トップページ - 横浜国立大学学術情報リポジトリ

全文

(1)On the Eigenvalues of Laplacian By. Masao MAEDA* Let M be an P-dimensional compact Riemannian manifold with or without. boundary and g its Riemannian metric. The Laplace-Beltrami operator A (Laplacian) acting on Cco-functions on M is locally given by A-- ,,;l=), i-G ･-o-a., (vt;-gtj oa.,). where {u,},="..., is a local coordmate of M) g,2'==g(oOu,, oOu,), G:==det(gtJ). and (g`j) is the inverse matrix of (gij). We can consider the following two eigenvalue problems:. (1) Af==2L fiO on OM when aM4￠, (2) Ah==rk when OM=ip. These R and pt are called the eigenvalues of A. Cgy(M) denotes the set of all. COO-functions on M vanishing on OM when OM¥￠ and Cco(M) the set of all Cco-functions on M. Then as is well known, all eigenvalues of A counting multiplicity form a discrete sequence. O<Ri;S22;S''';$2n;$''',2.Too when OMtip (O==pto<pti:gpt2:El'''.<.ptn;S'･',pt.Too when OM=￠) and there exist functions giEC,oo(M), i=1, 2, ･･･ (resi. p. ￠,ECoo(M), i=O, 1, 2, ･･･). forming a basis of L2(M) (=the space of square integrable functions) such that'. Agi=2igi (resp･[x￠i:pti￠i). and S.gotg)jdvM==Sij (resp･S.gbigbJdvM=Sto)･ Here dvM is the canonical measure induced from the Riemannian metric of M. We have the so called minimum principle:. Minimum Principle. 2n=inf{ S. <d9, dg>/S. g2; gGCoOO(M), S. ggt =O, 1:-{i=<n-1 }, ptn==inf{S. <dip,dip>/S. ip2; ￠ECOO(M), S. ￠￠i==O, O;sli--<n-1}.. " Department of Mathematics, Faculty of Education, Yokohama National University..

(2) 30 M. MAEDA. Now in [2], S. Y. Cheng proved the following theorem:. , Let Mbe a compact domain on a minimal hypersurface in the (P+1)dimensional Euclidean space EP+i, then. n Zn+i:$ln+4i..,Zi/Pn Or ln+i/ln:Sl+4/P. t. '. The purpose of this note is to get' a similar result for a compact minimal. submanifold with or without boundary in the standard unit sphere. We only discuss the case 6M#ip. The case OM=￠ can be discussed analogously.. Let M be a P-dimensional compact Riemannian manifold with smooth boundary OM immersed minimally and isometrically in SM(EM"i. Let (x. cr==1, ･･･, m+1) denotes an orthonormal coordinatie system in EM'i. We denote x.IM also by x.. In the following, we will estimate R.+i by using 2i,･",2n･. The next is well known, see [3]. ' LEMMA1. zNx.==Px., cr=1,･･-,m+1. ' we write S.fdvM as Sf, fGCco(M). Set a.ile:-=Sx.gigle. For the simplicity, Then the functions. n Unt:=x.qi￡ a.ilegleECooo(M), 1.<.a$m+1, lgi.Sn le=1. are all orthogonal to gi, ''', 9n, i･ e･. SUalgj=O,' for IS.i,j-f{:n,IS.aSm+1. ' LEMMA2･ -2i,S<dXa',d9i>Ua'i=P.];,],,aZik'. ' PROOF･ p2 .]E, ),, J'<dXa･, d{pz>Ua･t= -2 i, S<dXev, dSPt> (Xa･SDim :i] aa'zkSPle). ' ='2 :i.] S< :i) Xa'dXev d9i>9t+2.,,;i.?,,"arileS<dXa" d9i>qle ,. ==2.,li.l),,aa'ileS<dXat,d9i>gk,. because .V;,x&==1. Now. q. 1. ' S<dXa･, d9t>9le==S<dXa" d(9z9le)>-S<dXa', d9k>9t. = SAXaqz9k-S<dXew d9h>9t.

(3) On the Eigenvalues of Laplacian 31 = PSX.9igk- S<dx., dg,>g,. ' =' Pa.tk' S<dx., dgk>gi･. ' SoPa2ct,le=a.ileS<dxwdgt>gk+aatleS<dXa,d9le>9i･ '. '. Since aalh==a.ki, we get. ". 2.,;.?,,a.ikS<dx.,dgi>gk=P.,;.?,,aZile. q.e.d.. ' From this lemma, we see that there exist some numbers 1' and P such that UpjEIO. For if U.iEiO for all i and a, then. Xa9i=Zaaifegle for all i and cr.. le. So SxZgi=S(2i] a.ikgk) (]i) a.itgi)=IE? a?aik;. '. ' by Lemma 2, And hence li.)S>ilxkg:･==.?l,],,a2aife. That is n=.?;,,,a2aik. And this is a contradiction. Thus there exist some numbers i and P such that. UpjiEO. For these Upj, -･ AUpj=A(xpgj-:i)apjle9le) ･ == zN x,ggj+xpA gj-2<dxp, dgj>- IIi] a&le tCN gk. =1).xpspj+Rjxpgoj-2<dxp,dsoj>- ]Rfeapjkspk =(p+2,)xpq,-2<dxp, dgi>- ]!i) 2feaBjle9k･. '. So SU,,,AUp,=-(P+2,)Sxpg,Upj-2S<dx.o,dg,>Up, '. ･=(p+2,)SUZ,-2S<dxp,dg,>Up,.. By the minimum principle. 2..,s-Sus,Aupj/Suzj =-P+R,-2S<dxp,dsp,>Up,/SU2,,. .t. So ln+i-2n-PS'-2S<dXp,d9j>Up,/SUZi''. Thus we have -. ' '. (1) 2n+i-2n-P;:ll-2i,S<dXa,d9i>Uai/.]2,],SUk'.

(4) 32 M. MAEDA We. P"t. A:. i･k,A"2at,;'l'k4(i/ipl =ctt. i"u 1.<dLil dMq?.>)2,a"d Schwartz's inequality,. ;:g 4( .]2,), SUZ,)( .E, ), S<dx., dg,>2) 1. Hence -2 i, S<dx., dq,> U.,/ .E, ), SUZi ,. ;El(4/PA) .E,), S<dx.. dgi>2, when A40.. Since. '. '. )S<dx. dgi>2==S<dg,, dqi>==SgiAq, =2i, we have. n (2) 2n+i-Rn-P;$(4/PA)ZRi, when AtO. i=1 If A=O, then by Lemma 2 and (1), we have. 2n+i-2n-P:$O And this estimation is sharper than the case A7EO because A>O, see So hereafter we consider the case A#O.. Nowsince SU2.i=SxZg3-li)aZik,wehave '. ' =n- A. .Z,i SUki=n- .]Eit],ka&rk '. Then from Lemma 2 and (1), we have. (3) 2.+i---R.-P:$PA/(n-A)==-P+nP/(n-A).. '. ' Put K:= ]$b 2i, and X:---R..i-2.-P. Then (2) and (3) are expressed as i=1. (2i) Xg4K/PA, (3') X+Pf!nP/(n-A). From (2') and (3'), we have. , npX2-4KX-4KP$O. From this, we easily see. '. XS2(K+VK2+Knp2)/np S2(K+toK+nP2/2to)/np. =2(to+1)K/np+P/w, where w is a free parameter such that toll.1. Thus we get. (2)..

(5) On the eigenvalues of Laplacian 33THEoREM. Let M be a P-dimensional comPacC Riemannian manifold with or without boundary immersed isometrically and minimally in m-dimensional ' unit sPhere. Then. ln+i-<=2n+2(tu.p+1) t?.,li+("+1)P when 6M#di where tu>=1 is a free Parameter and when OM==￠ the same ineqzaaloity is t･rue replacing Ri by Lti. L. References [1] M. BERGER, P. GAuDocHoN and E. MAzET, Le spectre d'une vari6t6 rierh'anniene, [2] [3]. Springer-Verlag (1971). S. Y. CHENG, Eigenfunctions and eigenvalues of laplacian, Proceedings of Symposia in Pure Math. Vol. 27 (1975), 185-193. T. TAKAHAsHi, Minimal immersions of Riemannian manifolds, JT. Math. Soc. Japan 18 (1966), 380-385.. ,. s. i.

(6)