Some remarks for the spectrum of the $p$-laplacian on sasakian manifolds

SOME REMARKS FOR THE SPECTRUM OF THE p-LAPLACIAN ON SASAKIAN MANIFOLDS

TAE Ho KANG AND JIN SUK PAK

1. Introduction

Let (M, g) be a compact manifold of dimension n with metric tensor g. Let t:i. ^P = d6 + 6d be the Laplace-Beltrami operator acting on the space of smooth p-forms. Then we have the spectrum of t:i. ^P for each

O~p~n

SpecP(M,g) = {O ::;

i +oo},

where each eigenvalue is repeated according to its multiplicity. Many authors have studied the relationship between the spectrum of M and the geometry of M. And also, Z.Olszak[ 1 ], J.S.Pak, J.C.Jeong and W.T.Kim[2], S.Yamaguchi and G.Chiiman[ 7] and others studied the spectrum of Sasakian manifolds. In this paper we shall prove

THEOREM A. Let M = (M,fj>,e,TJ,g) and M ' = (M',q/,e',TJI,g') be compact c-Einstem Sasakian manifolds with SpeeP M = SpeeP M ' for an arbitrazy fixed p ~ 1 (which implies dimM=dimM'=n). If (n,p) rt {(15, 1),(15,2),(15, 13),(15, 14)}, then M is of constant fj>- sectional curvature c if and only if M ' ^is of constant fj>'-sectional cur- vature c' =c.

Received March 25, 1994.

1991 AMS Subject Classifications: 53C12j 53A50.

Key words: Sasakian manifold, spectrum, contact Bochner curvature tensor

This research was partially supported by TGRC-KOSEF and the Basic Science

Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-

1404.

342 Tae Ho Kang and Jin Suk Pak

THEOREM B. Let M = (M,4>,e,Tf,g) and M' = (M',4>',e,Tf,g) be compact Sasakian manifolds witb SpeeP(M)

SpeeP(M')(whicb impliesdimM=c#mM' =n). If n is. given, tbere exists an integer p(O ~ p ~ n) such" tbat M is of conStant </J- sectional curvature c if and only

if M' is of constant 4>'- sectional curvature c = c'.

2. Preliminaries

By R = (Ri~A:)' P ⁼ (RjA:) = (R,lA:) and

= (gjA: RjA:) we de- note the Riemannian curvature tensor, the Ricci tensor and the scalar curvature, respectively, and 9 = (gij) is a Riemannian metric ten- sor on M, (gij ) = ^(gij)

For the tensor field T on M we denote

ITI the norm of T with respect to g. Then for each p _~ 2m +

l( =dimM) the Minakshisundaram-Pleijel-Gaifney asymptotic expan-

sion iorSpeeP-(M~-gYisgi~enby - - - . --.

00

~ 2m+l

N

L.J exp( -Aa,pt) = ^(47rt)-

^[ao,p + ^tal,p + ... + ^{t aN,p]}

a=O

+ ^{o(t N - m +} i ) as t 1 0,

where ao,p, al,p, a2,p, ... are numbers' which can be expressed by ( see [ 3])

a2,p

= 2.. r [{s(2rn+ 1) _60(2m - 1) +180(2m - 3)}

360JM P p-l p-2

(2.3) + _{-2emp+~) +.180e:_-/) -720e:_-23) } IpI

-t. { ~e:p~.':);:~~C:-e--/) t~~~.e;~~2~)} ^1!l12J ~¥,

where dM denotes the volume element of M, and (:) = 0 fork ~ 0 or _

r < o.

Let M = (M,4>,e,7],9) be a compact Sasakian manifold (cf. [ 8 D. This means that M is a (2m + I)-dimensional compact differ- entiable manifold with a normal contact metric structure (4),e,7],9), where 4> = (4)f),e = (e i ),7] = (7]i) are tensor fields of type (1,1), (1,0), (0,1) respectively.

Now we introduce the tensor fields H = ^{(Hkjih) and Q = (Qij) on}

M defined by

c+3 c - l

H kjih = Rkjih - -4-(9kh9ji - 9ki9jh) - -4-(4)kh4>ji - 4>ki4>jh - 24>kj4>ih - 9kh7]j7]i + 9ki7]j7]h - gji7]k7]h + gjh7]k7]i),

Qij = Rij - agij - b7]i7]j, where c =

m(3m + 1),a = ..!!- - ¹ and

m(m + 1) 2m

Then we have

b = 2m+ 1 - - . q

2m (2.4)

IHI2 = IRI 2 _ 2 q2 + 4(3m + 1) q _ 4m(2m + 1)(3m + 1),

m( m + 1) m + 1 m + 1

(2.5) IQI ² = Ipl2 - 2~ ^{q2 + 2q - 2m(2m + 1).}

A Sasakian manifold M = (M, 4>,~, 7],g) is called a space of constant 4>-sectional C'UM1ature c(resp.c-Einstein) if H(resp.Q) vanishes identi- cally. It is well known that a space of constant 4>-sectional curvature is c-Einstein. For any c-Einstein manifold of dimension ~ 5, the scalar curvature is necessarily constant. A 3-dimensional c-Einstein manifold means that the scalar curvature is constant. On any 3-dimensional Sasakian manifold the tensor field H vanishes, but in this case the scalar curvature may be non-constant. Therefore, in dimension 3, it is of constant 4>-sectionaJ. curvature if and only if q is constant.

We also consider the so-called contact Bochner curvature tensor field B = (Bkjih) defined on M by (cf. [1 ,7])

B kjih = Rkjih - 2m 1 + 4 (gkhRji - gkiRjh - gjhRki + 9jiRkh - 4>khRjl4>~ + 4>ki R jl4>i - rPjiRkl4>i + rPjhRkl4>~

+ 24>kjRilrPi + 24>ihRklrP~ - Rkh7]j7]i + Rki'T}j'T}h

344 Tae Ho Kang and Jin Suk Pak

(2.8) (2.7)

. . ) r-4 ( )

- Rji''lk''lh + Rjh"lk"li + 2m + 4 9kh9ji - 9ki9jh

+ 2m r+2m + 4( <Pkh<Pji - <Pki<Pjh - 2<Pkj<Pih)

- 2m r + 4(9kh T/j"li - 9ki"ljT/h + 9ji"lk"lh - 9jhT/kT/i),

0'+ 2m .

where r = 2m + 2· Then we also obtaIn (2.6)

I BI 2 _ IRI 2 _ _ 8_

1 1 ² 2 ²

- m+2 P + (m+ l)(m + 2)0-

4(3m2 + 3m - 2) 2 8(13m + 14)

+ (m+1)(m+2) 0'-24m +36m-56+ (m+1)(m+2)"

Moreover, it may be easily seen that H = 0 if and only if B = 0 and Q = o. From (2.4)"'(2.6), we have

IRI ² = IBI ² + _8_ IQI2 + 2 0'2 _ 4(3m + 1) 0- m + 2 m( m + 1) (m + 1) 24 ² 36 56 8(4m ^{2 -} 2m - 7)

+ m - m+ + .

m+1

For p <I. {I, 2, 3, 2m, 2m + I}, substituting (2.7) into (2.3) yields a2,p = ^a lM f [4PI IBI ² + ~2P21Q12 ^{m+ . + m m+ (4 1)P30- 2}

16 . 16m

+ - - 1 P ⁴ 0' + - - 1Ps] dM,

m+ m+' .

where

PI := PI(m,p) = 8m4 - (60p + 8)m3 + (210p2 -120p - 2)m2 + ( -180p3 + 225p2 - 75p + 2)m

+ 45p4 - 90p3 + 60p2 - 15p,

.,', P 2 : - :P ₂ (m;p),:,: '-4m

+(180p-'f'28)fu

(450p2 -300}J~23)m3 + (360p3 - 465p2 + 15p - 7)m2

- (90p4 - 180p3 + 45p2 -15p - 6)m - 30p2 + ^30p,

P3 := P

(m,p) = 20m

(120p + 4)ms + (240p2 - 9)m

- (180p3 + 30p2 - 120p + 1l)m ³

+ (45p ⁴ + 90p3 - 180p2 - 15p + 1)m ² - (45p ^{4 -} 90p3 + 15p ^{2 -} 3)m - 15p ² + 15p,

._ (2;n_-2

3 )

Q . - ,

360p(p - 1)(2m - p + ^{1)(2m - p)}

P 4 and Ps are also constant depending only on m and p.

For pE {I,2,3,2m,2m + I}, the formula (2.3) is of the form

(2.9)

f[ ² 8 2 4 2

a2,p = f3 lM 4Q11BI + m + 2 Q21QI + m(m + 1) Q3(7

16 16m

+ - - 1 Q4(7 + - - 1 Qs] dM,

m+ m+

where for i = 1,2,3,4,5 (i) if p = I,m ~ 2, then

1

f3 = 360'

P i (m,l)

Qi = Qi(m) = 2m(2m -1)(2m - 2)' while for (m,p) = (1,1), Q1 = -6, Q2 = 1:5, Q3 = 9,

(ii) if p = 2,m ~ 2, then

while for (m,p) = (1,2), (iii) if p = 3, m ~ 2, then

1

f3=2x360' ^{Q Q}

Pi(m,2)

i = i(m) = (2m -1)(2m _ 2)' Q1 = -12,Q2 = 2,Q3 165 = 18,

1

f3 = 6 x 360' while for (m,p) = (1,3),

Q . ⁼ Q.(m) = Pi(m,3)

I I

2m _ 2 '

Q1 = 3,Q2 = -,Q3 15 = 18,

2

346 Tae Ho

and Jin Suk Pak

(iv) if p = 2m,m ~ 2, then

f3 _{. = 360'} ^{1 Q} _i= ^{Q (} i m)= 2m(2m-1)(2m-2)' ^Pi(m,2m)

(v) if p = 2m + I,m ~ 2, then

f3 = 360' ^1.

Pi(m,2m + 1)

Qi = ^Qi(m) = ^(2m + 1)(2m)(2m _ 1)(2m - 2)

REMARK 1. The signs of the coefficients of IBI ^{2 ,} IQI ² and q2 in the formulae (2.8) and (2.9) are respectiveiy determined by the polynomials Ph P 2 and P a when (m,p) =I (1,1), (1, 2), (1, 3).

REMARK 2. In the folioWlng'table we list some particular values of

m for p _~ 100.

p I the values of m such that PI, P 2 , P a > 0 1 I [8,51] .

2 I [2,4] 6 [8,93]

3 I [2,6] [9,136]

4 I [3,8] [12,178]

5 I [2,10] [14,221]

6 I [4,12] [17,263]

7 I [3,14] [19,305]

8 I [5,16] [22,348]

9 I 4 [6,19] [25,390]

10 I [6,9] [11,21]' [27,433]

20 I [10,11] [13,17] [24,43] [52,857]

30 I [15,17] [19,25] [36,66] [77,1281]

40 I [20,23] [26,33] [49,89] [101,1705]

50 I [25,30] [32,41] [62,112] [126,2129]

60 I [30,36] [39,50] [75,135] [150,2553]

70 I [35,42] [45,58] [87,158] [174,2976]

... SQ.++40,48} ",[5~1&1]"".[198,,3400] , 90 I [280,3824]

100 I [300,4248]

We obtain all the values found in [1,7] when p = 1,2.

(2.10)

(2.11)

From now on we shall write (2.8) and (2.9) in the following form;

f[ ^{2 8 2 4 2}

a2,p = I lM 4RI IBI + m + 2R21QI + m(m + 1)R30"

16 16m

+ --1~0" + --lRs] dM,

m+ m+

where I is either ^et or f3, and Ri is either Pi or Qi (i=1,2,3,4,5).

REMARK 3. The equation (2m

p + 1) _ 6(2;_-/) = 0 does not admit the natural roots, In fact, (2m + 1) _ 6 (2m - 1) = 0 if and

. p p - 1 ,

only if m(2m + 1) - 3p(2m - p + 1) = 0 if and only if m =

u-2 u-1

-2-'P = - 2 - ± v, where u

-12v

= 1. Therefore m can not be a natural number, because u is an odd number.

REMARK 4. Let M = (M,</>,~,'T},g) and M' = (M',</>',e,r!',g') be compact Sasakian manifolds with SpecP(M) = SpecP(M') for an arbitrary fixed p ~ 1. Then for any m E N(2m + 1 ? p) such that the polynomials RI, R2 and Ra are strictly positive(for example, some particular values listed in Remark 2), M is of constant </>-seetional curvature c if and only if M' is of constant </>'-sectional curvature c = c'.

Proof. Assume that M' has constant </>'-sectional-curvature c'. Then our assumption SpecP(M) == SpecP(M') and Remark 3 imply

fM ^{[4R I IBI} 2

+ m ~ ^{2R21QI2 + m(m 4 +} 1)R30"2] dM

•

= _lM' f _m (4 _m+1 )R~O"'2dM,

On the other hand

f 0"2 dM? f 0"'2 dM' ,

lM lM'

because J ^M 0" dM = fM' 0"' dM' ,0"' = constant, J M dM = fM' dM'.

Hence from (2.11) we obtain B = 0 = Q. Q.E.D.

348 Tae Ho Kang and Jin Suk Pak

3. Proof of Theorems

Proof of Theorem A. H M and M' are c-Einstein manifolds, then Q = 0 = Q', and u, u' are constants. By Remark 3 and (2.2), wehave u = u'. The assumption SpecP(M) = SpeeP(M') implies

But for (n,p) ~ {(15, 1), (15, 2), (15, 13), (15, 14)}, RI =I 0 (d. The- orem 3.1(i) in [ 4 D. Hence B = ^{0 if} and only if B' = ^O. Q.E.D.

Let Snbe an odd dimensional unit sphere with constant curvature 1.

Then S;:::: (sn,~,e,fj,g) admits a Sasakian structure (~,e,fj,g) which is ~~ed a nat'lJ,ral Sasaki.an structure on sn. Using our THEOREM A, Remark 4 and Theorem 2([ 5 D, we c;an deduce the characterization.

COROLLARY. Let M = (M,,p,e,TJ,g) be a compact Sasakian mani- fold with SpeeP(M) = SpecP(S) for a given p ? 1. If (i) .the functions RI,R2,R3 are strictly positive, or (ii) M is c-Einstein and (n,p) ~

{(IS, 1), (15,2), (15,13), (15, 14)}, then M is isomozphic to S, that is, there is an isometry f: (M, g) ---+ (sn,g) such that f*e = [,!*ii = TJ and f*

4> = ~

f*·

Proof of Theorem B. By Remark 1, for (rn, p) f/. (1,1), (1,2), (1, 3), it is sufficient to show that there exists an integer p such that PI, P 2 , P 3 >

O. This can be done as follows (2m + 1 =: n);

H n = 3,5,7,9,11, we choosep = 0 ([1,7)). H n = 13,17:5 n:5 187, we choose p = 2 (Remark 2). H n = 15, we choose p = 4 (Remark 2). H n ~ 47(n = ^{16k - 1 or 16k + 1 or 16k +3 or 16k +}

5 or 16k+7 or 16k+90r 16k+ll or 16k+13(k ~ 3», we always choose p = k.

To see the last statement, we calculated the following polynomials PIt P 2 , P 3 , which can be obtained from (2.3) with 2m + 1 =: n.

Pl(n,p):= 4PI(rn,p) ,; 2n(n - !)(n - 2)(n - 3) - 30(n - 2)(n - 3)p(n - p)

+ 180p(p -l)(n - p)(n - p - 1),

P2(n,p) := 8P2(m,p) = - n(n - 1)(n - 2)(n - 3)(n + 3)

+ 90(n - 2)(n - 3)(n + 3)(n - p)p - 360p(p - 1)(n - p)(n - p -1)(n + 3)

+ 16n(n - l)(n - 2)(n - 3) - 240p(n - p)(n - 2)(n - 3)

+ 1440p(p - 1)(n - p)(n - p -1),

P;(n,p) := 16P

(m,p)

= ⁵⁽ⁿ + l)n(n -1)2(n - 2)(n - 3)

- 60(n + l)(n - l)(n - 2)(n - 3)p(n - p)

+ 180p(p -l)(n - p - 1)(n - p)(n - 1:)(n + 1)

+ 16n(n - l)(n - 2)(n - 3) - 240(n - 2)(n - 3)p(n - p)

+ 1440p(p - 1)(n - p - l)(n - p) - 2(n + l)n(n - 1)(n - 2)(n - 3)

+ ¹⁸⁰⁽ⁿ + ^l)(n ~ 2)(n - 3)p(n - p)

-720p(p -l)(n - p - l)(n - p)(n + 1). Q.E.D.

References

1. Z. Olszak, The spectrum of the Laplacian and the curvature of SasaJ:i.an man- ifolds, Lecture Notes in Mathematics, vol. 838, Springer-Verlag, 1979.

2. J. S. Pale, J. C. Jeong and W. T. Kim, The contact conformal cunJature tensor field and the spectrum pf Laplacian, J.Korean Math. Soc. 28 (1991),267-274..

3. V. K. Patodi, CUnJ4ture and the fundamental solution of the heat operator, J.Indian Matb. Soc. 34 (1970).

4. M. Puta and A. Torok, The spectrum of the p- Laplacian on KiihJer manifold, Rendiconti di Matematica 11 (1991), 257-271, Serie VII, Roma.

5. T. Taka.hashi, SasaJ:i.an manifold with pseudo-Riemannian metric, Toboku Ma th. J. 21 (1969),271-290.

6. Gr. Tsagas, The spectrum of the Laplace operator for a special comple:r: mani- fold, Lecture Notes in Mathematics, vol. 838, Springer-Verlag, 1979.

7. S. Yamaguchi and G. Chiiman, Eigenvalues of the Laplacian of Sasakian man-

ifolds, TRU Math. 15-2 (1979), 31-41.

350 Tae Ho Kang and Jin Suk Pak

8. K. Yano and M. Kon, Structures on manifolds, vol. 3, Series in Pure Math., World Scientific, 1984.

The Ho Kang

. Department of Mathematics University of Ulsan

Ulsan 680-749, KOREA Jin Suk Pak

Department of Mathematics

Kyungpook National University

Taegu 702-701, KOREA