We consider a two-dimensional torus T2 equipped with a flat metric go as well as a conformally equivalent metric g, g = h4go. Denote by âg the Lapla...

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arXiv:math/9812044v2 [math.DG] 9 Dec 1998

ILKA AGRICOLA, BERND AMMANN and THOMAS FRIEDRICH May 21, 2018

Abstract We compare the eigenvalues of the Dirac and Laplace operator on a twodimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples.

Subj. Class.: Differential geometry. 1991 MSC: 58G25, 53A05. Keywords: Dirac operator, spectrum, surfaces.

1

Introduction

We consider a two-dimensional torus T 2 equipped with a flat metric go as well as a conformally equivalent metric g, g = h4 go . Denote by ∆g the Laplace operator acting on functions and let Dg be the Dirac operator. The following estimates for the first positive eigenvalues µ1 (g) and λ21 (g) of ∆g and Dg2 are known (see [2] and [6]): a.)

λ21 (go ) λ21 (go ) 2 ≤ λ (g) ≤ 1 h4max h4min

,

µ1 (go ) µ1 (go ) ≤ µ1 (g) ≤ 4 , h4max hmin

where hmin (hmax ) denotes the minimum (maximum) of the conformal factor.

b.) µ1 (g) ≤ ∗

16π . vol (T 2 , g)

Supported by the SFB 288 of the DFG.

1

In case the spin structure of the torus T 2 is nontrivial, the Dirac operator has no kernel and, moreover, there exists a constant C depending on the conformal structure fixed on T 2 and on the spin structure such that λ21 (g) ≥

C vol (T 2 , g)

(see [7]). However, explicit formulas for the constants are not known. In this respect, the situation on T 2 clearly differs from the case of the two-dimensional sphere S 2 , where 4π λ21 (g) ≥ vol (S 2 , g) holds for any metric g (see [3], [7]). In this paper we compare µ1 (g) and λ21 (g) for the trivial spin structure and a metric with S 1 -symmetry. We will construct deformations gE of the flat metric such that vol (gE ) ≡ vol (go ) and µ1 (gE ) < λ21 (gE ) holds for any parameter E 6= 0 near zero. For this purpose we calculate, in complete generality, the formulas for the first and second variation of the spectral functions µ1 and λ21 . It turns out that for any local deformation of the flat metric, the first minimal eigenvalue of the Laplace operator is always smaller than the corresponding eigenvalue of the Dirac operator up to second order. The question whether or not there exists a Riemannian metric g on the two-dimensional torus such that λ21 (g) < µ1 (g) remains open. Denote by λ21 (g; l) and µ1 (g; k) the first eigenvalue of the Dirac and Laplace operator, respectively, such that its eigenspace contains the representation of weight l respectively k. The discussion in the final part of this paper suggests the conjecture that for same index l = k, the eigenvalues λ21 (g, l) and µ1 (g, l), are closely related, more precisely, that the Laplace eigenvalue is always smaller than the Dirac eigenvalue and that their difference should be measurable by some other geometric quantity. The Dirac equation for eigenspinors of index l = 0 can be integrated explicitly. In case of index l 6= 0, the Hamiltonian describing the Dirac equation is a positive Sturm-Liouville operator. First, this observation yields an upper bound for λ21 (g, l). On the other hand, it proves the existence of many eigenspinors without zeros. We furthermore apply the general variation formulas in order to study the eigenvalues of the Laplace and Dirac operator for the family of metrics gE = (1 + E cos(2πN t))(dt2 + dy 2 ) in more detail. In case N = 2, the Laplace equation is reduced to the classical Mathieu equation. A similar reduction of the Dirac equation yields a special SturmLiouville equation whose solutions we shall therefore call Mathieu spinors. We investigate the eigenvalues of this equation and compute (for topological index l = 1) the first terms in the Fourier expansion of these Mathieu spinors. These computer calculations have been done by Heike Pahlisch and our grateful thanks are due to her for this. Furthermore, we thank M. Shubin for interesting discussions on SturmLiouville equations.

2

2

The first positive eigenvalue of the Dirac operator and Laplace operator

Let g and go be two conformally equivalent metrics on T 2 . Then the Laplace and Dirac operators are related by the well-known formulas ∆g =

1 ∆o , h4

grad (h) 1 Do + , 2 h h3 where grad (h) denotes the gradient of the function h with respect to the metric go . Let us fix the trivial spin structure on T 2 . In this case the kernel of the operator Do coincides with the space of all parallel spinor fields, in particular, any solution of the equation Do (ψo ) = 0 has constant length. The kernel of the operator Dg is given by Dg =

ker(Dg ) =

1 ψo : Do (ψo ) = 0 . h

The square Dg2 of the Dirac operator preserves the decomposition S = S + ⊕ S − of the spinor bundle S. Moreover, Dg2 acts on the space of all sections of S ± with the same eigenvalues. Therefore, the first positive eigenvalue λ21 (g) of the operator Dg2 can be computed using sections in the bundle S + only. Any section ψ ∈ Γ(S + ) is given by a function f and a parallel spinor field ψo ∈ Γ(S + ): ψ = (f · h)ψo . The spinor field ψ is L2 -orthogonal to the kernel of the operator Dg2 if and only if Z

(ψ,

2 1 h ψo )dTg

T2

2

= |ψo |

Z

f h4 dTo2 = 0

T2

holds, where dTo2 and dTg2 = h4 dTo2 are the volume forms of the metrics go and g. The Rayleigh quotient for the operator Dg2 is given by Z

T2

|Dg (ψ)|2 dTg2 = Z

T2

Z

T2

|h · grad (f ) + 2f grad (h)|2 dTo2 . Z

|ψ|2 dTg2

f 2 h6 dTo2

T2

Finally, we obtain the following formulas for the first positive eigenvalue µ1 (g), λ21 (g) of the Laplace operator ∆g and the square Dg2 of the Dirac operator with respect to the trivial spin structure:

µ1 (g) = inf

Z |grad (f )|2 dTo2 T2

Z

2 4

f h

dTo2

T2

3

:

Z

T2

f h4 dTo2 = 0

λ21 (g) = inf

Z |h · grad (f ) + 2f · grad (h)|2 dTo2 T2

Z

f 2 h6 dTo2

:

Z

f h4 dTo2 = 0 .

T2

T2

A direct calculation yields the formula Z

T2

|h·grad (f )+2f grad (h)|2 dTo2 =

Z

Z

h2 f ∆o (f )dTo2 + (4|grad (h)|2 + 21 ∆o (h2 ))f 2 dTo2 .

T2

T2

Let us use this formula in case that f is an eigenfunction of the Laplace operator, i.e., ∆o (f ) = µ1 (g)h4 f. Then it implies the following inequality between the first eigenvalues of the Laplace and Dirac operator:

λ21 (g) ≤ µ1 (g) +

Z

4|grad (h)|2 + 12 ∆o (h2 ) f 2 dTo2

T2

Z

.

f 2 h6 dTo2

T2

We are now looking for L2 -estimates in case the metric g admits an S 1 -symmetry. Indeed, let us suppose that the metric g is defined on [0, 1] × [0, 1] by g = h4 (t)go = h4 (t)(dt2 + dy 2 ), where the conformal factor h4 depends on the variable t only. Moreover, we assume that the function h (t) has the symmetry h(t) = h(1 − t). Then any function f (t) with f (t) = −f (1 − t) satisfies the condition Z

f h4 dTo2 = 0

T2

and, consequently, yields upper bounds for µ1 (g) and λ1 (g):

µ1 (g) ≤

Z1

0

Z1

|f ′ (t)|2 dt := BLu (g; f )

f 2 (t)h4 (t)dt

0

4

Z1

2

h(t)f ′ (t) + 2f (t)h′ (t) dt

λ21 (g) ≤

0

Z1

u := BD (g; f ) .

f 2 (t)h6 (t)dt

0

3

The first and second variation of µ1(g) and λ21 (g)

We consider a Riemannian metric g = h4 (t)go = h4 (t)(dt2 + dy 2 ) on T 2 (0 ≤ t ≤ 1, 0 ≤ y ≤ 1) and denote by E(µ1 (g)) and E(λ21 (g)) the eigenspaces of the Laplace operator and the Dirac operator corresponding to the first positive eigenvalue. The isometry group S 1 acts on these eigenspaces and therefore they decompose into irreducible representations E(µ1 (g)) = where

P

P

(k1 ) ⊕ · · · ⊕

P

(km ) and

E(λ21 (g)) =

P

(l1 ) ⊕ · · · ⊕

(k) denotes the 1-dimensional S 1 -representation of weight k.

P

(ln ),

Proposition 1: The weights kα2 of the first positive eigenvalue µ1 (g) of the Laplace operator are always bounded by one: kα2 ≤ 1. The weights lβ2 of the first positive eigenvalue λ21 (g) of the Dirac operator are bounded by one under the condition ′

h (t) : 0 ≤ t ≤ 1 ≤ 3π. max h(t)

Proof: Suppose that f (t, y) = A(t)e2πkα iy is an eigenfunction of the Laplace operator, ∆g f = µ1 (g)f . Then the function A(t) is a solution of the Sturm-Liouville equation n

o

−A′′ (t) = µ1 (g)h4 (t) − 4π 2 kα2 A(t). Then consider the function

F (t, y) = A(t)e2πiy and remark that 5

∆g F = µ1 (g)F + 4π 2 (1 − kα2 ) Since

Z

F . h4

F dTg2 = 0, we obtain, in case of the first positive eigenvalue, that

T2

µ1 ≤

Z

∆g (F )F¯ dTg2

T2

Z

= µ1 + 4π 2 (1 − kα2 )

T2

|F |2 dTg2

Z

|F |2 dTo2

T2

Z

T2

. |F |2 h4 dTo2

The latter inequality yields kα2 ≤ 1 immediately. The corresponding result for the Dirac operator follows from the formula λ21 ≤ λ21 +

4π 2 Z

T2

2 4

|F | h

dTo2

(1 − l2 )

Z

T2

|F |2 dTo2 +

l−1 π

Z1 0

|F (t)|2

h′ (t) h(t)

dt ,

where we have already used the differential equation (∗∗) for A that will be derived in the next paragraph. ✷ Solutions of the Laplace equation ∆g f = µ1 (g)f are given by solutions of the SturmLiouville equation −A′′ (t) = {µ1 (g)h4 (t) − 4π 2 k2 }A(t)

(∗)

with k = 0, ±1. In a similar way we can reduce the Dirac equation to an ordinary differential equation. The Dirac operator Dg acts on spinor fields via the formula 1 Dg = 2 h (t)

0 i i 0

!

h′ (t) ∂t + 3 h (t)

0 i i 0

!

1 + 2 h (t)

0 −1 1 0

!

∂y .

Suppose that a spinor field ψ ∈ Γ(S + ) is a solution of the equation Dg2 (ψ) = λ21 (g)ψ. Then ψ is given by a solution of the Sturm-Liouville equation ′′

−A (t) =

(

λ21 (g)h4 (t)

h(t)h′′ (t) − 2(h′ (t))2 h′ (t) 2 2 + − 4π l + 4πl h2 (t) h(t)

)

A(t)

(∗∗)

with l = 0, ±1. In case l = 0, this equation can be solved. Proposition 2: The eigenvalues of the Sturm-Liouville equation (∗∗) for l = 0 are given by (n ∈ Z)

6

λ2 =

4π 2 n2 Z1 0

2 .

h2 (t)dt

Proof: The Sturm-Liouville operator H=−

h(t)h′′ (t) − 2(h′ (t))2 1 d2 − h4 (t) dt2 h6 (t)

admits a square root, namely √

Since we have

d 2 ¯ dt h (t)A(t)A(t)

H(−) =

i h3 (t)

d (h(t) −). dt

= 0, any solution of the equation i

h3 (t)

d (h(t)A(t)) = λA(t) dt

satisfies the condition |A(t)| =

const . h(t)

Consequently, it makes sense to define a function f : R1 → R1 by the formula h(t)A(t) = eif (t) , for which we easily obtain the differential equation f ′ (t) = λh2 (t). But since A(t) is a periodic solution, we have the condition 2πn =

Z1

′

f (t)dt = λ

0

Z1

h2 (t)dt

0

for some integer n ∈ Z, thus yielding the result. Corollary: Let λ2 (g) be an eigenvalue of the Dirac operator on the two-dimensional torus T 2 with respect to the trivial spin structure and a Riemannian metric g = h4 (t)(dt2 + dy 2 ) with isometry group S 1 . Moreover, suppose that the eigenspinor is S 1 -invariant (l = 0). Then λ2 (g)vol (T 2 , g) ≥ 4π 2 holds.

7

Proof: Since the volume is given by

vol (T 2 , g)

=

Z1

Z1

h4 (t)dt, the inequality follows

0

directly from the Cauchy-Schwarz inequality previous Proposition.

0

2

2

h (t)dt ≤

Z1

h4 (t)dt

and the

0

✷

Remark: This corollary should be compared with the following fact. Fix a representation Σ(k) and denote by µ1 (g; k) the first eigenvalue of the Laplace operator such that its eigenspace contains the representation Σ(k). In case k 6= 0 the solution A(t) of equation (∗) is positive (see [8], page 207) and consequently the inequality Z1 0

4πk2 − µ1 (g; k)h4 (t) dt ≥ 0

is valid. We thus obtain the estimate 4π 2 k2 ≥ µ1 (g; k)vol (T 2 , g) and equality holds if and only if the metric is flat. In particular (k = ±1) we have 4π 2 ≥ µ1 (g)vol (T 2 , g) for the first positive eigenvalue of the Laplace operator in case that its eigenspace contains the representation Σ(±1).

Let us introduce the Hamiltonian operator Hl defined by the Sturm-Liouville equation (∗∗) for λ2 = 0: Hl = −

d2 h′ (t) h(t)h′′ (t) − 2(h′ (t))2 2 − . + 4πl − 4πl dt2 h(t) h2 (t)

Proposition 3: For l 6= 0, the Hamiltonian operators Hl are strictly positive. Ho is a non-negative operator. Proof: A direct calculation yields the formula Z1 0

Hl

ϕ(t) h(t)

ϕ(t) dt = h(t)

Z1

ϕ(t) ϕ′ (t) − 2πl h(t) h(t)

0

2

dt ≥ 0

where ϕ(t) is any periodic function. The equation 2πlϕ(t) − ϕ′ (t) = 0 does not admit a periodic, non-trivial solution in case l 6= 0. Consequently, Hl is a strictly positive operator for l 6= 0. Corollary: Fix an S 1 -representation (l). Let λ21 (g, l) be the first eigenvalue of the Dirac operator on the two-dimensional torus T 2 with respect to the trivial spin structure and an S 1 -invariant metric P

8

g = h4 (t)(dt2 + dy 2 ) such that the representation (l) occurs in the decomposition of the eigenspace. Then P the multiplicity of (l) is one and the eigenspinor does not vanish anywhere (l 6= 0). P

Proof: Since Hl is strictly positive, the eigenvalue λ2 (g) is the unique positive number λ2 such that inf spec(Hl − λ2 h4 ) = 0. The corresponding real solution of this Sturm-Liouville equation is unique and positive (see [8], page 207). Corollary: For a fixed S 1 -representation (l) denote by λ21 (g, l) the first eigenvalue of the Dirac operator such that the eigenspace E(λ21 (g, l)) contains the representation P (l). Then the inequality P

λ21 (g, l) ≤

Z1

0

(2πlϕ(t) − ϕ′ (t))2 dt h2 (t) Z1

h2 (t)ϕ2 (t)dt

0

holds for any periodic function ϕ(t). Proof: Since inf spec(Hl − λ21 (g, l)h4 ) = 0, we have Z1 0

Hl

ϕ h

ϕ (t)dt − λ21 (g, l) h

Z1 0

h2 (t)ϕ2 (t)dt ≥ 0

for any periodic function ϕ(t). In case of the flat metric go = dt2 + dy 2 we have µ1 (go ) = λ21 (go ) = 4π 2 and E(µ1 (go )) =

P

(0) ⊕

P

(0) ⊕

P

(1) ⊕

(−1) = E(λ21 (go )).

P

The spaces (±1) correspond to the case that k = l = ±1 and are generated by P the constant function. The two spaces (0) are generated by the functions sin(2πt), cos(2πt). P

Notation: We introduce now a few notations which will be used throughout this article. Let us consider a deformation gE = h4E (t)go of the flat metric go depending on some parameter E. We assume that h4E (t) = h4E (1 − t) 9

holds for all parameters of the deformation. The eigenvalues µ1 (go ) and λ21 (go ) of multiplicity four split into three eigenvalues µ1 (go ) 7→ {µ1 (E), µ2 (E), µ3 (E)}

λ21 (go ) 7→ {λ21 (E), λ22 (E), λ23 (E)}.

,

The eigenvalue µ3 (E) corresponds to the case that k = ±1, has multiplicity two, and its eigenfunction is a deformation of the constant function. The eigenvalues µ1 (E) 6= µ2 (E) correspond to solutions of the Sturm-Liouville equation (∗) and their eigenfunctions are deformations of sin (2πt) and cos (2πt), respectively. The situation is different for the Dirac equation: there, according to Proposition 2, the trivial S 1 -representation (l = 0) yields one eigenvalue λ21 (E) of multiplicity two and the non-trivial representations (l = ±1) define in general two distinct eigenvalues λ22 (E), λ23 (E) of multiplicity one. However, in case hE (t) = hE (1 − t), the spectral functions λ22 (E) and λ23 (E) coincide. Obviously, for small values E ≈ 0 we have µ1 (gE ) = min {µ1 (E), µ2 (E), µ3 (E)}

λ21 (gE ) = min {λ21 (E), λ32 (E), λ23 (E)}.

,

We will compute the first and second variation of µα (E) and λ2α (E) at E = 0. For this purpose we introduce the following notation: Let A be a function depending both on E and t. Then A˙ denotes the derivative with respect to E and A′ the derivative with respect to t. Moreover, we expand the function h4E (t) in the form h4E (t) = 1 + EH(t) + E 2 G(t) + O(E 3 ).

Theorem 1: Consider a deformation gE = (1 + EH(t) + E 2 G(t) + O(E 3 ))go = h4E (t)go of the flat metric on the torus T 2 such that h4E (t) = h4E (1 − t). Moreover, suppose that for E 6= 0 and k = 0 the eigenvalues µ1 (E) 6= µ2 (E) are simple eigenvalues of the Sturm-Liouville equation (∗). Then −8π 2

Z1

H(t) sin (2πt)dt ,

µ˙ 3 (0) = −4π 2

Z1

H(t)dt.

=

λ˙ 23 (0)

a.) µ˙ 1 (0) =

b.)

λ˙ 21 (0)

=

λ˙ 22 (0)

2

0

µ˙ 2 (0) = −8π

2

Z1 0

0

=

−4π 2

Z1

H(t)dt.

0

In particular, we obtain µ˙ 1 (0) + µ˙ 2 (0) = 2µ˙ 3 (0) = 2λ˙ 2α (0).

10

H(t) cos2 (2πt)dt

Corollary: Suppose that the deformation gE = (1 + EH(t) + E 2 G(t) + O(E 3 ))go of the flat metric go satisfies the condition H(t) = H(1 − t) as well as Z1 0

2

H(t) sin (2πt)dt 6=

Z1

H(t) cos2 (2πt)dt.

0

Then, for all parameters E 6= 0 near zero we have the strict inequality µ1 (gE ) < λ21 (gE ). Next we compute the second variation of our spectral functions under the assumption that the first variation is trivial. Theorem 2: Consider a deformation gE = (1 + EH(t) + E 2 G(t) + O(E 3 ))go = h4E (t)go of the flat metric go on the torus T 2 and suppose that the conditions h4E (t) = h4E (1 − t) and Z1

2

H(t) sin (2πt)dt =

0

Z1

H(t) cos2 (2πt)dt = 0

0

are satisfied. Moreover, suppose that for E 6= 0 and k = 0 the eigenvalues µ1 (E) 6= µ2 (E) are simple eigenvalues of the Sturm-Liouville equation (∗). Then a.) µ ¨1 (0) =

−16π 2

Z1 0

2

G(t) sin (2πt)dt − 16π

2

Z1

H(t)C(t) sin (2πt)dt,

0

where C(t) is the periodic solution of the differential equation C ′′ (t) = −4π 2 H(t) sin (2πt) − 4π 2 C(t). b.) µ ¨2 (0) =

−16π 2

Z1 0

2

G(t) cos (2πt) − 16π

2

Z1

H(t)C(t) cos(2πt)dt,

0

where C(t) is the periodic solution of the differential equation C ′′ (t) = −4π 2 H(t) cos(2πt) − 4π 2 C(t). 11

c.) µ ¨3 (0) =

−8π 2

Z1 0

G(t)dt − 8π

2

Z1

H(t)C(t)dt,

0

where C(t) is the periodic solution of the differential equation C ′′ (t) = −4π 2 H(t). d.)

¨ 2 (0) λ 1

=

−8π 2

Z1

G(t)dt + 2π

2

Z1

H 2 (t)dt

0

0

¨ 2 (0) = λ ¨ 2 (0) = −8π 2 e.) λ 2 3 −2π

Z1

0 Z1

G(t)dt + 4π

2

Z1 0

2

H (t)dt − 8π

2

Z1

H(t)C(t)dt

0

H ′ (t)C(t)dt,

0

where C(t) is the periodic solution of the differential equation C ′′ (t) = −4π 2 H(t) − πH ′ (t). Proof of Theorem 1 and Theorem 2: The formulas for the derivatives of λ21 (E) are a direct consequence of Proposition 2. We will prove the variation formulas for λ23 and just remark that one can investigate the other spectral functions in a similar way. Moreover, since all the calculations we make are up to order two with respect to E, we may assume for simplicity that h4E (t) = 1 + EH(t) + E 2 G(t). We compute H ′′ + EG′′ 5 2 (H ′ + EG′ )2 1 hE h′′E − 2(h′E )2 E − E = h2E 4 (1 + EH + E 2 G) 16 (1 + EH + E 2 G)2 and, consequently, we obtain the formulas d dE d2 dE 2

hE h′′E − 2(h′E )2 h2E

!

hE h′′E − 2(h′E )2 h2E

!

= E=0

= E=0

1 ′′ H 4

,

d dE

h′E hE

E=0

=

1 ′ H 4

1 ′′ 5 (G − H ′′ H) − (H ′ )2 . 2 8

The spectral function λ23 (E) is defined by a periodic solution AE (t) of the SturmLiouville equation A′′E (t) = −λ23 (E)h4E (t)AE (t)−

h′ (t) hE (t)h′′E (t) − 2(h′E (t))2 AE (t)+4π 2 AE (t)−4π E AE (t) 2 hE (t) hE (t) 12

with the initial conditions λ23 (0) = 4π 2 , Ao (t) ≡ 1. Therefore, we obtain

1 A˙ ′′o (t) = −λ˙ 23 (0) − 4π 2 H(t) − 4π 2 A˙ o (t) − H ′′ (t) + 4π 2 A˙ o (t) − πH ′ (t) 4 in this case and, consequently, λ˙ 23 (0) = −4π 2

Z1

H(t)dt.

0

Let us now compute the second variation in case that λ˙ 21 (0) = 0 = λ˙ 23 (0). We differentiate the Sturm-Liouville equation twice at E = 0: ¨ 2 (0) − 8π 2 G(t) − 8π 2 H(t)A˙ o (t) + 5 (H ′ (t))2 − 1 G′′ (t) − H ′′ (t)H(t) A¨′′o (t) = −λ 3 8 2 ! 2 1 ′′ d d ′ − H (t)A˙ o (t) − 2πH (t)A˙ o (t) − 4π (ln (hE (t)))E=0 . 2 dt dE 2

Then we obtain

¨ 2 (0) = −8π 2 λ 3 −

1 2

Z1

Z1 0

0

G(t)dt − 8π

2

Z1

H(t)A˙ o (t)dt +

0

H ′′ (t)A˙ o (t)dt − 2π

Z1

5 1 − 8 2

Z1

(H ′ (t))2 dt

0

H ′ (t)A˙ o (t)dt.

0

Since A˙ o (t) is a solution of the differential equation 1 A˙ ′′o (t) = −4π 2 H(t) − H ′′ (t) − πH ′ (t), 4 we have 1 − 2

Z1 0

1 H (t)A˙ o (t)dt = − 2

Z1

= 2π 2

Z1

′′

0

0

1 H(t)A˙ ′′o (t)dt = + 2 H 2 (t)dt −

1 8

Z1

Z1 0

1 H(t) 4π H(t) + H ′′ (t) dt 4

2

(H ′ (t))2 dt

0

and, consequently, we obtain

¨ 2 (0) = −8π 2 λ 3

Z1 0

Z1

= −8π 2

Z1

Z1

0

G(t)dt + 2π

2

0

G(t)dt + 4π 2

0

2

H (t)dt − 8π

2

H 2 (t)dt − 8π 2 13

Z1

H(t)A˙ o (t)dt − 2π

Z1

H(t)C(t)dt − 2π

0

0

Z1

0 Z1 0

H ′ (t)A˙ o (t)dt

H ′ (t)C(t)dt,

where C(t) := A˙ o (t) + 41 H(t) is a solution of the differential equation C ′′ (t) = −4π 2 H(t) − πH ′ (t).

Corollary:

¨ 2 (0) = µ ¨3 (0) + 2π 2 λ 3

Z1

H 2 (t)dt.

0

In particular, for all parameters E 6= 0 near zero we have the inequality µ3 (E) < λ23 (E). Moreover, the first positive eigenvalue µ1 (gE ) of the Laplace operator is always smaller then the corresponding eigenvalue λ21 (gE ) of the Dirac operator for any metric gE near E ≈ 0, i.e., µ1 (gE ) < λ21 (gE ). Remark: The explicit formulas in Theorem 1 and Theorem 2 can be generalized to the case of an arbitrary conformal deformation. Fix a Riemannian metric go on a surface M 2 and consider the deformation gE = (1 + EH + E 2 G + O(E 3 ))go of the metric. Moreover, suppose that µ1 (E) is the deformation of the eigenvalue of the Laplace operator and fE is the corresponding family of eigenfunctions. Then the following formulas hold:

a.) µ˙ 1 (0) = −µ1 (0)

Z

Hfo2 dMo2

M2

Z

fo2 dMo2

Z

(Gfo2 + Cfo )dMo2

M2

b.) µ ¨1 (0) = −2µ1 (0)

M2

, Z

fo2 dMo2

M2

where the function C is the solution of the differential equation ∆o C = µ1 (0)Hfo + µ1 (0)C. The corresponding expression for the variation of the eigenvalue λ1 (E) of the Dirac operator can also be computed:

14

Z

H · |ψo |2 dMo2

M2 c.) λ˙ 1 (0) = −λ1 (0)

2

Z

M2

|ψo |2 dMo2

and a similar formula holds for the second variation. Once again, a similar, though even more intricate computation yields the fourth variation of λ23 (E) under the assumption that all previous variations of λ23 (E) vanish. This is needed for the discussion of the example in Section 5. Theorem 3: Consider a deformation gE = (1 + EH(t))go of the flat metric go on the torus T 2 and suppose that the following conditions are satisfied: a.) H(t) = H(1 − t); ···

¨ 2 (0) = λ2 (0) = 0 . b.) λ˙ 23 (0) = λ 3 3 Then the fourth derivative [λ23 (0)](IV) of the spectral function λ23 (E) at E = 0 is given by the formula

[λ23 (0)](IV)

= 6

Z1 0

45 H (t)H (t)dt + 2 3

+

Z1

−

Z1

0

′′

Z1 0

2

′

2

H (t)(H (t)) dt −

Z1

16π 2 H(t) + H ′′ (t) C3 (t)dt

0

5 ′ (H (t))2 + 2H(t)H ′′ (t) C2 (t)dt 2

15(H ′ (t))2 + 6H ′′ (t)H(t) H(t)C1 (t)dt,

0

where the functions C1 (t), C2 (t), C3 (t) are periodic solutions of the equations 1 C1′′ (t) = −4π 2 H(t) − H ′′ (t) − πH ′ (t) 4 1 5 ′ 1 ′′ ′′ ′′ 2 ′ 2 ′ C2 (t) = H(t)H (t) + (H (t)) + 2πH (t)H(t) − 8π H(t) + H (t) + 2πH (t) C1 (t) 2 8 2 15 3 H ′′ (t)H 2 (t) + H(t)(H ′ (t))2 + 6πH ′ (t)H 2 (t) C3′′ (t) = − 2 4 3 5 ′′ ′ 2 ′ + H(t)H (t) + (H (t)) + 6H(t)H (t) C1 (t) 2 8 3 ′′ 2 ′ − 3πH (t) + H (t) + 4π H(t) C2 (t). 4 15

Remark: In the special case of H ′′ (t) = −16π 2 H(t) the derivative [λ23 (0)](IV) does not depend on C3 (t) and the formulas become much simpler. Such a metric will be the object of Section 5.

4 4.1

Examples The variation gE = (1 + E cos(2πt))go

The volume vol (T 2 , gE ) = 1 of this variation of the flat metric go is constant and all first derivatives at E = 0 vanish since Z1

2

cos(2πt) cos (2πt) =

0

Z1

cos(2πt) sin2 (2πt) = 0.

0

A computation of the second derivatives yields the following numerical values: 10 2 ¨2 (0) = π 2 , µ ¨3 = −4π 2 µ ¨1 (0) = − π 2 , µ 3 3 ¨ 2 (0) = π 2 λ 1

,

¨ 2 (0) = λ ¨ 2 (0) = −3π 2 . λ 2 3

In particular, we obtain µ1 (gE ) < λ21 (gE ) for all parameters E 6= 0 near zero. The eigenspinor corresponding to the minimal positive eigenvalue of the Dirac operator does not vanish anywhere (Figure 1).

-0.1

λ2(Ε) 1 -0.05

µ2(Ε) 0.05 39.49

39.48

39.47

39.46

39.45

µ1(Ε)

λ22(Ε)=λ32(Ε)

µ3(Ε)

(Figure 1)

16

0.1

4.2

The Mathieu deformation gE = (1 + E cos(4πt))go of the flat metric

This deformation of the flat metric again preserves the volume, and the Laplace equation essentially reduces to the classical Mathieu equation u′′ (x) + (a + 16q cos(2x))u(x) = 0. In this case the first variation is trivial only for the Dirac equation. Indeed, we have µ˙ 1 (0) = 2π 2

µ˙ 2 (0) = −2π 2

,

,

µ˙ 3 (0) = 0

λ˙ 21 (0) = λ˙ 22 (0) = λ˙ 23 (0) = 0. Even for the Mathieu deformation we conclude that µ1 (gE ) < λ21 (gE ) for E = 6 0 near zero. A computation of the second derivatives yields the following result (Figure 2): µ ¨3 (0) = −π 2

,

¨ 2 (0) = π 2 λ 1

¨ 2 (0) = λ ¨ 2 (0) = 0. λ 2 3

,

For a detailed discussion of this metric, we refer to the next section.

µ1(Ε)

µ2(Ε) 40.25

λ 21(Ε)

40 39.75

2 2 λ 2(Ε)= λ 3(Ε)

39.5 39.25

µ3(Ε) -0.3

-0.2

-0.1

0.1 38.75

(Figure 2)

17

0.2

0.3

4.3

The variation gE = (1 + E cos(2πN t))go, N ≥ 3

Since Z1

2

cos(2πN t) cos (2πt)dt =

0

Z1

cos(2πN t) sin2 (2πt)dt = 0

0

for N ≥ 3, the first variations of our spectral function vanish. We compute the second variation using the algorithm in Theorem 2: µ ¨1 (0) = µ ¨2 (0) = − ¨ 2 (0) = π 2 λ 1

,

4π 2 N2 − 4

,

µ ¨3 (0) = −

4π 2 N2

¨ 2 (E) = λ ¨ 2 (0) = 1 − 4 π 2 . λ 2 3 N2

In particular, we obtain again λ21 (gE ) > µ1 (gE ) for all parameters E 6= 0 near zero (Figure 3).

2

2

2

λ2(Ε)=λ3(Ε)

λ1(Ε)

39.49

39.485

39.48

39.475

-0.1

-0.05 µ3(Ε)

0.05

µ1(Ε)=µ2(Ε)

(Figure 3)

18

0.1

5

The Mathieu deformation of the flat metric

In the previous examples, the deformation gE = (1 + E cos(4πt))go of the flat metric go plays an exceptional role, because the derivatives µ˙ 1 (0), µ˙ 2 (0) 6= 0 are non-zero. Therefore, we study the behaviour of the first positive eigenvalue for the Laplace and Dirac operator in more detail. First of all, the lower bound 4π 2 ≤ µ1 (E), λ21 (E) h4max yields the estimate 4π 2 ≤ µ1 (E), λ21 (E) 1 + |E| for all parameters −1 < E < 1. In case of the function f (t) = sin(2πt) the upper bound BLu (g, f ) of Section 2 leads to the estimate µ1 (E) ≤

8π 2 , 2 + |E|

i.e., for all parameters −1 < E < 1 the inequality 8π 2 4π 2 ≤ µ1 (E) ≤ 1 + |E| 2 + |E| holds. On the other hand, for the Dirac operator the function 1

f (t) = (1 + E cos(4πt)) 4 sin(2πt) u (g , f ) for its first eigenvalue with the property gives an upper bound BD E u lim BD (gE , f ) = 5π 2 .

E→−1

We will thus investigate the limits lim µ1 (E) as well as lim λ21 (E). The eigenvalue E→−1

E→−1

µ1 (E) is related with a periodic solution of the Sturm-Liouville equation

A′′ (t) = −µ1 (E) 1 + E cos(4πt) A(t) + 4π 2 k2 A(t),

1 where k = 0, ±1 (see Proposition 1). Let us introduce the function B(x) := A 2π x where 0 ≤ x ≤ 2π. Then the Sturm-Liouville equation is equivalent to the classical Mathieu equation

B ′′ (x) + (a + 16q cos(2x))B(x) = 0, where the parameters a and q are given by 19

a=

µ1 (E) − k2 4π 2

,

q=

Eµ1 (e) 16(4π 2 )

,

k = 0, ±1.

For E → −1 the parameters of the Mathieu equation are related by a = −16q − k2

,

k = 0, ±1.

Using the estimates for µ1 (E) we obtain 2π 2 ≤ lim µ1 (E) ≤ E→−1

1 1 ≤q≤− 24 32

i.e., −

8 2 π , 3

in case E = −1.

A numerical computation shows that, under these restrictions, the Mathieu equation has a unique periodic solution for k = 0 and q ≈ 0, 04113. This solution B(x) is the first Mathieu function se1 (x, q), which is the deformation of the function sin(x). Consequently, we have lim µ1 (E) = −16 · q · 4π 2 ≈ 2, 6323π 2 .

E→−1

The limits of the spectral functions µ2 (E) and µ3 (E) can be computed in a similar way: lim µ2 (E) ≈ 1, 79 · (4π 2 ) ,

E→−1

lim µ3 (E) ≈ 0, 9 · (4π 2 ).

E→−1

These limits correspond to the Mathieu functions ce1 (x, q) and ceo (x, q) for the parameters q ≈ −0, 112 in case of µ2 (E) and q ≈ −0, 056296 in case of µ3 (E). 4π2 µ3(Ε)

(3,6)π2

35

u BL

30

(2,66)π2 (2,63)π2

25

µ1(Ε)

lower bound

2π2 -1

-0.8

-0.6

-0.4

(Figure 4) 20

-0.2

Approximation of the periodic solution for µ1 (E), E → −1 : NDSolve[{y’’[x] + 32(0.04113)(Sin[x])^2 y[x] == 0, y[x] == 0 , y’[0] == 1} , y , {x , 0 , 10 Pi}] Plot[Evaluate[y[x]/.% , {x , 0 , 10 Pi}]

1

0.5

5

10

15

20

25

30

-0.5

-1

(Figure 5)

Approximation of the periodic solution for µ2 (E), E → −1 : NDSolve[{y’’[x] + 32(0.1112)(Sin[x])^2 y[x] == 0, y[x] == 1 , y’[0] == 0} , y , {x , 0 , 10 Pi}] Plot[Evaluate[y[x]/.% , {x , 0 , 10 Pi}] 1

0.5

5

10

15

-0.5

-1

(Figure 6)

21

20

25

30

Approximation of the periodic solution for µ3 (E), E → −1 : NDSolve[{y’’[x] + 32(0.056296)(Sin[x])^2 - 1) y[x] == 0, y[0] == 1 , y’[0] == 0} , y , {x , 0 , 10 Pi}] Plot[Evaluate[y[x]/.% , {x , 0 , 10 Pi}]

1.5 1.4 1.3 1.2 1.1

5

15

10

20

25

30

(Figure 7)

The eigenvalues λ2α (E) of the Dirac operator are related with the periodic solutions of the Sturm-Liouville equation −A′′ (t) =

(

h′ (t) h(t)h′′ (t) − 2(h′ (t))2 2 2 − 4π l + 4πl λ2 h4 (t) + h2 (t) h(t)

)

A(t).

For the Mathieu deformation we have 1 2 h(t)h′′ (t) − 2(h′ (t))2 2 E + cos(4πt) + 4 E sin (4πt) = −4π E . h2 (t) (1 + E cos(4πt))2

First we discuss the case that l = 0. Then the first positive eigenvalue of the Dirac equation is given by λ2 =

4π 2 Z1 0

2 .

h2 (t)dt

In case of the Mathieu deformation we obtain lim

E→−1

Z1 0

2

h (t)dt =

√ 2 2 1 − cos(4πt)dt = π

Z1 q 0

22

and, finally, 1 lim λ2 (E) = π 4 ≈ (4, 92)π 2 E→−1 2

5 π² (4,92) π²

50

Bu

D

λ21 (E)

45 40 4 π² 35 30 25

lower bound 2 π²

-1

-0.8

-0.6

-0.4

-0.2

(Figure 8)

We now investigate the case l = 1. Let us consider the Hamiltonian operator HE given by the Sturm-Liouville equation for λ2 = 0:

HE = −

h′ (t) h(t)h′′ (t) − 2(h′ (t))2 d2 d2 2 + 4π − 4π − = − + pE (t), dt2 h(t) h2 (t) dt2

where the potential pE (t) is given by the formula pE (t) = 4π 2 + Eπ 2

4 cos(4πt) + 4 sin(4πt) + E sin2 (4πt) + 2E(2 + sin(8πt)) . (1 + E cos(4πt))2

For all parameters −1 < E ≤ 0 the Hamiltonian operator HE is strictly positive (see Proposition 3). Consequently, the eigenvalue λ23 (E) is the first number such that inf spec (HE − λ2 (1 + E cos(4πt))) = 0, and the corresponding solution of the Sturm-Liouville equation A′′E (t) = (pE (t) − λ23 (E)(1 + E cos(4πt)))AE (t) is unique and everywhere positive. In particular, the solution satisfies the condition AE (t + 21 ) = AE (t).

23

Since AE (t) is a positive periodic solution of the Sturm-Liouville equation, we obtain the condition Z1 0

(pE (t) − λ23 (E)(1 + E cos(4πt)))dt > 0

and thus an upper bound for λ23 (E): λ23 (E)

<

Z1

pE (t)dt.

0

41 40.75 2 upper bound for λ 3 (Ε)

40.5 40.25

-0.5

-0.4

-0.3

-0.2

-0.1 39.75 39.5

(Figure 9)

We notice that this upper bound for λ23 (E) grows and reflects, indeed, the real behaviour of λ23 (E) near E = 0. To see this, we use Theorem 3 to compute the fourth variation of this spectral function (the third variation vanishes since λ23 (E) has to be a symmetric function in E). One obtains the following result: [λ23 (0)](IV) =

27 2 π > 0. 4

On the other hand, using well-known approximation techniques for Sturm-Liouville equations with periodic coefficients (see [9]) we can approximate λ23 (E) for a fixed parameter E. Indeed, one replaces the potential in the Sturm-Liouville equation by the first terms of its Fourier series. This reduces the computation of the approximative eigenvalue to a finite-dimensional eigenvalue problem. For example, in case of E = −0.3 the mentioned methods yields the result λ23 (−0.3) ≈ 39.6733. Let us study the behaviour of the spectral function λ23 (E) for E → −1. More generally, denote by λ2 (E, l) the first eigenvalue of the Dirac operator such that the 24

corresponding eigenspace contains an S 1 -representation of weight l. In particular, 1) = λ2 (E, −1). We apply the Corollary of Proposition 3 to we have λ23 (E) = λ2 (E, p the function hE (t) = 4 1 + E cos(4πt) and conclude that Z1 0

Z1 q

(2πlϕ(t) − ϕ′ (t))2 p − λ2 (E, l) 1 + E cos(4πt)

1 + E cos(4πt)ϕ2 (t)dt ≥ 0

0

holds for any periodic function ϕ(t). Fix a test function ϕ(t) and consider the limit E → −1. Then we obtain the inequality

lim λ2 (E, l) ≤

E→−1

1 2

Z1

(2πlϕ(t) − ϕ′ (t))2 dt | sin(2πt)|

0

Z1

.

| sin(2πt)|ϕ2 (t)dt

0

We apply this estimate to the function ϕl (t) =

cos(2πt) + l sin(2πt) . 2(l2 + 1)π

Then 2πlϕl (t) − ϕ′l (t) = sin(2πt) and we obtain the following Proposition 4: lim λ2 (E, l) ≤ 6π 2

E→−1

(l2 + 1)2 . (1 + 2l2 )

Remark: At E = 0 we have λ2 (0, l) = 4π 2 l2 . On the other hand, for l ≥ 3 the inequality 6·

(l2 + 1)2 < 4l2 1 + 2l2

holds, i.e., lim λ2 (E, l) < λ2 (0, l)

E→−1

l ≥ 3.

The latter inequality means that the eigenvalue λ2 (E, l) decreases for E → −1 (l ≥ 3). The behaviour of λ23 (E) = λ2 (E, l) for l = 1 is completely different. This spectral function increases for E → −1. Using the formula

λ23 (E) =

Z1

0 inf 1 ϕ>0 Z 0

(2πlϕ(t) − ϕ′ (t))2 q

1 + E cos(4πt)

q

1 + E cos(4πt)ϕ2

25

we can approximate the positive minimizing Mathieu spinor MS (E, t) of topological index l = 1 by expanding it in its Fourier series. We thus obtain for example: E = −0.9:

λ23 (−0.9) ≈ 40.1464

MS(-0.9,t)=(Sqrt[Sqrt[1+ (-0.9)Cos[4 Pi t]]]) Sqrt[1+ (0.44)Sin[4 Pi t] + (0.15)Cos[4 Pi t] + (0.09)Sin[8 Pi t] + (0.17)Cos[8 Pi t] + (0.028)Sin[16 Pi t] + (0.051)Cos[16 Pi t] + (0.051)Sin[12 Pi t] + (0.085)Cos[12 Pi t] + (0.016)Sin[20 Pi t] + (0.026)Cos[20 Pi t] + (0.01)Sin[24 Pi t] + (0.014)Cos[24 Pi t] + (0.005)Sin[28 Pi t] + (0.007)Cos[28 Pi t] + (0.0033)Sin[32 Pi t] + (0.0044)Cos[32 Pi t]] E = −0.95:

λ23 (−0.9) ≈ 44.6024

MS(-0.95,t)=(Sqrt[Sqrt[1+ (-0.95)Cos[4 Pi + (0.049)Cos[4 Pi t] + ( 0.1)Sin[8 Pi + (0.063)Sin[12 Pi t] + (0.063)Cos[12 + (0.04)Cos[16 Pi t] + (0.026)Sin[20 + (0.017)Sin[24 Pi t] + (0.018)Cos[24 + (0.011)Cos[28 Pi t] + (0.008)Sin[32

t]]])Sqrt[1+ (0.585)Sin[4 Pi t] t] + (0.08)Cos[8 Pi t] Pi t] + (0.041)Sin[16 Pi t] Pi t] + (0.026)Cos[20 Pi t] Pi t] + (0.012)Sin[28 Pi t] Pi t] + (0.007)Cos[32 Pi t]]

Finally, we can compute the limit lim λ23 (E) replacing again the potential in the E→−1

Sturm-Liouville equation by the first terms of its Fourier series. For E = −1 this amounts to studying the differential equation 2

′′

sin (2πt)A (t) =

1 2 π 9 − 3 cos(4πt) − 4 sin(4πt) − 2λ23 sin4 (2πt) A(t) 2

and the finite-dimensional approximation yields the result lim λ23 (E) ≈ 47.2437.

E→−1

Remark: The second variation formulas prove that, in case of the family gE = (1+E cos(2πt))go (N = 1), the minimal positive eigenvalues of the Laplace and Dirac operator decrease (see Example 4.1) and are smaller than 4π 2 . The numerical evaluation of µ3 (E) and λ23 (E) yields the following table:

E

0

-0.1

-0.3

-0.5

-0.7

-0.9

-0.95

-0.99

-1

µ3

4π 2

39.284

37.897

35.741

33.378

31.09

30.5

30.1

30.013

λ23

4π 2

39.333

38.353

36.714

34.983

33.331

33.2830

36.04

≈ 36.2

26

6

Final remarks

As shown previously, any local deformation gE of the flat metric realizes the inequality µ1 (gE ) < λ21 (gE ) between the first eigenvalues of the Laplace and Dirac operator up to second order. We are not able to give an example of a Riemannian metric g on T 2 such that λ21 (g) < µ1 (g) holds. Moreover, denote again by λ21 (g; l) the first positive eigenvalue of the Dirac operator such that the eigenspace contains an S 1 -representation of weight l ∈ Z. The corresponding eigenvalue of the Laplace operator we shall denote by µ1 (g; l). It is a matter of fact that in all families of Riemannian metrics we have discussed these two eigenvalues are very close. Let us consider, for example, the metric gE by the function E

hE (t) = e π (sin(2πt)−2 cos(2πt)) . For the parameter E = 1 we obtain the following numerical values using the approximation method described before in the space spanned by the functions 1, sin(2πnt), cos(2πnt) (1 ≤ n ≤ 5): λ21 (g; 1) ≈ 6.11056

µ1 (g; 1) ≈ 5.19025.

,

However, even in this case we already have the inequality µ1 (gE ; 1) < λ21 (gE ; 1) and the following figure shows the graph of the two spectral functions for 0 ≤ E ≤ 1 (for the first and the second positive eigenvalue):

90

80

70

60

x 50

40

30

20

10 0

0.2

0.4

0.6 E

(Figure 10) 27

0.8

1

References [1]

I. Agricola, Th. Friedrich. Upper bounds for the first eigenvalue of the Dirac operator on surfaces, to appear in ”Journal of Geometry and Physics”.

[2]

B. Ammann. Spin-Strukturen und das Spektrum des Dirac-Operators, Dissertation Freiburg 1998, Shaker-Verlag Aachen 1998.

[3]

Chr. B¨ar. Lower eigenvalues estimates for Dirac operators, Math. Ann. 293 (1992), 39-46.

[4]

Th. Friedrich. Dirac-Operatoren in der Riemannschen Geometrie, Vieweg-Verlag Braunschweig/ Wiesbaden 1997.

[5]

P. Hartman. Ordinary differential equations, New York 1964.

[6]

P. Li, S.T. Yau. A new conformal invariant and its application to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269-291.

[7]

J. Lott. Eigenvalue bounds for the Dirac operator, Pac. Journ. Math. 125 (1986), 117-128.

[8]

M. Reed, B. Simon. Methods of modern mathematical physics, Part IV, Academic Press Boston 1978.

[9]

E.T. Whittaker, G.N. Watson. A course of modern analysis, Cambridge University Press 1927.

ILKA AGRICOLA Humboldt-Universit¨at zu Berlin, Institut f¨ ur Mathematik, Sitz: Ziegelstraße 13a, Unter den Linden 6, D-10099 Berlin e-mail: [email protected] BERND AMMANN Universit¨at Freiburg, Mathematisches Institut, Eckerstr. 1, D-79104 Freiburg e-mail: [email protected] THOMAS FRIEDRICH Humboldt-Universit¨at zu Berlin, Institut f¨ ur Mathematik, Sitz: Ziegelstraße 13a, Unter den Linden 6, D-10099 Berlin e-mail: [email protected]

28