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Upper bounds for the rst eigenvalue of the Dirac operator on surfaces.

Ilka Agricola and Thomas Friedrich

Humboldt-Universitat zu Berlin, Institut fur Reine Mathematik, Ziegelstra e 13a, D-10099 Berlin, Germany e-mail: agricola@mathematik.hu-berlin.de, friedric@mathematik.hu-berlin.de

Abstract

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface M 2 ,! R3 as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the di erent estimates of the eigenvalue of the Dirac operator for special families of metrics.

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

1 Introduction

The Dirac operator D acting on spinor elds de ned over a 2-dimensional, compact, oriented Riemannian manifold (M 2 ; g) with a xed spin structure has a non-trivial kernel in general. Therefore, lower bounds for the eigenvalues of D are not known in case the genus of M 2 is positive. The genus zero case is an exceptional one: using the uniformization theorem for simply-connected Riemann surfaces, we conclude that any metric g on S 2 is conformally equivalent to the standard metric go of S 2 . Since the dimension of the space of all harmonic spinors depends on the conformal structure only, it turns out that, for any metric g on S 2 , there are no harmonic spinors. This observation yields a lower bound for the rst eigenvalue 2 of D2 1 proved by J. Lott (1986) and Chr. Bar (1992): the inequality 4 2 vol (S 2 ; g) 1 holds for any Riemannian metric on S 2 (see 2], 12]).

Supported by the SFB 288 of the DFG.

1

On the other hand, several upper bounds for 2 depending on di erent geometric 1 data are known. Intrinsic upper bounds involving the injectivity radius and the Gaussian curvature have been obtained by H. Baum (see 6]) and Chr. Bar (see 3]). In case the Riemannian surface (M 2 ; g) is isometrically immersed into the 3-dimensional Euclidean space R3 , one has extrinsic upper bounds depending on the C 0 -norm of the principal curvatures 1 ; 2 of the surface (see 6]). Denote by H = ( 1 + 2 )=2 the mean curvature. Then the following estimate for 2 depending 1 on the L2 -norm of the mean curvature H is well-known (see 7], 4]):

Z

2 1

H 2 dM 2 vol (M 2 ; g) :

M2

In the present paper we will prove stronger extrinsic upper bounds for 2 in case 1 of an isometrically immersed surface M 2 ,! R3 of arbritrary genus as well as an intrinsic upper bound for genus zero and genus one. Moreover, we will compare the di erent estimates of the eigenvalue of the Dirac operator for special families of metrics. The extrinsic upper bound in case of a surface isometrically immersed into R3 depends on two smooth functions f : M 2 ! R and G : R ! R.

Theorem 1: The rst eigenvalue 2 of the square of the Dirac operator on a surface 1 M 2 ,! R3 is bounded by

Z

2 1

M

2

H 2 (f 2 + G2(f ))dM 2 +

Z

Z

M

2

jgrad f j2(1 + G0(f )]2)dM 2

M2

(f 2 + G2(f ))dM 2

;

where f : M 2 ! R, G : R ! R are smooth functions and G0 denotes the derivative of G.

Suppose now that (M 2 ; g) is a two-dimensional Riemannian manifold di eomorphic to S 2 . Denote by go the standard metric of S 2 . Then there exists a uniformization map, i.e., a conformal di eomorphism : S 2 ! M 2 . Let us introduce the function h : S 2 ! R by the formula (g) = h4 go : The set U (S 2 ; M 2 ) of all uniformization maps preserving the orientation can be parametrised by the elements of the connected component of the group of all conformal di eomorphisms of S 2 , i.e., U (S 2 ; M 2 ) SL(2; C ). We introduce a new Dir invariant c (M 2 ; g) de ned in a similar way as the conformal volume of a Riemann surface (see 11]):

Dir 2 c (M ; g) = inf : S2

8 <Z

jgrad(h )j2 dS 2 : 2 U (S 2 ; M 2 )= : ; h2

2

9

The vector eld grad(h ) is the gradient of the function h : S 2 ! R with respect to the standard metric of S 2 .

Theorem 2: Let (M 2 ; g) be a two-dimensional Riemannian manifold di eomorphic

to the sphere S 2 . Then

0

holds.

4 2? 1 vol (M 2 ; g)

vol (M 2 ; g)

2 Dir c (M ; g)

The same method applies to Riemannian metrics on the two-dimensional torus T 2 . The spin structures of T 2 are described by pairs ("1 ; "2 ) of numbers "i = 0; 1, the trivial spin structure corresponding to the pair ("1 ; "2 ) = (0; 0). Let ? be a lattice in R2 with basis v1 ; v2 and denote by v1 ; v2 the dual basis of the dual lattice ? . We will compare the at metric go on the torus T 2 = R2 =? with a conformally equivalent metric g = h4 go .

equivalent to the at torus T 2 and equipped with the trivial spin structure. Then the Dirac operator on (M 2 ; g) has a two-dimensional kernel. Moreover, the rst positive eigenvalue 2 (g) of D2 on (M 2 ; g) is bounded by 1 Z 2 (g ) + 4 jgrad (h)j2 1 dT 2 1 o 2 h2 h6 : T 2 (g) Z 1 1 2

Theorem 3: Let (M 2; g) be a two-dimensional Riemannian manifold conformally

trivial, the Dirac operator has a trivial kernel and 2 (D) is bounded by 1 Z 1 dT 2 2 h2 T 2 (D) 2 j" v + " v j2 Z : 1 1 2 2 1

T2

Theorem 4: Let (M 2; g) be a two-dimensional Riemannian manifold conformally equivalent to the at torus T 2 . In case the spin structure ("1 ; "2 ) = (0; 0) is non6

dT T 2 h2

h2 dT 2

Moreover, the inequality

2 (D)vol (M 2 ; g) 1 2 (g )vol (T 2 ; g ) + o 1 o

Z

T2

jgrad (h)j2 dT 2

h2

with

2 (g )vol (T 2 ; g ) = 2 q o 1 o

j"1 v1 + "2 v2 j2 jv1 j2 jv2 j2 ? hv1 ; v2 i

holds.

We shall apply the previous results to two families of surfaces of special interest. Let us rst consider the ellipsoid 3

lim vol (4E (a)) = 2 ; alim vol (4E (a)) = 0: !1 a!0 Using the upper bounds for 2 (a) already known, we cannot control the behaviour 1 of 2 (a) for small or large values of the parameter a. For example, the L2 -bound 1 given by the mean curvature H has the following limits:

z2 = 1 : E (a) = a2 A calculation of the volume yields that the lower bound 4 =vol (E (a)) for 2 (a) is 1 a monotone decreasing function of the parameter a:

(x; y; z ) 2 R3 : x2 + y 2 +

(

)

Now, a combination of our stronger extrinsic and intrinsic upper bounds for the rst eigenvalue of the Dirac operator yields the following improvement for the ellipsoid:

H 2 dE (a) 1 H 2 dE (a) (a) lim E (a) = 1 ; alim Evol (E (a)) = 2 : !1 a!0 vol (E (a))

2 1

Z

Z

Theorem 5: The rst eigenvalue

1 3.) 2 (a) < 2 ln(2)+3 a for a ! 1: 1 In the last part of this paper we apply our estimates to a tube of radius r around a circle of curvature , i.e., a "round" torus. Parametrizing the spin structure as before, the inequalities for 2 ( ; r) allow us to prove, in particular, 1

3 1.) 2 a!0 2 (a) 2 + ln 2 2; 2; lim 1 2.) alim 2 (a) = 0; !1 1

of D2 on the ellipsoid E (a) satis es

for the spin structure ("1 ; "2 ) = (1; 0) and lim ( r !1 1

2

lim ( r!0 1

2

; r)vol ( ; r) = lim0 2 ( ; r)vol ( ; r) = 0 ! 1 ; r)vol ( ; r)

2

for the spin structure ("1 ; "2 ) = (0; 1) (for these two spin structures, no upper bounds were available before). However, they turn out to yield no improvement for the induced spin structure ("1 ; "2 ) = (1; 1); thus, in this case, the classical bound involving the integral over H 2 divided by the volume is still the best one available. For a conformal change of the Riemannian metric g = h4 go on a surface, one easily proves the following C 0 -estimate for the rst non vanishing eigenvalue 2 (g) of the 1 Dirac operator (see for example 1], Thm. 4.3.1) 2 (go ) 2 (go ) 1 1 2 (g) : Altogether, one obtains the following asymptotic behaviour for radius r around a circle with curvature : 4

h4 max

1

h4 min

2 (g) 1

on a tube of

"1 = 0 = "2 "1 = 1 = "2 "1 = 1; "2 = 0 "1 = 0; "2 = 1

lim 2 (r; ) r!0 1 lim 2 (r; ) !0 1 lim 2 vol (r; ) r !0 1

2

1

1 4r2

1 2 4

1

1 4r2

0 0

0 0

1

1

2 Extrinsic upper bounds

Let M 2 be a compact, oriented surface isometrically immersed into the Euclidean ~ space R3 and denote by N (m) the unit normal vector of M 2 at the point m 2 M 2 . The restriction jM 2 of a spinor eld de ned on R3 is a spinor eld on the surface M 2 . Let be a parallel spinor on R3 . Then the spinor eld 1 ~ ' = 2 (1 ? i) jM 2 + 1 (?1 + i)N jM 2 2 is of constant length on M 2 and satis es the two-dimensional Dirac equation

D(' ) = H' ;

where H denotes the mean curvature of the surface (see 10]). Thus, starting with two parallel spinors 1 ; 2 with

j 1j = j 2 j = 1 and h 1; 2 i = 0 ;

we obtain two solutions '1 ; '2 of the Dirac equation

D(' ) = H'

;

= 1; 2

such that j'1 (m)j = j'2 (m)j = 1 and h'1 (m); '2 (m)i = 0 holds at any point m 2 M 2 . Given two real-valued functions f; g : M 2 ! R we consider the spinor eld = f'1 + g'2 : After applying the Dirac operator to

D( ) = H + grad (f ) '1 + grad (g) '2 ;

a direct calculation yields the formula 5

jD( )j2 = H 2(f 2 + g2 ) + jgrad (f )j2 + jgrad (g)j2 ? 2 Re (grad (f ) grad (g) '2 ; '1 ):

In case the vector elds grad (g) and grad (f ) are parallel, the last term in this formula vanishes since '1 and '2 are orthogonal. In this case the Rayleigh quotient coincides with

Z

M2

Z

jD( j

)j2

Z

The condition for the gradients of the functions f and g is satis ed for example if g is a function depending on f , i.e., g = G(f ). Finally, we have proved Theorem 1.

M2

j2

2 = M

H 2 (f 2 + g2 )dM 2 +

Z

Z

M2

jgrad (f )j2 + jgrad (g)j2 dM 2

M2

(f 2 + g2 )dM 2

:

3 Intrinsic upper bounds for a surface di eomorphic to S 2 or T 2

Let (M 2 ; go ) be a compact, oriented 2-dimensional Riemannian spin manifold and denote by Do its Dirac operator. Moreover, consider a conformally equivalent metric

g = h4 go :

The corresponding Dirac operator D is related with Do by the formula (see 5]) D = 1 D + grad (h) : Consequently, the equation D( ) = is equivalent to 1 Do ( ) = h2 ? h grad(h) : ( )j2 + jgrad 2 h)j h

2

h2

o

h3

For any spinor eld we compute the L2 -norm of D( ):

Z

M2

jD(

)j2 dM 2

=

Z (

Suppose now that is an eigenspinor of the Dirac operator Do with eigenvalue i . Then Re (grad (h) ; Do ( )) = 0 and we obtain the formula

Z

M2

jDo (

j

j2 +

h

2 Re (grad (h)

; Do ( )) dMo2 :

)

M2

jD(

)j2 dM 2

=

Z (

M2

Z

2 + jgrad (h)j i h2

(

2)

j j2 dMo2 :

)

Hence, the rst eigenvalue 2 (D) of the Dirac operator is bounded by 1

2 (D) 1

inf D ( inf i o )=

M2

i

2 2 + jgrad (h)j j j2 dM 2 i o h2

Z

M

j j2 h4 dMo2 2

:

6

Let us now discuss the special case that (M 2 ; go ) is the two-dimensional sphere with its standard metric and g a conformally equivalent metric. The rst eigenvalue of the Dirac operator on S 2 is 1 = 1. Moreover, the corresponding eigenspinor is a real Killing spinor satisfying the di erential equation 1 rX ( ) = ? 2 X ; X 2 T (S 2 ): In particular, the length of is constant and we obtain the inequality

Z

2 (D) 1

2 2 4 + S vol (h 2 ; g) vol (S 2 ; g) S

jgrad (h)j2 dS 2

:

Starting with a surface (M 2 ; g) di eomorphic to S 2 , the latter inequality holds for any uniformization, i.e., for any conformal di eomorphism : S 2 ! M 2 such that (g) = h4 go . In particular, we have proved Theorem 2.

Remark: For any conformal di eomorphism 2 SL(2; C ) of the two-dimensional sphere S 2 we denote by h : S 2 ! R the function de ned by the equation

(go ) = h4 go : Let f : S 2 ! R be a smooth function. Then we de ne the number

c (f ) = inf

Dir

Z

S

Z

2

jgrad(f

) + grad(log(h ))j2 dS 2 : 2 SL(2; C ) : (g) = h4 go , we have

In case of a uniformization : S 2 ! M 2 such that

S S

jgrad(h )j2 dS 2 = Z jgrad(log(h ))j2 dS 2 2 2 h2

Dir 2 Dir c (M ; g) = c (log(h )):

Dir and, consequently, for the quantity c (M 2 ; g) de ned in the introduction, the relation

We now consider the case that (M 2 ; g) is the at torus T 2 = (R2 =?; go ) given by a lattice ? in R2 with trivial spin structure. In this case there are two parallel spinor elds '+ and '? of constant length and the rst non-trivial eigenvalue 2 (go ) of the 1 square of the Dirac Do operator on T 2 is

2 (g ) = 4 2 min 1 o

n

jv j2 : 0 6= v 2 ? ;

o

where ? denotes the dual lattice (see 8]). Suppose now that g is a metric on M 2 conformally equivalent to go , g = h4 go . Then the kernel of the corresponding Dirac 1 1 operator is again two-dimensional and spanned by the spinor elds h '+ ; h '? . Fix a spinor eld such that Do ( ) = 1 (go ) . Then the length of is constant, i.e., j j 1. The spinor eld = =h3 is orthogonal to the kernel of the Dirac operator D with respect to the L2 -norm of the metric g. Indeed, we have 7

Z

M2

1 ; h'

dM 2 =

Z

T2

1 ; 1' h3 h

h4 dT 2 =

1 Z ?D ( ); ' dT 2 = = (g ) 2 o 1 o T This observation yields the inequality

Z

1 Z ? ; D (' ) dT 2 = 0: o 1 (go ) T 2

2 (g) 1

M2

Z

jD( )j2 dM 2 j j2 dM 2

M2

for the rst non-trivial eigenvalue of D2 on (M 2 ; g). Moreover, we have Z Z 1 h4 dT 2 = Z 1 dT 2 2 dM 2 = j j

M2 T 2 h6 T 2 h2

and

Z

M2

jD(

)j2 dM 2

=

Z

(

T2

jDo (

( )j2 + jgrad 2 h)j h

2

j

j2 +

2 Re (grad (h) ; D ( )) dT 2 : o h

)

Since the equation

Do (h3 ) = Do( ) = 1 (go ) = 1(go )h3

can be rewritten in the form 3 Do ( ) = 1 (go ) ? h grad (h) ;

we obtain the formulas 2 Re (grad (h) ; D ( )) = ? 6 jgrad (h)j2 j j o h h2 and

jDo ( )j2 =

Altogether, this implies

Z

2 (g ) + 9 1 o h2

Z

jgrad (h)j2 j j2 :

1 jgrad(h)j2 h6 dT 2

M

2

jD(

)j2 dM 2

=

T

2

2 (g ) + 4 1 o h2

and it proves Theorem 3, in particular. Let us now consider the case that the spin structure on (M 2 ; g) T 2 is non-trivial. Then the Dirac operator has no kernel and the eigenspinors of the Dirac operator Do on T 2 are again of constant length (see 8]). Then our method provides the inequality 8

Z

vol (M 2 ; g) The Gaussian curvature G of the metric g is given by

2 (g) 1

2 (go )vol (T 2 ; go ) 2 1 + T 2 ; g) vol (M

jgrad(h)j2 dT 2 h2

:

h4 G = ?2 (log(h));

where denotes the Laplacian with respect to the at metric. We integrate this latter equation: Z Z Z jgrad(h)j2 dT 2 2= 4 G log(h)dT 2 = ?2 G log(h)dM h

M2 T2 T2

h2

thus obtaining

2 (g) 1 2 (go )vol (T 2 ; go ) 1 vol (M 2 ; g) ?

2 1 M 2 G log(h)dM ; 2 vol (M 2 ; g)

Z

where G denotes the Gaussian curvature of (M 2 ; g). However, we can use a more delicate comparison for the Dirac operator depending on the spin structure ("1 ; "2 ). Consider the dual lattice ? with basis v1 ; v2 as well as the 1-form

! = i(dx; dy) ("1 v1 + "2 v2 ):

The Dirac operator D("1 ;"2 ) corresponding to the spin structure ("1 ; "2 ) on (M 2 ; g) is related to the Dirac operator D for the trivial spin structure by

D("1 ;"2) = D + it;

where the vector eld t is dual with respect to the metric g to the 1-form ! (see 8]). 1 Let '+ be the parallel spinor eld with respect to the at metric. Then = h '+ 2 ; g), i.e., D( ) = 0: Therefore, we obtain is a harmonic spinor on (M

jD("1 ;"2)( )j2 = jtj2 j j2 = j!j2 j j2 : g g

In dimension n = 2 the L2 -length of a 1-form depends only the conformal structure, i.e., if the metrics g = h4 go and go are conformally equivalent, then for any 1-form ! the formula

j!j2 dMg2 = j!j2o dMg2o g g

holds. Now we integrate:

Z

M

2

jD("1 ;"2)(

)j2 dM 2

1 = 2 h2 j!j2o dT 2 = 2 j"1 v1 + "2 v2 j2 g

T

Z

Z

T

2

1 dT 2 : h2

On the other hand, we have 9

Z

nally, we obtain

Z

M2

j

j2 dM 2

1 h4 dT 2 = Z h2 dT 2 ; = 2 h2 2

T T

Z

2

Z

M

2

jD("1 ;"2)(

Z

)j2 dM 2

This equality nishes the proof of Theorem 4.

M2

j

j2 dM 2

2 T = 2 j"1 v1 + "2 v2 j2 Z h

1 dT 2

T2

h2 dT 2

:

4 The rst eigenvalue of the Dirac operator on the ellipsoid with S 1-symmetry

We now discuss the rst eigenvalue of the Dirac operator on the ellipsoid E (a) with S 1 -symmetry de ned by the equation

R3

For the calculations we will use the following convenient parametrization of E (a):

z2 x2 + y2 + a2 = 1:

p

x = 1 ? w2 cos ' ; y = 1 ? w2 sin ' ; z = a w ; where the parameters (w; ') are restricted to the intervals ?1 w 1; 0 ' 2 .

For brevity we introduce the function

a (w) = (1 ? a2 )w2 + a2 :

p

Then the Riemannian metric ds2 , the Gaussian curvature G, the mean curvature H a and the volume form dE (a) are given by the formulas:

a w 1.) ds2 = 1?(w2) dw2 + (1 ? w2 )d'2 ; a 2.) H 2 = a42 ?3 (w)f a (w) + 1g2 ; a 3.) G = a2 ?2 (w); a

4.) dE (a) = 1=2 (w)dw ^ d'. a

4.1

Evaluation of the extrinsic upper bounds

We shall use the extrinsic upper bound for the eigenvalue of the Dirac operator for the family of functions f de ned by f = a (w) ; > 1 : 2 p Notice that f is just the -th power of (a multiple of) 1= G. The length of the gradient of the function f on the ellipsoid is given by 10

2 5.) jgrad (f )j2 = 4 2 (1 ? a2 )2 a ?3 (w)w2 (1 ? w2 ).

Let us rst discuss the case that the parameter a < 1 is small. Then a2 holds and we can estimate the rst integral appearing in Theorem 1 0

Z

a (w)

1

E (a)

H 2 f 2dE (a)

4

a2

Z1

0

2 ?5=2 (w)dw:

a

The latter integral may be rewritten using the transformation 1 ? a2 w = ax, thus yielding 0

Z

p

E (a)

H 2 f 2dE (a) 4 p

Z

a4 ?2

a Z1

1

p

?a2

1 ? a2

0

(1 + x2 )2 ?5=2 dx:

We shall prove that for all > 1 2 lim a!0

E (a)

H 2 f 2 dE (a) = 0:

5 , we have 2 ?5=2 (w) 1 and the result follows immediately. Indeed, in case a 4 2 ?5=2 (w) a4 ?5 . Finally, 5 , we use the inequality a2 3< If 4 a (w), i.e., a 4 3 . Then one has 1 5 3 1 consider the case that 2 < 4 2 ? 2 < 2 and, hence, (1 + x2 ) (1 + x2 )5=2?2 , which implies

a Z1

1

p

?a2

0

Z

(1 + x2 )2 ?5=2 dx

1 Z

0

dx < 1 1 + x2

Z1

and nishes the argument. In a similar way we show lim a!0

Z

E (a)

f 2 dE (a) =

lim 4 a!0

Z1

0

a

2 +1=2 (w)dw = 4

0

w4

+1 dw =

2 : 2 +1

Finally, we investigate the integrals

E (a)

jgrad(f

)j2 dE (a) = 16

(1 ? a2 )2 2

Z1

0

2 ?5=2 (w)w2 (1 ? w2 )dw:

a

Using the Lebesgue theorem ( > 1 ) we conclude 2

a!0 E (a)

lim

Z

jgrad(f

)j2 dE (a) = 16

2

Z1

0

w4 ?3 (1 ? w2 )dw = 4 2 ? 1 :

Since the rst eigenvalue 2 (a) of the square of the Dirac operator on E (a) is bounded 1 by the expression 11

Z

2 (a) 1

E (a)

H 2 f 2dE (a) +

Z

Z

E (a)

jgrad(f )j2 dE (a)

E (a)

f 2 dE (a)

;

we obtain lim (a) a!0 1

2

4

(2 + 1) = 2 (2 + 1) (2 ? 1)2 2 ?1

in the limit a ! 0. The latter inequality holds for any > 1 . For = 1 we obtain, 2 for example, the inequality lim (a) a!0 1

2

6

1 and the optimal parameter = 1 + p2 yields the estimate 2

lim 2 (a) 3 + 2 2 5; 8: a!0 1 Later, this result will be sharpened with the aid of the intrinsic bounds; however, we already get as a partial result that 2 remains bounded. 1 We now discuss the case of a large parameter a (a > 1). It is convenient to write i h 2 a a (w) in the form a (w) = (a2 ? 1) a2 ?1 ? w2 . The formulas 1.) - 5.) used before imply

Z

p

E (a)

jgrad(f )j2 dE (a)

E (a)

Z1 "

Z

f 2dE (a)

= 4 2 a2 1 1 ?

0

# a2 ? w2 2 ?5=2 w2 (1 ? w2 )dw a2 ? 1 Z1 "

0

Z

a2 ? w 2 a2 ? 1

#2

+1=2

dw

:

We compute again its limit for a ! 1:

alim !1 E (a)

jgrad(f )j2 dE (a)

E (a)

Z

f 2dE (a)

Z1

= 0:

Thus, the asymptotic behaviour is dominated by the second term of the estimate:

Z

alim !1

E (a)

Z

H 2 f 2 dE (a) f 2dE (a)

E (a)

10 = 4 Z1

0

1 ? w2 ]2 ?1=2 dw 1 ? w2 ]2

+1=2 dw

:

This yields the inequality 12

Z1

alim 1 (a) !1

2

1 4

Z1

0

1 ? w2 ]2 ?1=2 dw 1 ? w2 ]2

+1=2 dw

0

for any > 1 . The special value in case of the parameter = 1 can easily be 2 calculated to be 3 2 alim 1 (a) 10 : !1 1 However, the inequality holds for any > 2 ; for ! 1 we obtain the optimal result 1 2 alim 1 (a) 4 : !1 Remark: Let us point out that, for = 1, the integral approximation of 2 (a) is, on 1 both sides a ! 0; 1, not the best one among the extrinsic upper bounds considered, but we may come very close to the optimal value using the family of functions f . The exact formula holding for all parameters 0 < a < 1 is in this case:

2 (a) 1 3 7 3 2 + 13 a2 + 16 a4 + 2 a2 ? 3 a4 ? 16 a6 f (a) 8 2 5 6 1 5 2 5 4 3 + 12 a + 8 a ? 8 a f (a)

8 > > < > :

where the function f (a) is given by

f (a) = >

p 1 p 1 2 ln 1? a ?a2 1?a p2 ? pa1?1 arcsin aa?1 2

a<1 a>1

:

Figure 1 (a 2 0; 1 ) and gure 2 (a 2]1; 1 ) give an overview of the di erent extrinsic bounds. The lower solid line is the only known lower bound proportional to the inverse of the volume due to Lott and Bar; the upper solid line is the well known upper bound involving the integral over H 2 divided by the volume. The short dashed curve corresponds to = 1=2 in our family of functions; as seen before, this is the maximal value for for which the curve does not remain bounded as a ! 0. Its limit for a ! 1 is 1=3. Finally, the long dashed curve is the upper bound for = 1 as discussed previously.

13

10

8

6

y

4

2

0

0.2

0.4 a

0.6

0.8

1

Figure 1 (0 a 1)

1

0.8

0.6

y

0.4

0.2

0

2

4

6

8 a

10

12

14

Figure 2 (1 a 1)

14

4.2

Evaluation of the intrinsic upper bound

We now apply Theorem 2 to the ellipsoid E (a). We can nd a uniformization map : S 2 ! E (a) of the form (x; ') = (w(x); '). By formula 1.) for ds2 we obtain a )) a (ds2 ) = 1 ?(w(xx) w0 (x)]2 dx2 + (1 ? w2 (x))d'2 a w2 ( and the condition (ds2 ) = h4 (x) (1 +4x2 )2 fdx2 + x2 d'2 g a a implies the di erential equation

1=2 (w(x)) a 0 1 ? w2 w

as well as the boundary conditions w(0) = 1 and w(1) = ?1. The function h4 (x) a is then given by where wa (x) is the unique solution of the di erential equation ( ) depending on the parameter a. We calculate the gradient of ha (x) with respect to the standard metric go = (1 +4x2 )2 fdx2 + x2 d'2 g of the sphere S 2 and nally obtain

2 h4 (x) = (1 ? wa (x)) (1 +xx ) ; a 4 2 22

1 = ?x

()

jgrad(ha )j2 dS 2 = Z1 1 I1 (a) := h2 2 x a 2 0

Z

S

wa (x) + x2 ? 1 2 dx: 1=2 (w (x)) x2 + 1 a a

!

Theorem 2 then provides the inequality 4 I1 (a) 2 (a) 1 vol(E (a)) + vol(E (a)) :

1 The solution of the di erential equation ( ) has the symmetry wa (x) = ?wa x . 1 Indeed, suppose that wa (x) is a solution and consider w (x) = ?wa x . Then w solves again the di erential equation ( ) and w (0) = ?wa (1) = 1, w (1) = ?wa(0) = ?1. This implies that, for any parameter 0 < a < 1, the solution wa (x) of the equation ( ) vanishes at x = 1. Consequently wa (x) is a decreasing function and we have

8 > < > :

wa (x) 0 for 0 x 1; 0 < a < 1; wa (x) 0 for 1 x 1; 0 < a < 1:

15

In particular, I1 (a) may be reduced to an integral over the interval 0; 1]:

I1 (a) =

Z1

We study again the limits for a ! 0; 1. First we consider the case that a Then, for all points 0 x 1, we have

1=2 (w

0

x

1

wa (x) + x2 ? 1 2 dx: 1=2 (w (x)) x2 + 1 a a

1.

p

!

a

a (x)) =

q

2 (1 ? a2 )wa (x) + a2

1 ? a2 wa (x)

and, consequently, ( 0 1 ? a2 wa1x)wa2(x) : ? wa We integrate this inequality on the interval y; 1]. Using the fact that wa (1) = 0, we obtain the estimate

0 wa(x)

p

1=2 (w (x)) a a 2 (x) 1 ? wa

2 wa (y) 1 ? y

p12 a2 ?

1=2 (w

; 0 y 1:

On the other hand, we have 1=2 (wa (x)) 1. This inequality implies a

0 wa (x) 2 1 ? wa (x)

a

0 a (x))wa (x) = ? 1 2 1 ? wa (x) x

and, nally,

2 wa (y) 1 ? y2 ; 0 y 1: 1+y Altogether, for any x 2 0; 1], we obtain the inequalities 1 ? x2 lim w (x) lim w (x) 1 ? x2 : a!0 a 1 + x2 a!0 a Now we apply the following observation: Let wa be a sequence of numbers such that

a.) 0 < wa < 1; b.) lim wa > 0 .

a!0

2 Then the sequence wa = 1=2 with a = (1 ? a2 )wa + a2 converges to 1, i.e., a

lim wa 2 = 1: a!0 1=

a

In our situation we can conclude that

lim wa (x) = 1 a!0 1=2 (wa (x)) a 16

and nally we are able to calculate the limit:

lim I (a) = a!0 lim a!0 1 =

Z1

Z1

0

1 1 + x2 ? 1 x x2 + 1

Z1

0

x

1

wa(x) + x2 ? 1 2 dx = 1=2 (w (x)) x2 + 1 a a

!2

!

dx = 4

Z1

0

x3 dx: (1 + x2 )2

Using a!0 vol (E (a)) = 2 we obtain lim lim 2 (a) a!0 1

x3 3 2 + 2 (1 + x2 )2 dx = 2 + ln 2 2; 2:

0

In a similar way we handle the case that a 1. The inequalities (0 x 1) 1 allow us to prove the estimate 1 ? x2=a 1 + x2=a

2 wa (x) 1 ? x2 ; 1+x

a (wa (x))

a2

which is valid for all 0 x 1 and a 1. However, the function w= 1=2 (w) is a a monotone decreasing function for w > 0. Consequently, we have

1?x2=a 1+x2=a 1=2 1?x2=a a 1+x2=a

wa (x) 1=2 (w (x)) 1 a a

and from this inequality we can deduce

2 4a2 x2=a : 1 ? 1=wa (x) 2 (1 ? a2 )(1 ? x2=a )2 + a2 (1 + x2=a )2 a (wa (x)) We split the integral I1 (a) into three parts:

!

I1 (a) =

= 4

Z1

0

Z1

wa(x) ? 1 + 2x2 1=2 (w (x)) 1 + x2 a a x3 dx + 4 (1 + x2 )2 x

1

Z1

!

!2

dx =

!

0

+

Z1

0

wa (x) ? 1 2 dx: 1=2 (w (x)) a a

0

!

wa(x) 1=2 (w (x)) ? 1 xdx a a

We estimate the last term using the inequality for 1 ? the value of the integral 17

2 wa (x) and obtain as 1=2 a (wa (x))

I1 (a) 4

Z1

0

x3 dx + 4 (1 + x2 )2

Z1

0

wa(x) 1=2 (w (x)) ? 1 xdx+ a a

!

3 2 2 + 2 a p 2a ln 8a3 ? 6a2 + 2apa2 ? 1 : a ? 1 8a ? 6a ? 2a a ? 1 The volume vol (E (a)) of the ellipsoid behaves like 2 a, i.e., vol (E (a)) = 2 : alim !1 a Therefore, we can control the asymptotic behaviour of 2 (a) for a ! 1: 1 2 (a) 1

p

!

4

a I1 (a) a a vol (E (a)) + a vol (E (a))

2

x3 dx = 1 2 ln(2)+3]: 4+4 Z a a (1 + x2 )2 a

0

1

In particular, we have shown

alim 1 (a) = 0: !1

5 The rst eigenvalue of the Dirac operator on the tube around a circle

We consider a circle in a plane with curvature and length L = 2 = . Let r be a xed radius and denote by M 2 (r) its tube in R3 of radius r, r < 1. The induced metric on the surface M 2 (r) is given by the formula

g = (1 ? r cos ')2 ds2 + r2 d'2 ; where we use the length parameter 0 s L for the circle and 0

: 0; L] 0; A] ! 0; L] 0; 2 ]

parametrizes the angle of the tube. First of all we calculate a uniformization

'

2

of this metric on T 2 . Suppose is given by the condition (s; ) = (s; '( )). Then the equation (g) = h4 (ds2 + d 2 ) yields the di erential equation '0 ( ) =1 1 ? r cos('( )) r and the function h = h(s; ) is given by

h2 = r'0 ( ) = 1 ? r cos('( )):

Using the integral (a < 1)

Z

(1 + a)tg x dx p 22 = p 2 2 arc tg 1 ? a cos(x) 1?a 1?a 18

?

!

we obtain the solution '( )

p tg '(2 ) = 1 ? r tg 21r 1 ? r2 2 : 1+r Since '(p) maps the interval 0; A] bijectively onto 0; 2 ], we conclude A = 2 r= 1 ? r2 2 . Moreover, the function h2 is determined by s

h2

1 ? tg2 '(2 ) = = 1 ? r cos('( )) = 1 ? r 1 + tg2 '(2 ) 1 + tg2 21r 1 ? r2 2 2 2) = (1 ? r p 1 + r + (1 ? r )tg2 21r 1 ? r2

p

2

:

Hence, we obtain aiuniformization of the metric of the tube M 2 (r) parametrized on h 0; L] 0; p12 rr2 2 . The basis of the lattice is ? and thus the dual lattice has the basis p 1 ; 0 ; v = 0; 1 ? r2 v1 = L 2 2 r By Theorem 4 we obtain the estimate

2( 1

v1 = (L; 0) ; v2 = 0; p 2 r 2 1?r

2 2

;

!

:

; r)

1 4

2 2 2" + 1 ? r " 2 1 2 r

!

Z Z

T

2

T

h2 h2 dT 2 2

1 dT 2

for the rst eigenvalue of the Dirac operator on the tube M 2 (r) with respect to the spin structure ("1 ; "2 ). We compute these two integrals:

Z

T2

h2 dT 2

=

ZL ZA

0 0

h2 (s;

)dsd = L

A

ZA

0

r'0( )d = Lr

2 Z 0

d' = 2 rL;

2

and

Z

T 2 h2

1 dT 2 = L Z

A

r

0

Z 1 d = LZ r2 d' 0 ( )d = Lr '0 ( ) r (1 ? r cos('( )))2 ' (1 ? r cos('))2 :

0

0

Consequently, this ratio is equal to Z 1 dT 2 2 1 Z T 2 h2 = 2 (1 ? r d' '))2 = Z cos(

T

h2 dT 2 2

0

(1 ? r2 2 )3=2

1

;

19

i.e.,

2( 1

; r)

1 4

2 2 2" + 1 ? r " 1 r2 2

!

(1 ? r2 2 )3=2

1

:

The volume vol ( ; r) of the tube equals and we obtain the inequality

2( 1

vol ( ; r) = 4 2 r

!

; r)vol ( ; r)

2( 1

2

2 2 r "1 + 1 ?rr "2

(1 ? r2 2 )3=2

Z

1

:

()

Now we apply the inequality

; r)vol ( ; r)

2 (g )vol (T 2 ; g ) + o 1 o

to our situation. Since h2 = r'0 ( ) = 1 ? r cos('( )), we can calculate the gradient of h: jgrad (h)j2 = r 2 sin2('( ))'0 ( ) h2 4 1 ? r cos('( )) and, therefore, we obtain

Z

T2

jgrad (h)j2 dT 2

h2

T

jgrad (h)j2 = r 2 h2 2

2

2 Z 0

sin2 (') d' = 2 1 ? p1 ? r2 1 ? r cos(') r

!

2

:

Then we have proved the estimate

2( 1

2 2 2 p ; r)vol ( ; r) r "1 + 1 ?rr "2 p 1 2 2 + r (1 ? 1 ? r2 2 ) : ( ) 1?r We discuss the inequalities ( ) and ( ) for the three non-trivial spin structures on the tube. For all cases, we provide a gure in which the long dashed line represents the estimate( ), and the short dashed line the estimate ( ). The x-axis uses the variable a = r , the y-axis is to be understood in multiples of 2 . For comparison matters only, we have also drawn the line for constant value 2.

Case 1: "1 = 1; "2 = 0. In this case we obtain

2( 1 2( 1

; r)vol ( ; r)

2r

(1 ? r2 2 )3=2

2

1

() ( )

In particular, we conclude lim ( r!0 1

2

2 2 p ; r)vol ( ; r) p r 2 2 + r 1 ? 1 ? r2 1?r

; r)vol ( ; r) = lim0 2 ( ; r)vol ( ; r) = 0: ! 1

20

5

4

3

2

1

0

0.2

0.4 a

0.6

0.8

1

Figure 3 ("1 = 1; "2 = 0)

Case 2: "1 = 0; "2 = 1. In this case the inequalities are

2( 1

; r)vol ( ; r) r p 1 2 1?r

2( 1

2

2

() ( )

2:

and, in particular, we conclude

; r)vol ( ; r) r

2

2

lim ( r !1 1

; r)vol ( ; r)

21

10

8

6

4

2

0

0.2

0.4 a

0.6

0.8

1

Figure 4 ("1 = 0; "2 = 1)

Case 3: "1 = 1 = "2.

In this case we obtain the estimates 2 1 2 ( ; r)vol ( ; r) 1 r (1 ? r2 2 )3=2

2 2

p

()

2( 1

; r)vol ( ; r) r p 1 2 2 + r 1 ? 1 ? r2 1?r

2

:

( )

Let us compare these estimates obtained via the uniformization of the tube with the estimate using the embedding M 2 (r) R3 . Notice that the embedding induces the spin structure "1 = 1 = "2 on the tube. Then we obtain

2( 1

; r)vol ( ; r)

Z

H 2 dM 2 (r) r p 1 2 2 ; 1?r M 2 (r)

2 1

2

(

)

i.e., the extrinsic bound (drawn as a solid line in gure 5) for intrinsic estimates.

is better than the

22

10

8

6

4

2

0

0.2

0.4 a

0.6

0.8

1

Figure 5 ("1 = 1 = "2 ) In this case 0 (D) = 0 is an eigenvalue of the Dirac operator and Theorem 3 yields the following estimate for the rst non-trivial eigenvalue 2 ( ; r): 1

2 Z 2( 1

Case 4: "1 = 0 = "2

; r) min 2

2;

r2

1

0 2 Z 0

(1 ? r cos ')4 (1 ? r cos ')2

2

d'

d'

:

In particular, we obtain lim 2 ( ; r) = 0 ; rlim 2 ( ; r) 2 !0 1 !0 1 and lim ( r !0 1

2

; r)vol ( ; r) = 0:

23

References

1] B. Ammann. Spin-Strukturen und das Spektrum des Dirac-Operators, Dissertation Freiburg 1998. 2] Chr. Bar. Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39-46. 3] Chr. Bar. Upper eigenvalue estimates for Dirac operators, Ann. Glob. Anal. Geom. 10 (1992), 171-177. 4] Chr. Bar. Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Glob. Anal. Geom. 16 (1998). 5] H. Baum. Spin-Strukturen und Dirac-Operatoren uber pseudo-Riemannschen Mannigfaltigkeiten, Teubner-Verlag Leipzig 1981. 6] H. Baum. An upper bound for the rst eigenvalue of the Dirac operator on compact spin manifolds, Math. Zeitschrift 206 (1991), 409-422. 7] M. Bordoni. Spectral estimates for Schrodinger- and Dirac-type operators on Riemannian manifolds, Math. Ann. 298 (1994), 693-718. 8] Th. Friedrich. Zur Abhangigkeit des Dirac-Operators von der Spin-Struktur, Coll. Math. vol. XLVIII (1984), 57-62. 9] Th. Friedrich. Dirac-Operatoren in der Riemannschen Geometrie, ViewegVerlag Braunschweig/ Wiesbaden 1997. 10] Th. Friedrich. On the spinor representation of surfaces in Euclidean 3-space, to appear in "Journ. Geom. Phys.", dg-ga/9712021, SFB 288 Preprint No. 295. 11] P. Li and S.T. Yau. A new conformal invariant and its application to the Willmore conjecture and the rst eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269-291. 12] J. Lott. Eigenvalue bounds for the Dirac operator, Pac. Journ. Math. 125 (1986), 117-128. 13] T.J. Willmore. Riemannian Geometry, Clarendon Press Oxford 1996.

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