flat metric g̃. The vector space spanned by Ψ_{1}, …, Ψ_{i} will be denoted by U_{i}.

We can bound μ̃_{i} by the Rayleigh quotient

μ̃_{i} ≤ max_{_{Ψ}∈U_{i}−\{0\}} \frac{(D̃Ψ̃, D̃Ψ̃)g̃}{(Ψ̃, Ψ̃)g̃} .

We look at the numerator and the denominator separately:

(D̃Ψ̃, D̃Ψ̃)g̃ = ∫ e^{2u}⟨D̃Ψ, D̃Ψ⟩ dvolg̃
= ∫ e^{2u+(n−1)u}⟨DΨ, DΨ⟩ dvolg̃
= ∫ e^{u}⟨DΨ, DΨ⟩ dvol_{g}
≤ (DΨ, DΨ)_{g} max_{m∈M} e^{u}
≤ μ_{i} (Ψ, Ψ)_{g} max_{m∈M} e^{u}

(Ψ̃, Ψ̃)g̃ = ∫ ⟨Ψ̃, Ψ̃⟩ dvolg̃
= ∫ e^{−u} ⟨Ψ, Ψ⟩ dvol_{g}
≥ e^{− max u}(Ψ, Ψ)_{g}

Thus

μ̃_{i} ≤ μ_{i} max_{m∈M} e^{2u}

which is one of the inequalities stated in the proposition.

The other inequality can be proven in a completely analogous way. □

6 Systoles and norms on H^{1}(T^{2}, ℤ_{2})

In this section we define norms on the space of spin structures 𝔖𝔭𝔦𝔫(M). These norms are strongly related to systoles.

Recall that for any compact Riemannian manifold (M, g), the space H^{1}(M, ℝ) carries a natural L^{p}-norm defined to be the quotient norm of the L^{p}-norm on 1-forms

∥α∥_{L^{p}} := inf {∥ω∥_{L^{p}} | ω closed 1-form representing α} .

For p = ∞ this norm is the so-called stable norm and for p = dim M it is invariant under conformal changes of the metric.

In our special case M = T^{2}, we know that Γ^{∗} = H^{1}(T^{2}, ℤ) = Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ) is a lattice in H^{1}(T^{2}, ℝ) and that the surjective map

P : \frac{1}{2} Γ^{∗} → Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2}) = H^{1}(T^{2}, ℤ_{2})
α( .) ↦ (−1)^{2α( .)}

has kernel Γ^{∗}.

Definition. The L^{p}-norm on H^{1}(T^{2}, ℤ_{2}) is the quotient norm of the L^{p}-norm on Γ^{∗} with respect to the quotient map P, i. e. for η ∈ Hom_{ℤ}(H_{1}(T^{2}, ℤ), ℤ_{2})

∥η∥_{L^{p}} := inf \{∥α∥_{L^{p}} | α ∈ \frac{1}{2}Γ^{∗}, P (α) = η\} .

Therefore we have norms on the space of spin structures on T^{2}. The L^{2}-norm is of particular interest as it is invariant under conformal changes and therefore it is a spin-conformal invariant. In the following section it will turn out that the smallest eigenvalue of D^{2} on a flat torus with spin structure χ is

\frac{4π^{2} ∥χ∥_{L^{2}}^{2}}{area} .

Another quantity will be used for our estimate of osc u (section 9): The cosystole σ_{1} is defined to be

σ_{1}(T^{2}, g) := inf {∥α∥_{L^{2}} | α ∈ Γ^{∗} − {0}} .

For flat tori the first positive eigenvalue of the Laplacian is

\frac{4π^{2} σ_{1}^{2}}{area} .

The aim of the rest of this section is to relate the L^{2}-norms to some systolic data.

Definition. For a Riemannian 2-torus (T^{2}, g) with spin structure χ we define the systole sys_{1}(T^{2}, g) ∈ ℝ, the spinning systole spin-sys_{1}(T^{2}, g, χ) ∈ ℝ ∪ {∞} and the non-spinning systole nonspin-sys_{1}(T^{2}, g, χ) ∈ ℝ to be

sys_{1}(T^{2}, g) := inf {length(γ) | γ is a non-contractible loop.}
spin-sys_{1}(T^{2}, g, χ) := inf {length(γ) | γ is a loop with χ([γ]) = −1.}
nonspin-sys_{1}(T^{2}, g, χ) := inf {length(γ) | γ is a non-contractible loop with χ([γ]) = 1
and [γ] is a primitive element in H_{1}(T^{2}, ℤ).}

An element α ∈ H_{1}(T^{2}, ℤ) is called primitive if there are no k ∈ ℕ, k ≥ 2, β ∈ H_{1}(T^{2}, ℤ) with α = k · β.

These quantities have the following relationships

sys_{1}(T^{2}, g) = min{spin-sys_{1}(T^{2}, g, χ), nonspin-sys_{1}(T^{2}, g, χ)}
sys_{1}(T̂^{2}, g) = min{2 · spin-sys_{1}(T^{2}, g, χ), nonspin-sys_{1}(T^{2}, g, χ)}

where T̂^{2} is the covering of T^{2} associated to ker χ ⊂ π_{1}(T^{2}). This covering is 2-fold for non-trivial χ, and T̂^{2} = T^{2} for χ ≡ 1.

PROPOSITION 6.1. Let g be any Riemannian metric on T^{2} and let χ be any spin homomorphism. There is a flat metric g_{0} which is conformal to g. This metric g_{0} is unique up to a multiplicative constant.

Furthermore, the following inequalities hold:

(a) \frac{sys_{1}(T^{2}, g)^{2}}{area(T^{2}, g)} ≤ \frac{sys_{1}(T^{2}, g_{0})^{2}}{area(T^{2}, g_{0})} = σ_{1}(T^{2}, g_{0})^{2} = σ_{1}(T^{2}, g)^{2}

(b) \frac{nonspin-sys_{1}(T^{2}, g, χ)^{2}}{area(T^{2}, g)} ≤ \frac{nonspin-sys_{1}(T^{2}, g_{0}, χ)^{2}}{area(T^{2}, g_{0})}

= 4 ∥χ∥_{L^{2}(T^{2},g_{0})}^{2} = 4 ∥χ∥_{L^{2}(T^{2},g)}^{2}

(c) \frac{spin-sys_{1}(T^{2}, g, χ)^{2}}{area(T^{2}, g)} ≤ \frac{spin-sys_{1}(T^{2}, g_{0}, χ)^{2}}{area(T^{2}, g_{0})}

(d) \frac{spin-sys_{1}(T^{2}, g_{0}, χ)^{2}}{area(T^{2}, g_{0})} ≥ \frac{1}{4 ∥χ∥_{L^{2}(T^{2},g_{0})}^{2}}

(e) For any η ∈ H^{1}(T^{2}, ℤ_{2}) and 1 ≤ p ≤ q ≤ ∞

∥η∥_{L^{p}(T^{2},g_{0})} area(T^{2}, g_{0})^{−(1/p)} = ∥η∥_{L^{q}(T^{2},g_{0})} area(T^{2}, g_{0})^{−(1/q)}
∥η∥_{L^{p}(T^{2},g)} area(T^{2}, g)^{−(1/p)} ≤ ∥η∥_{L^{q}(T^{2},g)} area(T^{2}, g)^{−(1/q)}
∥η∥_{L^{2}(T^{2},g_{0})} = ∥η∥_{L^{2}(T^{2},g)}

(f) For any η ∈ H^{1}(T^{2}, ℤ_{2}) and 1 ≤ p ≤ 2 ≤ q ≤ ∞

∥η∥_{L^{p}(T^{2},g)} area(T^{2}, g)^{(\frac{1}{2} − \frac{1}{p})} ≤ ∥η∥_{L^{p}(T^{2},g_{0})} area(T^{2}, g_{0})^{(\frac{1}{2} − \frac{1}{p})}
∥η∥_{L^{q}(T^{2},g)} area(T^{2}, g)^{(\frac{1}{2} − \frac{1}{q})} ≥ ∥η∥_{L^{q}(T^{2},g_{0})} area(T^{2}, g_{0})^{(\frac{1}{2} − \frac{1}{q})}

We have equality in the inequalities of (a)–(c) if and only if g is flat.

For the characterization of the equality case in (d) we choose a lattice Γ together with an isometry I : ℝ^{2}/Γ → (T^{2}, g_{0}). Then equality in (d) is equivalent to the fact that there are generators γ_{1}, γ_{2} for the lattice Γ satisfying γ_{1} ⊥ γ_{2}, I^{∗}(χ)(γ_{1}) = 1 and I^{∗}(χ)(γ_{2}) = −1.

Proof. The existence and uniqueness of g_{0} follows from the uniformization theorem for 2-dimensional tori. The equations for the flat metric g_{0} follow directly from elementary calculations. As already stated previously, the L^{2}-norm is invariant under conformal changes, thus the last equations in (a), (b) and (e) hold. The inequality in (e) follows from the Hölder inequality.

The first equation in (e) then follows from the fact, that η is represented by a real harmonic 1-form ω with ∥η∥_{L^{1}(T^{2},g_{0})} = ∥ω∥_{L^{1}(T^{2},g_{0})}. The pointwise norm |ω|_{g_{0}} is constant and therefore

∥ω∥_{L^{1}(T^{2},g_{0})} = ∥ω∥_{L^{∞}(T^{2},g_{0})} area(T^{2}, g_{0}) ≥ ∥η∥_{L^{∞}(T^{2},g_{0})} area(T^{2}, g_{0}).

The inequalities in (f) follow from (e).

The remaining inequalities in (a), (b) and (c) are direct consequences from Lemma 6.2 below.

The discussion of the equality case is straightforward. □

LEMMA 6.2 ([Amm98, Prop. 3.7.2]). With the notation of the previous proposition we define for v ∈ H_{1}(T^{2}, ℤ)

L_{g}(v) := min {length_{g}(c) | c : S^{1} → 𝕋^{2} represents v} ,

and L_{g_{0}}(v) similarly. Then

\frac{L_{g}(v)^{2}}{area(T^{2}, g)} ≤ \frac{L_{g_{0}}(v)^{2}}{area(T^{2}, g_{0})}.

We have equality for v ≠ 0 if and only if g is flat.

Proof of lemma. The proof of the lemma follows the pattern of the proof of Loewner’s theorem in [Gro81, 4.1].

Let g = e^{2u}g_{0}. We start with a minimizer c of L_{g_{0}}(v). There is an isometric torus action on (T^{2}, g_{0}) acting by translations. Translation by x ∈ T^{2} will be denoted by L_{x}. An easy calculation shows that

∫_{T^{2},g_{0}} dx length_{g}(L_{x}(c)) = L_{g_{0}}(v) ∫_{T^{2},g_{0}} dx e^{u(x)} ≤ L_{g_{0}}(v) area(T^{2}, g_{0})^{1/2} area(T^{2}, g)^{1/2}.

Because the left hand side is an upper bound for L_{g}(v) area(T^{2}, g_{0}), the inequality of the lemma follows. The case of equality is then obvious. □

7 Spectra of flat 2-tori

In this section we recall the well-known formulas for the spectrum of the Laplacian and of the Dirac operator on flat 2-tori.

Because it is clear how the eigenvalues change under rescaling we will restrict to the case

T^{2} = \frac{ℝ^{2}}{Γ_{xy}} Γ_{xy} = span \{( \begin{matrix}1 \\ 0 \end{matrix} ) , ( \begin{matrix}x \\ y \end{matrix} )\} , y > 0

where T^{2} carries the metric g_{0} induced by the Euclidean metric of ℝ^{2}. The dual lattice Γ_{xy}^{∗} := H^{1}(T^{2}, ℤ) = Hom_{ℤ}(Γ_{xy}, ℤ) is generated by the vectors

γ_{1} := ( \begin{matrix}1 \\ −x/y \end{matrix} ) und γ_{2} := ( \begin{matrix}0 \\ 1/y \end{matrix} ) .

The function

f_{γ} : T^{2} → ℂ f_{γ}(x) := exp (2πi ⟨γ, x⟩) γ ∈ Γ_{xy}^{∗}

is an eigenfunction of the Laplace operator Δ on complex valued functions to the eigenvalue 4π^{2}|γ|^{2} where | . | denotes the Euclidean norm on ℝ^{2}. Moreover, the family (f_{γ}|γ ∈ Γ_{xy}^{∗}) is a complete system of eigenfunctions. Note that γ can also be viewed as a 1-form on T^{2} and if ∥ . ∥_{L^{2}} is the L^{2}-norm defined in the previous section then

∥γ∥_{L^{2}}^{2} = |γ|^{2}area.

Therefore we obtain

PROPOSITION 7.1. The spectrum of the Laplacian on T^{2} is given by the family

\{\frac{4π^{2} ∥γ∥_{L^{2}}^{2}}{area} | γ ∈ Γ_{xy}^{∗}\}

where each eigenvalue appears with the correct multiplicity.

The first three eigenvalues can be easily expressed using the invariants of the previous section

λ_{0} = 0 λ_{1} = λ_{2} = \frac{4π^{2} σ_{1}^{2}}{area} .

The eigenfunctions and eigenvalues of the square of the Dirac operator are very similar if the spin structure is trivial. Let ψ_{1} and ψ_{2} be parallel orthonormal spinors on T^{2}, then (f_{γ}ψ_{j}|j = 1, 2; γ ∈ Γ_{xy}^{∗}) is a complete system of eigenfunctions to the eigenvalues 4π^{2}|γ|^{2}. Therefore the eigenvalues μ_{1} ≤ μ_{2} ≤ μ_{3} … are the same as for the Laplace operator, but the multiplicities are doubled. In particular

μ_{1} = μ_{2} = 0 μ_{3} = μ_{4} = μ_{5} = μ_{6} = \frac{4π^{2} σ_{1}^{2}}{area} .

Now we assume that T^{2} carries a non-trivial spin structure. After a rescaling of the metric and an orthonormal

transformation of ℝ^{2} we can assume that the spin structure is trivial on ( \begin{matrix}1 \\ 0 \end{matrix} ) and non-trivial on ( \begin{matrix}x \\ y \end{matrix} ) and that

0 ≤ x ≤ \frac{1}{2},  (x − \frac{1}{2})^{2} + y^{2} ≥ \frac{1}{4},  y > 0.  (6)

The set of all (x, y) satisfying (6) is called the spin-conformal moduli space Mspin. The elements of Mspin correspond to equivalence classes of tori with non-trivial spin structures under the equivalence relation of conformal diffeomorphisms preserving the spin structure.

Let (ψ_{1}, ψ_{2}) be a basis of parallel sections of the spinor bundle on ℝ^{2} and assume that they are pointwise orthogonal. Then

Ψ_{j,γ} := exp(2πi ⟨γ, x⟩) ψ_{j}, γ ∈ Γ_{xy}^{∗} + \frac{γ_{2}}{2}

is a spinor that is invariant under the action of Γ_{xy}. Thus it defines an eigenspinor for D^{2} : ΣT^{2} → ΣT^{2} with eigenvalue 4π^{2}|γ|^{2} and the family (Ψ_{j,γ}|j = 1, 2; γ ∈ Γ_{xy}^{∗} + (γ_{2}/2)) is a complete system of eigenspinors.

We obtain a similar proposition as above.

PROPOSITION 7.2 ([Fri84]). Assume that T^{2} carries a non-trivial spin structure as above. Then the spectrum of the square of the Dirac operator D^{2} on T^{2} is given by the family

\{\frac{4π^{2} ∥γ∥_{L^{2}}^{2}}{area} | γ ∈ Γ_{xy}^{∗} + \frac{γ_{2}}{2} \} ,

and the multiplicity of each eigenvalue in the spectrum of D^{2} is twice the multiplicity in the family.

We want to prove that Γ_{xy}^{∗} + (γ_{2}/2) contains no vector that is shorter than γ_{2}/2. For this we need a lemma.

LEMMA 7.3. If linearly independent vectors v_{1}, v_{2} ∈ ℝ^{2} satisfy

0 ≤ ⟨v_{1}, v_{2}⟩ ≤ |v_{1}|^{2} ≤ |v_{2}|^{2},

then for any integers a, b with a ≠ 0 and b ≠ 0 the following inequality holds

|av_{1} + bv_{2}| ≥ |v_{2} − v_{1}|.

If |av_{1} + bv_{2}| = |v_{2} − v_{1}|, then |a| = |b| = 1.

Proof of lemma. Let |av_{1} +bv_{2}| ≤ |v_{2} −v_{1}|. Without loss of generality we can assume that a and b are relatively prime. We obtain

a^{2}|v_{1}|^{2} − 2 |ab| · ⟨v_{1}, v_{2}⟩ + b^{2}|v_{2}|^{2} ≤ |v_{1}|^{2} − 2 ⟨v_{1}, v_{2}⟩ + |v_{2}|^{2}

and therefore

(a^{2} + b^{2} − 2) |v_{1}|^{2} ≤ (a^{2} − 1) |v_{1}|^{2} + (b^{2} − 1) |v_{2}|^{2}
≤ 2(|ab| − 1) ⟨v_{1}, v_{2}⟩ ≤ 2(|ab| − 1) |v_{1}|^{2}.

Thus (|a| − |b|)^{2} ≤ 0 holds, i. e. |a| = |b|, and as we assumed that a and b are relatively prime we obtain |a| = |b| = 1. Because of |v_{1} + v_{2}| ≥ |v_{2} − v_{1}| the lemma holds. □

COROLLARY 7.4. If (x, y) ∈ Mspin, then:

(a) There is no vector in Γ_{xy}^{∗} + (γ_{2}/2) that is shorter than γ_{2}/2.

(b) The shortest vectors in Γ_{xy}^{∗} − {0} have length

min \{\frac{1}{y} , \frac{\sqrt{x^{2} + y^{2}}}{y} \} .

Proof.

(a) Because of relations (6) the vectors v_{1} := γ_{1}/2 and v_{2} := (γ_{1} + γ_{2})/2 satisfy the conditions of the lemma. Any element γ of Γ_{xy}^{∗} + (γ_{2}/2) can be written as av_{1} + bv_{2}, a, b ∈ ℤ − {0}. The lemma yields

|γ| ≥ |v_{2} − v_{1}| = \frac{|γ_{2}|}{2}.

(b) This time we set v_{1} = γ_{1} and v_{2} = γ_{1} + γ_{2}. As before 0 ≤ ⟨v_{1}, v_{2}⟩ ≤ |v_{1}|^{2} ≤ |v_{2}|^{2}. Any γ ∈ Γ_{xy}^{∗} − {0} is either a multiple of v_{1} or v_{2} (then |γ|^{2} ≥ |v_{1}|^{2} = |γ_{1}|^{2} = 1 + (x^{2}/y^{2})) or

|γ| ≥ |v_{2} − v_{1}| = \frac{1}{y} .

□

Thus the smallest eigenvalue μ_{1} of D^{2} satisfies

μ_{1} = π^{2}|γ_{2}|^{2} = \frac{π^{2}}{y^{2}} .  (7)

Using the notations of the previous section we see easily that the L^{2}-norm of the spin-structure χ satisfies

∥χ∥_{L^{2}}^{2} = \frac{1}{4y} .

With area = y we obtain

μ_{1} area = 4π^{2} ∥χ∥_{L^{2}}^{2} .

Analogously, we see for the cosystole that

σ_{1}^{2} = min \{\frac{1}{y} , \frac{x^{2} + y^{2}}{y} \} .

8 Regular bipartitions of 2-tori

Definition. A regular bipartition of T^{2} is a pair (X_{1}, X_{2}) of disjoint open subsets X_{i} ⊂ T^{2} such that ∂X_{1} = ∂X_{2} is a smooth 1-manifold, i. e. ∂X_{1} = ∂X_{2} is a disjoint union of finitely many smooth circles. In particular this implies T^{2} = X_{1} ˙∪X_{2} ˙∪∂X_{1}.

PROPOSITION 8.1. Let (X_{1}, X_{2}) be a regular bipartition of T^{2}. Then exactly one of the following conditions is satisfied (i) The inclusion X_{1} → T^{2} induces the trivial map π_{1}(X_{1}) → π_{1}(T^{2}).

(ii) The inclusion X_{2} → T^{2} induces the trivial map π_{1}(X_{2}) → π_{1}(T^{2}).

(iii) The boundary ∂X_{1} has at least two components that are non-contractible in T^{2}.

Proof. Assume a regular bipartition (X_{1}, X_{2}) satisfies (iii), then ∂X_{1} contains a non-contractible loop. By a small perturbation we can achieve that this loop lies completely in X_{1}. Therefore π_{1}(X_{1}) → π_{1}(T^{2}) is not trivial. Hence (X_{1}, X_{2}) does not satisfy (i). Similarly we prove that it does not satisfy (ii).

Now assume that a regular bipartition (X_{1}, X_{2}) satisfies both (i) and (ii). Van-Kampen’s theorem implies π_{1}(T^{2}) = 0. Therefore we have shown that at most one of the three conditions is satisfied.

It remains to show that at least one condition is satisfied. For this we assume that neither (i) nor (ii) is satisfied, i. e. there are continuous paths c_{i} : S^{1} → X_{i} that are non-contractible within T^{2}. Obviously ∂X_{1} is homologous to zero. We will show that at least one component of ∂X_{1} is non-homologous to zero. Then there has to be a second component that is non-homologous to zero, because [∂X_{1}] = 0 is the sum of the homology classes of the components.

We argue by contradiction. Assume that each component of ∂X_{1} is homologous to zero. Let π : ℝ^{2} → T^{2} be the universal covering. Then π^{−1}(∂X_{1}) is diffeomorphic to a disjoint union of countably many S^{1}. We write

π^{−1}(∂X_{1}) = ˙ ⋃_{i∈ℕ}Y_{i}

with Y_{i} ≅ S^{1}. We choose lifts c̃_{i} : ℝ → ℝ^{2} of c_{i}, i. e. π (c̃_{i}(t + z)) = c_{i}(t) for all t ∈ [0, 1], z ∈ ℤ and i = 1, 2. Then we take a path γ̃ : [0, 1] → ℝ^{2} joining c̃_{1}(0) to c̃_{2}(0). We can assume that γ̃ is transversal to any Y_{i}. We define I to be the set of all i ∈ ℕ such that Y_{i} meets the trace of γ̃. The set I is finite. Using the Theorem of Jordan and Schoenfliess about simple closed curves in ℝ^{2} we can inductively construct a compact set K ⊂ ℝ^{2} with boundary ⋃_{i∈I} Y_{i}. The number of intersections of γ̃ with ⋃_{i∈I} Y_{i} is odd. Thus, either c̃_{1}(0) or c̃_{2}(0) is in the interior of K. But if c̃_{i}(0) is in the interior of K, then the whole trace c̃_{i}(ℝ) is contained in K. Furthermore, c̃_{i}(ℝ) = π^{−1} (c_{i}([0, 1])) is closed and therefore compact. This implies that c_{i} is homologous to zero in contradiction to our assumption. □

9 Controling the conformal scaling function

Let T^{2} carry an arbitrary metric g. According to the uniformization theorem we can write g = e^{2u}g_{0} with a real function u : T^{2} → ℝ and a flat metric g_{0}. The function u is unique up to adding a constant.

The aim of this section is to estimate the quantity osc u := max u − min u. The estimate is similar to an estimate of the author in a previous publication [Amm00, Theorem 3.1]. The main difference is that the previous estimate needed the assumption

∥K_{g}∥_{L^{p}(T^{2},g)} (area(T^{2}, g))^{1−(1/p)} < 4π

which is no longer needed in the estimate presented here.

THEOREM 9.1. We assume

∥K_{g}∥_{L^{1}(T^{2},g)} < 4π.

Then for any p ∈ ]1, ∞[ we obtain a bound for the oscillation of u

(a) osc u ≤ S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} (area(T^{2}, g))^{1−(1/p)} , p, σ_{1}(T^{2}, g)^{−2}),

(b) osc u ≤ S(∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} (area(T^{2}, g))^{1−(1/p)} , p, \frac{area(T^{2},g)}{sys_{1}(T^{2},g)^{2}} ), where we use the definition

S(K_{1}, K_{p}, p, V) := \frac{p}{p − 1} [\frac{K_{p}}{4π} + \frac{1}{2}| log (1 − \frac{K_{1}}{4π} ) | + \frac{K_{1}}{8π − 2K_{1}} log (\frac{2K_{p}}{K_{1}} )] + \frac{K_{1}V}{8}

for K_{1} > 0 and S(0, K_{p}, p, V) := \frac{p}{p−1} \frac{K_{p}}{4π}.

Note that for sequences (K_{i}) and (K′_{i}), satisfying K_{i} → 0 and K′_{i} ≤ K_{i}, we have S(K_{i}, K′_{i}, p, V) → 0.

COROLLARY 9.2. Let F be a family of Riemannian metrics conformal to the flat metric g_{0}. Assume that there are constants K_{1} ∈ ]0, 4π[ and K_{p} ∈ ]0, ∞[, p ∈ ]1, ∞[ with

∥K_{g}∥_{L^{1}(T^{2},g)} ≤ K_{1} and ∥K_{g}∥_{L^{p}(T^{2},g)} (area(T^{2}, g))^{1− \frac{1}{p}} ≤ K_{p} for any g ∈ F.

Then the oscillation osc u_{g} of the scaling function corresponding to g is uniformly bounded on F by

osc u_{g} ≤ S(K_{1}, K_{p}, p, V(T^{2}, g_{0})).

Before proving the theorem we will present some examples showing that the theorem and the corollary no longer hold if we drop one of the assumptions ∥K_{g}∥_{L^{1}(T^{2},g)} ≤ K_{1} < 4π or ∥K_{g}∥_{L^{p}(T^{2},g)} (area(T^{2}, g))^{1− \frac{1}{p}} ≤ K_{p}.

Example. For any K_{1} > 0 there is a sequence (g_{i}) of Riemannian metrics with fixed conformal type, bounded volume, constant systole, with

∥K_{g_{i}}∥_{L^{1}(T^{2},g_{i})} ≤ K_{1} and osc u_{g_{i}} → ∞.

In order to construct such a sequence we take a flat torus and replace a ball by a rotationally symmetric surface which approximates a cone for i → ∞ (see [Amm00] for details).

Example. For any ε > 0 there is a sequence (g_{i}) of Riemannian metrics with fixed conformal type, bounded volume, constant systole, −1 ≤ K_{g_{i}} ≤ 1, ∥K_{g_{i}}∥_{L^{1}(T^{2},g_{i})} ≤ 4π + ε, ∥K_{g_{i}}∥_{L^{p}(T^{2},g_{i})} ≤ const and osc u_{g_{i}} → ∞. In order to construct such a sequence we take a ball out of a flat torus and replace it by a hyperbolic part, a cone of small opening angle, and a cap as indicated in the following picture. While the injectivity radius of the hyperbolic part shrinks to zero, the oscillation of u tends to infinity.

A

φ(A)

A_{−} A_{+} A_{#} Area_{g} := ; area(T^{2}, g)

min u

v_{−}

v_{+}

max u

In the picture the dots in the “limit space” indicate the hyperbolic part with injectivity radius tending to 0 and diameter tending to ∞.

Proof of Theorem 9.1. As Morse functions form a dense subset of the space of C^{∞}-functions with respect to the C^{∞}-topology, we can assume without loss of generality that u is a Morse function. We set Area_{g} := area(T^{2}, g) and Area_{0} := area(T^{2}, g_{0}). We define

G_{<}(v) := {x ∈ T^{2} | u(x) < v} G_{>}(v) := {x ∈ T^{2} | u(x) > v}

φ : [0, Area_{g}] → ℝ
A ↦ inf {sup_{x∈X} u(x) | X ⊂ T^{2} open, area(X) ≥ A}  (8)
= sup { inf_{x∈X^{c}} u(x) | X^{c} ⊂ T^{2} open, area(X^{c}) ≥ Area_{g} − A}  (9)

The infimum in (8) is actually a minimum and, as u is a Morse function, the only minimum is attained exactly for X = G_{<}(φ(A)). Similarly the supremum in (9) is attained exactly in X^{c} = G_{>}(φ(A)). The function φ is strictly increasing and is continuously differentiable. The inverse of φ is given by

φ^{−1}(v) = area(G_{<}(v)).

The differential φ′(A) is zero if and only if φ(A) is a critical value of u.

Now let v ∈ [min u, max u] be a regular value of u. We obtain

(φ^{−1})′ (v) = ∫_{∂G_{<}(v),g} \frac{1}{|du|_{g}} ≥ \frac{length(∂G_{<}(v), g)^{2}}{∫_{∂G_{<}(v),g} |du|_{g}}  (10)

where length(∂G_{<}(v), g) is the length of the boundary of ∂G_{<}(v) with respect to g. This inequality will yield an upper bound for φ′ which will provide in turn an upper bound for osc u = φ(Area_{g}) − φ(0) = ∫_{0}^{Area_{g}} φ′. We transform

∫_{∂G_{<}(v),g} |du|_{g} = ∫_{∂G_{<}(v)} ∗ du = − ∫_{G_{<}(v),g} Δ_{g}u = − ∫_{G_{<}(v),g} K_{g}.  (11)

The last equation follows from the Kazdan-Warner-equation Δ_{g}u = K_{g} [KW74]. We define κ using the Gaussian curvature function K_{g} : T^{2} → ℝ

κ : [0, Area_{g}] → ℝ, κ(A) := inf {sup_{x∈X} K_{g}(x) | X ⊂ T^{2} open, area(X) ≥ A} .

Any open subset X ⊂ T^{2} satisfies

∫_{0}^{area(X,g)} κ ≤ ∫_{X,g} K_{g} ≤ ∫_{Area_{g}−area(X,g)}^{Area_{g}} κ

and for X = T^{2} we have equality. Using Gauss-Bonnet theorem we see that

∫_{0}^{Area_{g}} κ = 0.

The right hand side of equation (11) now can be estimated as follows.

− ∫_{G_{<}(φ(A)),g} K_{g} ≤ − ∫_{0}^{A} κ = ∫_{A}^{Area_{g}} κ  (12)

Putting (10), (11) and (12) together, we obtain

φ′(A) ≤ \frac{∫_{A}^{Area_{g}} κ}{length(∂G_{<}(φ(A)), g)^{2}} .

Our next goal is to find suitable lower bounds for length(∂G_{<}(φ(A)).

Note that for any regular value v of u, (G_{<}(v), G_{>}(v)) is a regular bipartition of T^{2}. According to Proposition 8.1 exactly one of the following conditions is satisfied

(i) The inclusion G_{<}(v) → T^{2} induces the trivial map π_{1}(G_{<}(v)) → π_{1}(T^{2}).

(ii) The inclusion G_{>}(v) → T^{2} induces the trivial map π_{1}(G_{>}(v)) → π_{1}(T^{2}).

(iii) The boundary ∂G_{<}(v) has at least two components that are non-contractible in T^{2}.

If condition (i) is satisfied by v, it is obvious that it is also satisfied by v′ ∈ [0, v]. Similarly, if condition (ii) is satisfied by v, then it is also satisfied by v′ ∈ [v, Area_{g}].

v_{−} := sup{v ∈ [0, Area_{g}] | (i) is satisfied forv}
v_{+} := inf{v ∈ [0, Area_{g}] | (ii) is satisfied forv}
A_{±} := φ^{−1}(v_{±}).

In each of the three cases we derive a different estimate for length(∂G_{<}(v), g) and therefore we obtain a different bound for φ′.

(i) In this case G_{<}(v) can be lifted to the universal covering ℝ^{2} of T^{2}. We will also write g and g_{0} for the pullbacks of g and g_{0} to ℝ^{2}. The isoperimetric inequality of the flat space (ℝ^{2}, g_{0}) yields

length(∂G_{<}(v), g_{0})^{2} ≥ 4π area(G_{<}(v), g_{0}).

Using the relations

length(∂G_{<}(v), g) = e^{v} length(∂G_{<}(v), g_{0})  (13)
area(G_{<}(v), g) ≤ e^{2v} area(G_{<}(v), g_{0})  (14)

we obtain

length(∂G_{<}(v), g)^{2} ≥ 4π area(G_{<}(v), g).  (15)

Together with the Hölder inequality

− ∫_{0}^{A} κ ≤ ∥K_{g}∥_{L^{p}(T^{2},g)} A^{1−(1/p)}

we get

φ′(A) = \frac{1}{(φ^{−1})′(φ(A))} ≤ \frac{− ∫_{0}^{A} κ}{length(∂G_{<}(φ(A)), g)^{2}}
≤ \frac{1}{4π} ∥K_{g}∥_{L^{p}(T^{2},g)} A^{− \frac{1}{p}}

Integration yields

v_{−} − min u = φ(φ^{−1}(v_{−})) − φ(0)
≤ \frac{p}{p − 1} \frac{1}{4π} ∥K_{g}∥_{L^{p}(T^{2},g)} (φ^{−1}(v_{−}))^{1−(1/p)}
≤ \frac{p}{p − 1} \frac{1}{4π} ∥K_{g}∥_{L^{p}(T^{2},g)} (Area_{g})^{1−(1/p)}  (16)

(ii) This case is similar to the previous one, but unfortunately because of opposite signs some estimates do not work as before. For example (14) and (15) are no longer true for G_{<}(v) replaced by G_{>}(v). Instead we use Topping’s inequality [Top98a, Top99].

(length(∂G_{>}(v), g))^{2} ≥ 4π Â − 2 ∫ Â ; 0(Â − a) κ(Area_{g} − a) da  (17)

with Â = area(G_{>}(v), g). Using the estimate

∫ Â ; 0(Â − a) κ(Area_{g} − a) da ≤ Â ∫ Â ; 0 max{0, κ(Area_{g} − a)} da
≤ \frac{Â}{2} ∥K_{g}∥_{L^{1}(T^{2},g)}

we obtain

(length(∂G_{>}(v), g))^{2} ≥ (4π − ∥K_{g}∥_{L^{1}(T^{2},g)}) Â.  (18)

The obvious inequality

∫_{Area_{g}−Â}^{Area_{g}} κ ≤ ∥max{0, K_{g}}∥_{L^{1}(T^{2},g)} ≤ (1/2) ∥K_{g}∥_{L^{1}(T^{2},g)}

yields

φ′(Area_{g} − Â) ≤ \frac{1}{Â} \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{8π − 2 ∥K_{g}∥_{L^{1}(T^{2},g)}} .

Integration yields

φ(Area_{g} − Â) − φ(A_{+}) ≤ log (\frac{Area_{g} − A_{+}}{Â} ) \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{8π − 2 ∥K_{g}∥_{L^{1}(T^{2},g)}} .

The right hand side converges to ∞ for Â → 0. Thus we have to improve our estimates for small Â. The integral in (17) also has the following bound.

∫ Â ; 0(Â − a) κ(Area_{g} − a) da ≤ (∫ Â ; 0(Â − a)^{q} da)^{1/q} · (∫ Â ; 0 |κ(Area_{g} − a)|^{p} da)^{1/p}

= (\frac{Â^{q+1}}{q + 1})^{1/q} · ∥K_{g}∥_{L^{p}(T^{2},g)}  (19)

where we wrote q := p/(p − 1) in order to simplify the notation.

We obtain a second lower bound on the length

(length(∂G_{>}(v), g))^{2} ≥ 4π Â − cÂ^{1+ \frac{1}{q}} ∥K_{g}∥_{L^{p}(T^{2},g)}  (20)

for any c ≥ 2/ ^{q} \sqrt{q + 1}, e. g. c = 2. Note that our assumption ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π does not imply that the right hand side of the above inequality is always positive. Although (20) is better for small Â, it is not strong enough to control the length for larger Â. However, for

Â < (\frac{4π}{c · ∥K_{g}∥_{L^{p}(T^{2},g)}} )^{q}

we use (20) and

∫_{Area_{g}−Â}^{Area_{g}} κ ≤ Â^{1/q} ∥K_{g}∥_{L^{p}(T^{2},g)}

to obtain the estimate

φ′(Area_{g} − Â) ≤ \frac{Â^{−1/p} ∥K_{g}∥_{L^{p}(T^{2},g)}}{4π − cÂ^{1/q} ∥K_{g}∥_{L^{p}(T^{2},g)}} .

With the substitution

w = w(A) = 4π − c(Area_{g} − A)^{1/q} ∥K_{g}∥_{L^{p}(T^{2},g)}

integration yields

φ(Area_{g}) − φ(A_{#}) = ∫_{A_{#}}^{Area_{g}} φ′(A) dA

≤ ∫_{w(A_{#})}^{w(Area_{g})} \frac{q}{c} \frac{1}{w} dw = \frac{q}{c} log \frac{w(Area_{g})}{w(A_{#})}
= \frac{q}{c} log \frac{4π}{4π − c(Area_{g} − A_{#})^{1/q} ∥K_{g}∥_{L^{p}(T^{2},g)}}

for any A_{#} between Area_{g} − (4π/(c · ∥K_{g}∥_{L^{p}(T^{2},g)}))^{q} and Area_{g}. We choose

A_{#} := max \{Area_{g} − (\frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{2 ∥K_{g}∥_{L^{p}(T^{2},g)}} )^{q} , A_{+}\} .

Finally we obtain the estimates

max u − φ(A_{#}) ≤ \frac{q}{c} log \frac{8π}{8π − c ∥K_{g}∥_{L^{1}(T^{2},g)}}  (21)

φ(A_{#}) − v_{+} ≤ q \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{8π − 2 ∥K_{g}∥_{L^{1}(T^{2},g)}} log (\frac{2 Area_{g}^{1/q} ∥K_{g}∥_{L^{p}(T^{2},g)}}{∥K_{g}∥_{L^{1}(T^{2},g)}} ) .  (22)

For c = 2 the right hand sides of these inequalities contribute two summands to the formula for S.

(iii) If v = φ(A) is a regular value of u between v_{−} and v_{+}, then ∂G_{<}(v) contains at least two components that are non-contractible in T^{2}. Hence, for any metric g̃ on T^{2} we get

length(∂G_{<}(v), g̃) ≥ 2 sys_{1}(T^{2}, g̃).

In order to prove (a) of Theorem 9.1 we apply this equation to g̃ := g_{0}. Using ∫_{A}^{Area_{g}} κ ≤ (1/2) ∥K_{g}∥_{L^{1}(T^{2},g)} and length(∂G_{<}(v), g) = e^{v} length(∂G_{<}(v), g_{0}) we obtain

φ′(A) ≤ e^{−2φ(A)} \frac{∫_{A}^{Area_{g}} κ}{4 sys_{1}(T^{2}, g_{0})^{2}}
≤ \frac{1}{8} e^{−2φ(A)} \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{sys_{1}(T^{2}, g_{0})^{2}} .  (23)

Integration yields

v_{+} − v_{−} = ∫_{A_{−}}^{A_{+}} φ′(A) dA

≤ \frac{1}{8} \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{sys_{1}(T^{2}, g_{0})^{2}} ∫_{A_{−}}^{A_{+}} e^{−2φ(A)} dA

≤ \frac{1}{8} \frac{∥K_{g}∥_{L^{1}(T^{2},g)}}{sys_{1}(T^{2}, g_{0})^{2}} Area_{0}  (24)

where we used Area_{0} = area(T^{2}, g_{0}) = ∫_{0}^{Area_{g}} e^{−2φ(A)} dA.

Together with inequalities (16), (21) and (22) we obtain statement (a) of the theorem.

Similarly, setting g̃ := g we get statement (b). □

10 Some “inverse” inequalities

In Proposition 6.1 and Lemma 6.2 we proved some inequalities relating the metric g to g_{0}. It is easy to prove that they also hold in the other direction if we add a factor like e^{2 osc u}.

Explicitely we obtain:

(a) \frac{sys_{1}(T^{2}, g)^{2}}{area(T^{2}, g)} ≥ e^{−2 osc u} \frac{sys_{1}(T^{2}, g_{0})^{2}}{area(T^{2}, g_{0})}

(b) \frac{nonspin-sys_{1}(T^{2}, g, χ)^{2}}{area(T^{2}, g)} ≥ e^{−2 osc u} \frac{nonspin-sys_{1}(T^{2}, g_{0}, χ)^{2}}{area(T^{2}, g_{0})}

(c) \frac{spin-sys_{1}(T^{2}, g, χ)^{2}}{area(T^{2}, g)} ≥ e^{−2 osc u} \frac{spin-sys_{1}(T^{2}, g_{0}, χ)^{2}}{area(T^{2}, g_{0})}

(d) For any η ∈ H^{1}(T^{2}, ℤ_{2}) and 1 ≤ p ≤ 2 ≤ q ≤ ∞

∥η∥_{L^{p}(T^{2},g)} area(T^{2}, g)^{(\frac{1}{2} − \frac{1}{p})} ≥ e^{(1− \frac{2}{p}) osc u} ∥η∥_{L^{p}(T^{2},g_{0})} area(T^{2}, g_{0})^{(\frac{1}{2} − \frac{1}{p})}
∥η∥_{L^{q}(T^{2},g)} area(T^{2}, g)^{(\frac{1}{2} − \frac{1}{q})} ≤ e^{(1− \frac{2}{q}) osc u} ∥η∥_{L^{q}(T^{2},g_{0})} area(T^{2}, g_{0})^{(\frac{1}{2} − \frac{1}{q})}

A combination of these inequalities together with our upper bound for osc u in the previous section enables us to compare the quantities under consideration for a flat and an arbitrary metric in the same (spin-)conformal class.

11 Proof of the main results

Combining the inequalities derived in the previous sections, we are now able to derive our main results.

Theorem 2.1 is a consequence of Proposition 7.2 together with Proposition 5.2 and Theorem 9.1. Theorem 2.2 then follows from the calculation of the first eigenvalue of D^{2} on flat tori at the end of section 7. Using the inequalities in Proposition 6.1 and section 10 we can derive Corollaries 2.3 and 2.4.

Similarly, Theorem 2.5 is a consequence of Proposition 7.1 together with Proposition 5.1 and Theorem 9.1. Theorem 2.6 then follows from the calculation of the first positive eigenvalue of Δ on flat tori at the end of section 7. Using the inequalities in Proposition 6.1 we obtain Corollary 2.7.

12 An application to the Willmore functional

In this section S^{3} always carries the metric g_{S^{3}} of constant sectional curvature 1. For any immersion F : T^{2} → S^{3} we define the Willmore functional

W(F) := ∫_{(T^{2},F^{∗}g_{S^{3}} )} |H_{T^{2}→S^{3}}|^{2} + 1

where H is the relative mean curvature of F (T^{2}) in S^{3} and integration is the usual integration of functions T^{2} → ℝ over the Riemannian manifold (T^{2}, F^{∗}g_{S^{3}}). Note that the mean curvature H of F (T^{2}) in ℝ^{4} satisfies

|H|^{2} = |H_{T^{2}→S^{3}}|^{2} + 1.

The Willmore conjecture states that W(F) ≥ 2π^{2}. Li and Yau [LY82, Fact 3] proved that the conjecture holds if F is not an embedding.

Any immersion F : T^{2} → S^{3} induces a spin structure φ_{F} on T^{2}. The spin structure φ_{F} is non-trivial if and only if F is regularly homotopic to an embedding. Thus for any immersion F which is regularly homotopic to an embedding, the pair (F^{∗}g_{S^{3}}, φ_{F}) defines an element (x, y) in the spin-conformal moduli space Mspin (defined in section 7). In order to shorten our notation we write [F] := (x, y) ∈ Mspin. If T^{2} already carries a spin structure, we say that F is spin iff φ_{F} = φ.

Li and Yau proved:

THEOREM 12.1 ([LY82, Theorem 1]). Let F : (T^{2}, g) → (S^{3}, g_{S^{3}}) be a conformal embedding. Let Area_{g} be the area of (T^{2}, g) and let λ_{1}^{g} be the first positive eigenvalue of the Laplacian Δ of (T^{2}, g) then

W(F) ≥ \frac{1}{2} λ_{1}^{g}Area_{g}.

0.5 1.0  0.5 x

y

Clifford torus Li and Yau proved the Willmore conjecture for these spin-conformal classes

For these spin-conformal classes we prove the Willmore conjecture under a curvature assumption

The spin conformal moduli space Mspin

Figure 1: The spin conformal moduli space

From this theorem the conjectured inequality W(F) ≥ 2π^{2} follows, if [F] lies in a compact subset of Mspin with positive measure (see Figure 1).

A similar lower bound for W(F) in terms of Dirac eigenvalues has been given by Bär.

THEOREM 12.2 ([Bär98b ]). Let F : (T^{2}, g, φ) → (S^{3}, g_{S^{3}}) be an isometric spin immersion. Then for the first eigenvalue μ_{1} of the square of the Dirac operator the inequality

W(F) ≥ μ_{1}Area_{g}

holds.

Note that this estimate is non-trivial only if F is regularly homotopic to an embedding.

Remark. At the end of this section we will show by example that in general “isometric” can not be replaced be “conformal” in this theorem.

Our goal now is to apply our previous estimates and derive lower bounds for W(F). One way to deduce such bounds is to combine the theorem with our lower estimates for the first eigenvalue of the square of the Dirac operator. These lower estimates for the Willmore functional are weaker than the ones derived by the author in [Amm00], therefore we skip this approach.

In this article, our approach is to modify the techniques of Theorem 12.2. This yields together with Theorem 9.1 new results about the Willmore functional.

As in the previous sections we define

S(K_{1}, K_{p}, p, V) := \frac{p}{p − 1} [\frac{K_{p}}{4π} + \frac{1}{2}| log (1 − \frac{K_{1}}{4π} ) | + \frac{K_{1}}{8π − 2K_{1}} log (\frac{2K_{p}}{K_{1}} )] + \frac{K_{1}V}{8}

for K_{1} > 0 and S(0, K_{p}, p, V) := 0.

THEOREM 12.3. Let F : T^{2} → S^{3} be an immersion of the 2-dimensional torus in S^{3} endowed with the standard metric g_{S^{3}}. Let F be regularly homotopic to an embedding. We set g := F^{∗}g_{S^{3}}. Let (x, y) = [F] ∈ Mspin. Then

W(F) ≥ \frac{π^{2}}{y} − \frac{1}{8} (osc u) ∥K_{g}∥_{L^{1}(T^{2},g)} .

In particluar if ∥K_{g}∥_{L^{1}(T^{2},g)} < 4π and any p > 1

W(F) ≥ \frac{π^{2}}{y} − \frac{1}{8} S ∥K_{g}∥_{L^{1}(T^{2},g)}

with S := S (∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, \frac{Area_{g}}{sys_{1}(T^{2},g)^{2}} ) or

S := S (∥K_{g}∥_{L^{1}(T^{2},g)} , ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)}, p, σ_{1}(T^{2}, g)^{−2}).

Proof. We write the induced metric g on T^{2} in the form g = e^{2u}g_{0} with g_{0} flat. Any Killing spinor on S^{3} with the Killing constant α = (1/2) induces a spinor ψ on (T^{2}, g) satisfying

D_{g}ψ = Hψ + νψ,

where

ν = γ(e_{1})γ(e_{2}) = ( \begin{matrix}−i & 0 \\ 0 & i \end{matrix} ) ∈ End (Σ^{+}T^{2} ⊕ Σ^{−}T^{2})

(see e. g. [Bär98b]). There is an isomorphism of vector bundles [Hit74],[Hij86, 4.3.1]

ΣT^{2} → Σ̃T^{2}
Ψ ↦ Ψ̂

with

e^{u} ˆ \overbrace{D_{g}Ψ} = D_{g_{0}} Ψ̂ + \frac{1}{2}γ_{g_{0}}(grad_{g_{0}}u)Ψ̂

and

|Ψ̂| = |Ψ|.

Here γ_{g_{0}} means Clifford multiplication corresponding to the metric g_{0}. Note that Ψ̃ from section 5 satisfies Ψ̃ = e^{(u/2)}Ψ̂.

We apply this transformation for Ψ = ψ and we obtain

D_{g_{0}} ψ̂ = −\frac{1}{2}γ_{g_{0}}(grad_{g_{0}}u)ψ̂ + e^{u}Hψ̂ + e^{u}νψ̂.

As ν, γ(V) and νγ(V) are skew-hermitian for any vector V, this yields

|D_{g_{0}} ψ̂|^{2} = \frac{1}{4} |γ_{g_{0}}(grad_{g_{0}}u)ψ̂|^{2} + e^{2u}H^{2} |ψ̂|^{2} + e^{2u} |νψ̂|^{2}
= \frac{1}{4} |du|_{g_{0}}^{2} + e^{2u}H^{2} + e^{2u}.

Integration over (T^{2}, g_{0}) provides

λ̃_{1}area(T^{2}, g_{0}) ≤ \frac{1}{4} ∫_{T^{2}} |du|_{g_{0}}^{2} dvol_{g_{0}} + W(F),

where λ̃_{1} denotes the smallest eigenvalue of the square of the Dirac operator on (T^{2}, g_{0}).

On the other hand

∫_{T^{2}} |du|_{g_{0}}^{2} dvol_{g_{0}} = ∫_{T^{2}} uΔ_{g_{0}}u dvol_{g_{0}}
= ∫_{T^{2}} e^{2u}uK_{g} dvol_{g_{0}}
= ∫_{T^{2}} uK_{g} dvol_{g}
≤ \frac{1}{2} (osc u) ∥K_{g}∥_{L^{1}(T^{2},g)}.

Together with Theorem 9.1 and the results of section 7 we get the statement. □

COROLLARY 12.4. For any κ_{1} ∈ ]0, 4π[, any p > 1 and any κ_{p} > 0 there is a neighborhood U of the (y → 0)-end of Mspin with the following property: If F : T^{2} → S^{3} is an immersion such that the induced metric g := F^{∗}g_{S^{3}} and the induced spin structure φ_{F} represent a spin-conformal class in U and if the curvature conditions

∥K_{g}∥_{L^{1}(T^{2},g)} < κ_{1} and ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)} < κ_{p}

are satisfied, then the Willmore conjecture

W (F) ≥ 2π^{2}

holds.

COROLLARY 12.5. Let F_{i} : T^{2} → S^{3} be a sequence of immersions. The induced metrics g_{i} := F_{i}^{∗}g_{S^{3}} together with the induced spin structures define a sequence (x_{i}, y_{i}) in the spin-moduli space Mspin. Assume that y_{i} → 0 and that the curvature conditions

∥K_{g_{i}}∥_{L^{1}(T^{2},g_{i})} < κ_{1} < 4π and ∥K_{g}∥_{L^{p}(T^{2},g)} Area_{g}^{1−(1/p)} < κ_{p}

are satisfied for some p > 1 and κ_{p} < ∞. Then

W(F_{i}) → ∞.

The conclusion of the second corollary is false if we drop the curvature conditions. To see this we construct a sequence of immersions with y_{i} → 0 and W(F_{i}) < const. We start with an embedding F : T^{2} → S^{3} which looks in a neighborhood of some point like a cylinder. Now we “strangle” the torus as in the picture below:

a b c b a a b c b a a b  b a

We get a sequence F_{i} : T^{2} → S^{3} of C^{1}-embeddings with the following properties:

(i) F_{i}(T^{2}) coincides with F (T^{2}) in region a

(ii) F_{i}(T^{2}) coincides with a part of a half-sphere in region b,

(iii) F_{i}(T^{2}) coincides with a minimal surface in region c.

Note that the regions a, b and c depend on i. In the limit i → ∞, region c disappears. After smoothing we get a family of smooth embeddings satisfying both y_{i} → 0 and W(F_{i}) < const and area(T^{2}, F_{i}^{∗}g_{S^{3}}) → const.

Hence, the first eigenvalue of D^{2} is bounded from above. But the first eigenvalue of the spin-conformally equivalent flat torus with unit volume converges to ∞. This implies that there are spin-conformal classes in which the optimal constants in Lott’s inequality (2) are not attained by flat metrics.

From this example we can also conclude that Theorem 12.2 does no longer hold, if we replace the condition “isometric spin immersion” by “conformal spin immersion”.

13 Comparing spectra for different spin structures

In this section we remove the assumption dim M = 2 and assume that a compact Riemannian spin manifold (M, g) of arbitrary dimension carries at least two different spin structures ϑ and ϑ′. The space of spin structures on M is an affine space associated to the vector space H^{1}(M, ℤ_{2}) which will be identified with Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}) and Hom(π_{1}(M), ℤ_{2}).

For r ∈ ℝ, let H_{ℝ}^{1} (M, rℤ) be the set of all [ω] ∈ H1_{deRham}(M, ℝ) satisfying

∫_{X} ω ∈ rℤ for any closed 1-chain X.

Generalizing our definition in section 6 we define

P : H_{ℝ}^{1} (M, \frac{1}{2}ℤ) → Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}) = H^{1}(M, ℤ_{2})
[ω] ↦ ([X] ↦ exp(2πi ∫_{X} ω)) .

The kernel of P is H_{ℝ}^{1} (M, ℤ). We now define the stable norm for elements of χ of H^{1}(M, ℤ_{2})

∥χ∥_{L^{∞}} := inf {∥ω∥_{L^{∞}} | P ([ω]) = χ} .

In general P is not surjective, hence this norm takes values in [0, ∞]. The elements in the image of P are called realizable by a differentiable form. A homomorphism χ ∈ Hom_{ℤ}(H_{1}(M, ℤ), ℤ_{2}) is realizable by a differentiable form if and only if χ vanishes on the torsion subgroup of H_{1}(M, ℤ).

Definition. Two families (λ_{i}|i ∈ ℤ) and (λ′_{i}|i ∈ ℤ) of real numbers are said to be δ-close if there is a bijective map h : ℤ → ℤ with the property

|λ_{h(i)} − λ′_{i}| ≤ δ.

PROPOSITION 13.1. Assume that (M, g) carries two spin structures whose difference χ is realizable as a differential form. Then the spectra of D for the two spin structures are 2π ∥χ∥_{L^{∞}}-close.

Proof. We modify a technique used by Friedrich [Fri84] for calculating the spectrum of the Dirac operator on a flat torus.

Let us assume that the difference χ of the spin structures is realizable as a differentiable form. We take ω ∈ H_{ℝ}^{1} (M, (1/2)ℤ) with P ([ω]) = χ and ∥ω∥_{L^{∞}} ≤ ∥χ∥_{L^{∞}} + ε for a small number ε > 0. Then there is a complex line bundle L_{ω} on M which is trivialized by a section τ and a connection ∇ on L_{ω} such that

∇_{X}τ = 2πiω(X)τ.

The holonomy of the bundle (L_{ω}, ∇) is χ. Therefore the square of (L_{ω}, ∇) admits a parallel trivialization. Let L_{ω} carry the hermitian metric characterized by |τ| ≡ 1.

Denote by ΣM and Σ′M the spinor bundles to the two spin structures. Then

Σ′M ≅ ΣM ⊗ L_{ω}

where the isomorphism preserves the connection, the hermitian metric and the Clifford multiplication. Now we define

H : Γ(ΣM) → Γ(Σ′M)
Ψ ↦ Ψ ⊗ τ

The Dirac operators D and D′ for the two spin structures then satisfy

D′Ψ′ = H ∘ D ∘ H^{−1}Ψ′ + 2πiω · Ψ′

where · denotes the Clifford multiplication. Multiplication by 2πiω is a bounded operator on the space of L^{2}- sections of Σ′M. Its operator norm is 2π ∥ω∥_{L^{∞}}. The following well-known lemma completes the proof. □

LEMMA 13.2. Let D and D′ be two self-adjoint densely defined endomorphisms of a complex separable Hilbert space. We assume that the spectra of D and D′ are discrete with finite multiplicities. Suppose that D − D′ is a bounded operator of operator norm K. Then the spectra of D and D′ (with multiplicities) are K-close.

The lemma is well-known in perturbation theory. For example it can be deduced from considerations in [Kat66]. The eigenvalues (λ_{i}(t) | i ∈ ℤ) of

A_{t} := (1 − t)D + tD′, t ∈ [0, 1]

can be numbered such that λ_{i}(t) is a Lipschitz function in t with Lipschitz constant K. From this observation the lemma is evident.

Acknowledgement

I want to thank Christian Bär for many interesting and stimulating discussions about the subject. Christian Bär was the adviser of my PhD thesis and many results in this article are based on the work in my PhD thesis. The article was finally completed while I enjoyed the hospitality of the Graduate Center of CUNY, New York. It is a pleasure to thank Jozef Dodziuk for many helpful comments.

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Author’s address: Bernd Ammann, Universität Hamburg, Bundesstraße 55, 20 146 Hamburg, Germany

E-Mail: ammann@math.uni-hamburg.de
WWW: http://www.math.uni-hamburg.de/home/ammann