On the Spectrum of Riemannian Manifolds with Attached Thin Handles
The behavior as ε → 0 of the spectrum of the Laplace Beltrami operator Δε is studied on Rieinannian manifolds depending on a small parameter ε . They consist of a fixed compact manifold with attached handles whose radii tend to zero as ε → 0. We consider two cases: when the number of the handles is...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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irk-123456789-1065382016-10-01T03:01:51Z On the Spectrum of Riemannian Manifolds with Attached Thin Handles Khrabustovskyi, A. The behavior as ε → 0 of the spectrum of the Laplace Beltrami operator Δε is studied on Rieinannian manifolds depending on a small parameter ε . They consist of a fixed compact manifold with attached handles whose radii tend to zero as ε → 0. We consider two cases: when the number of the handles is fixed and their lengthes are also fixed and when the number of the handles tend to infinity and their lengthes tend to zero as ε → 0 . For these cases we obtain the operators whose spectrum attracts the spectrum of Δε as ε → 0 . 2009 Article On the Spectrum of Riemannian Manifolds with Attached Thin Handles / A. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 145-169. — Бібліогр.: 9 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106538 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The behavior as ε → 0 of the spectrum of the Laplace Beltrami operator Δε is studied on Rieinannian manifolds depending on a small parameter ε . They consist of a fixed compact manifold with attached handles whose radii tend to zero as ε → 0. We consider two cases: when the number of the handles is fixed and their lengthes are also fixed and when the number of the handles tend to infinity and their lengthes tend to zero as ε → 0 . For these cases we obtain the operators whose spectrum attracts the spectrum of Δε as ε → 0 . |
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Article |
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Khrabustovskyi, A. |
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Khrabustovskyi, A. On the Spectrum of Riemannian Manifolds with Attached Thin Handles Журнал математической физики, анализа, геометрии |
| author_facet |
Khrabustovskyi, A. |
| author_sort |
Khrabustovskyi, A. |
| title |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles |
| title_short |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles |
| title_full |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles |
| title_fullStr |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles |
| title_full_unstemmed |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles |
| title_sort |
on the spectrum of riemannian manifolds with attached thin handles |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/106538 |
| citation_txt |
On the Spectrum of Riemannian Manifolds with Attached Thin Handles / A. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 145-169. — Бібліогр.: 9 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrabustovskyia onthespectrumofriemannianmanifoldswithattachedthinhandles |
| first_indexed |
2025-07-07T18:37:13Z |
| last_indexed |
2025-07-07T18:37:13Z |
| _version_ |
1837014381525729280 |
| fulltext |
Journal of Mathemati
al Physi
s, Analysis, Geometry2009, vol. 5, No. 2, pp. 145�169On the Spe
trum of Riemannian Manifoldswith Atta
hed Thin HandlesA. KhrabustovskyiMathemati
al Division, B. Verkin Institute for Low Temperature Physi
s and EngineeringNational A
ademy of S
ien
es of Ukraine47 Lenin Ave., Kharkiv, 61103, UkraineE-mail:andry�ukr.netRe
eived May 5, 2008The behavior as "! 0 of the spe
trum of the Lapla
e�Beltrami operator�" is studied on Riemannian manifolds depending on a small parameter ".They
onsist of a �xed
ompa
t manifold with atta
hed handles whose radiitend to zero as " ! 0. We
onsider two
ases: when the number of thehandles is �xed and their lengthes are also �xed and when the number ofthe handles tend to in�nity and their lengthes tend to zero as " ! 0. Forthese
ases we obtain the operators whose spe
trum attra
ts the spe
trumof �" as "! 0.Key words: homogenization, Lapla
e�Beltrami operator, spe
trum, Rie-mannian manifold.Mathemati
s Subje
t Classi�
ation 2000: 35B27, 35P20, 58G25, 58G30.Introdu
tionThe aim of the paper is to study the behavior as " ! 0 of the spe
trum ofthe Lapla
e�Beltrami operator �" on the Riemannian manifolds M " dependingon a small parameter ". We
onsider two di�erent problems.In Se
tion 1 we
onsider a manifoldM " that
onsists of a �xed two-dimensional
ompa
t Riemannian manifold without boundary
and an atta
hed "thin" mani-fold �". The last one
onsists of several tubes with �xed lengthes and radii " (seeFig. 1 below). Thus �" "
onverges" to some graph � as "! 0.Let �
be the Lapla
e�Beltrami operator on
and L be the Lapla
e operatoron �, i.e., L is de�ned by the operation d2ds2 on the edges of � (s is a naturalparameter on the edge), Diri
hlet boundary
onditions on the ends of � andKir
hho�
onditions on the verti
es of �. We prove that the spe
trum of �"
A. Khrabustovskyi, 2009
A. Khrabustovskyi
onverges in some suitable sense to the union of the spe
trum of �
and thespe
trum of L. Also we study the behavior of
orresponding eigenvalues.These results generalize the results by C. Anne [1℄. The behavior of spe
trumis studied on a manifold with one atta
hed handle having a �xed length anda vanishingly small radius in [1℄. These results are extended to the
ase of theLapla
ian a
ting on di�erential p-forms in [2℄. The
onvergen
e of spe
tra onmanifolds whi
h
ollapse to a graph was studied in [6℄.In Se
tion 2 we
onsider the manifold M " whose topologi
al genus in
reasesas " ! 0. It is
onstru
ted in the following way. Let
be a
ompa
t two-dimensional Riemannian manifold without boundary, and D"i , i = 1 : : : N(") =3N1(") be a system of noninterse
ting balls ("holes") in
depending on ". Let
" =
nN(")Si=1 D"i . Suppose that the set f1 : : : N(")g is divided into subsets that
onsist of three elements. If the indexes i, j, k lie on one subset we
onne
t the"holes" D"i ;D"j ;D"k by means of a manifold that
onsists of the tubes G"i ; G"j ; G"kand a trun
ated sphere B"ijk (see Fig. 2 below). As a result, we obtain the manifoldM " =
" [i;j;k �G"i [G"j [G"k [B"ijk� :We suppose that the number of "holes" in
reases as " ! 0, while their radiitend to 0. It is supposed that the radii of the "holes" are mu
h smaller than thedistan
es between them. We also suppose that, in
ontrast to the manifold �" inSe
t. 1 and in
ontrast to [1℄, the metri
is su
h that the lengthes of the tubes
onverge to 0.We obtain the following result: if some
onditions on a distribution of the"holes" and on the metri
s on the tubes and the trun
ated spheres are hold,then the spe
trum of the operator ��"
onverges in some suitable sense to thespe
trum of the operator L de�ned by the formula[Lu℄(x) = ��
u(x) + Z
W (x; y)(u(x)� u(y))dy:Here W (x; y) is a positive symmetri
fun
tion. We present an example for whi
hW (x; y) is
al
ulated expli
itly.The behavior of the spe
trum of manifolds with
omplex mi
rostru
ture wasstudied in [5, 8℄ for another type of manifolds. We note that the behavior ofspe
trum of manifold with the atta
hed one handle, having a vanishingly smallradius and (in
ontrast to [1℄) a vanishingly small length, was studied in [4℄.The proof of main results is based on the abstra
t s
heme proposed in [7℄.Throughout the paper, we will denote by C various
onstants independentfrom ".146 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin Handles1. Riemannian Manifold with Atta
hed "Graph"1.1. Problem Setting and Main ResultLet
be a two-dimensional
ompa
t Riemannian manifold without boundaryand with a metri
g. By �
we denote the
orresponding Lapla
e�Beltramioperator. Let D"i , i = 1 : : : N be a system of balls in
with the
enters xi 2
and the radii ". We
onsider the following domain with holes:
" =
n N[i=1D"i :To
" we glue the manifold �" illustrated on Fig. 1 and
onstru
ted as follows.Let � be a graph in R3 . We denote the verti
es of this graph by pi, i = 1 : : : m(m > N) and the edges of the same graph by
"ij.
"ij
onne
ts the verti
es pi andpj. We introdu
e the symmetri
matrix fAijgmi;j=1 su
h that Aij = 1 if p"i and p"jare
onne
ted and Aij = 0 otherwise. We suppose that for the �rst N verti
espi, i = 1 : : : N there is only one pj su
h that Aij = 1. These are the ends of thegraph.Let zij be the natural parameter on
ij , zij 2 [0; lij ℄. We denote by p(zij) thepoint on
ij that
orresponds to the natural parameter zij .We denote by G"ij the
ylinder with the axis b
ij = fp(zij) 2
ij : zij 2 [Æ"; lij�Æ"℄; Æ" � 0g and with the radius ". The length of G"ij is equal to l"ij = lij � 2Æ".We
hoose the standard
ylindri
al
oordinates on G"ijG"ij = �('ij ; zij) : 'ij 2 [0; 2�℄; zij 2 [Æ"; lij � Æ"� :Clearly, Æ"
an be
hosen su
h that:1. G"ij are pairwise disjoint,2. jÆ"j � C � ".The boundary of G"ij
onsists of two
ir
les S"ij and S"ji. Here we suppose that S"ijis
loser to the vertex pi, and S"ji is
loser to the vertex pj .For i 2 fN + 1 : : : mg, let B"i be the sphere of the radius b" =p"2 + Æ"2 withthe
enter pi. It is
lear that S"ij � B"i . Let D"ij be a part of B"i that lies insidethe
ylinder G"ij, and let B"i = B"i n [j:Aij=1Dij:We obtain a two-dimensional manifold (see Fig. 1):�" = m[i=124 [i;j:Aij=1;i<jG"ij35 m[i=N+1B"i :Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 147
A. Khrabustovskyi
Fig. 1: Manifold �".The boundary of �"
onsists of S"ij, i, j : i = 1 : : : N , Aij = 1.Now we suppose that S"ij , i, j : i = 1 : : : N , Aij = 1 are di�eomorphi
to �D"i .Using this di�eomorphisms, we glue �" to
" and obtain a manifold withoutboundary M " =
" [ �":We denote by ~x the points of this manifold. Clearly, M "
an be
overed bya system of
harts and suitable lo
al
oordinates fx1; x2g
an be introdu
ed.It is supposed that M " is equipped with the metri
g" that
oin
ides with themetri
g on
" and with the Eu
lidean metri
s indu
ed from R3 on �". By g"�� ,we denote the
omponents of the metri
tensor in lo
al
oordinates.Let L2(B) be a Hilbert spa
e of the real-valued fun
tions on B � M " withthe s
alar produ
t and the norm(u; v)L2(B) = ZB u(~x) � v(~x)d~x; kuk2L2(B) = ZB �u(~x)�2d~x;where d~x =qdetg"��dx1dx2 is the volume form.We denote H" := L2(M "); H0 := L2(
)� L2(�).Let �" be a Lapla
e�Beltrami operator on M ". It is well known that thespe
trum of the operator ��" is purely dis
rete. Let 0 = �"1 < �"2 � �"3 � ::: ��"k !k!1 1 be the eigenvalues of ��" written with a
ount of their multipli
-ity, u"1; u"2; u"3 ::: be the
orresponding eigenve
tors normalized by the
ondition(u"i ; u"j)H" = Æij .In this se
tion we study the behavior of �"k as "! 0.148 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesLet L : L2(�) ! L2(�) be a Lapla
e operator on the graph � with Diri
hletboundary
onditions, i.e., L is de�ned by the operation[Lu℄(x) = � d2udzij (x); x = p(zij) 2
ijand by a de�nitional domain
onsisting of the fun
tions u 2 H2(
ij) 8i; j andsu
h that if we denote by uij the restri
tion of u on
ij , thenfor i = 1; N : u(xi) = 0;for i = N + 1;m : 8><>:uij(pi) are equivalent for all j : Aij = 1;Xj:Aij=1 �uij�� (pi) = 0;where ��� means the derivative in the dire
tion outward to
ij . In short, u isa
ontinuous fun
tion on � that satis�es the Diri
hlet
onditions on the endsof the graph as well as Kir
hho�
onditions in the verti
es (for more pre
isedes
ription of di�erential operators on the graphs and its properties see, e.g., [6℄).To des
ribe the behavior of eigenfun
tions we introdu
e the operator R" :H0 ! H": [R"f ℄(~x) = 8><>:f0(~x); ~x 2
";fij(zij)"�1=2; ~x = (zij ; 'ij) 2 G"ij;0; ~x 2 B"i ;f = (f0; fij; i; j : Aij = 1) 2 L2(
)� L2(�):Let L : H0 !H0: L = ���
00 L� ;and let �0; �1; �2::: be the eigenvalues of L written with a
ount of their multi-pli
ity. It is
lear that the spe
trum of L is the union of the eigenvalues of theoperator ��
and the eigenvalues of the operator L that are taken with a
ountof their multipli
ity.Theorem 1.1. For any k=1,2,3. . .�"k ! �k; "! 0:Theorem 1.2. Let �k < �k+1 = �k+2 = : : : = �k+m < �k+m+1 (i.e.,the multipli
ity of �k+1 is equal to m). Let N(�k+1) be the eigenspa
e of theJournal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 149
A. Khrabustovskyieigenvalue �k+1: Then for any w 2 N(�k+1) there exists a linear
ombination �u"of the eigenfun
tions u"k+1 : : : u"k+m su
h thatk�u" �R"wkH" ! 0; "! 0: (1.1.1)1.2. Proof of Theorems 1.1 and 1.2We prove Theorems 1.1 and 1.2 for the
ase N = 3;m = 4, i.e., �"
onsists ofthree tubes G"14; G"24; G"34 and the trun
ated sphere B"4 that
onne
ts these tubes.For the general
ase the theorems are proved in a similar way. We introdu
e newnotations: l"i := l"i4; zi := zi4; 'i := 'i4; G"i := G"i4;S"j := S"4j ; C"i := S"i4; B" := B"4; B" := B"4; i; j = 1; 2; 3;(i.e. ��" = Si=1;2;3C"i ).For simpli
ity we suppose that the metri
g is Eu
lidean in some neighbour-hood of the holes D"i (and thus g" is
ontinuous). For the general
ase the proofneeds small modi�
ations.We denote byA" andA0 the operators inverse to��"+I and L+I, respe
tively(I is an identi
al operator).Now we study the behavior of A" as "! 0.Theorem 1.3. The following
onditions are ful�lled:C1. For any f 2 H0 kR"fkH" ! kfkH0 ; "! 0: (1.2.1)C2. The operators A";A0 are positive,
ompa
t, self-adjoint and bounded inL(H") uniformly with respe
t to ".C3. For any f 2 H0kA"R"f �R"A0fkH" ! 0; "! 0: (1.2.2)C4. For any sequen
e f " 2 H" su
h that sup kf "kH" < 1 there exists thesubsequen
e "0 and w 2 H0 su
h thatkA"f " �R"wkH" ! 0; " = "0 ! 0: (1.2.3)P r o o f 1. The
ondition C1 follows dire
tly from the de�nition of theoperator R".150 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin Handles2. The
ondition C2 follows easily from the properties of the resolvent, namelythe following estimate is valid kA"kL(H") � 1:3. Let f 2 H. We denote u" = A"R"f , f " = R"f . To des
ribe the behaviorof u" on
" we introdu
e the operator �"0 : H1(M ")! H1(
) with the followingproperties:1) k�"u"kH0 + kr"�"u"kH0 � Cnku"kH" + kr"u"kH"o, C > 0,2) �"0u"(~x) = u"(~x) on
".(Here kr"u"kH" := ZM" 2X�;�=1 g��" �u�x� �u�x� d~x, where g��" are the
omponents of thetensor inverse to g"). This operator exists, see, e.g, [3℄.Due to C1�C2 we have ku"kH" � kf "kH" ! kfkH0 . Moreover, using varia-tional methods, we obtainkr"u"k2H" � 2kf "kH" � ku"kH" :Using these inequalities and the properties of the operator �"0, we
on
lude that�"0u" is bounded in H1(
) uniformly with respe
t to ", and therefore there existsa subsequen
e (still denoted by ") su
h that�"0u" !"!0 u0 2 H1(
) weakly in H1(
) and strongly in L2(
): (1.2.4)To des
ribe the behavior of u" on the tubes G"i , i = 1; 2; 3, we represent u" inthe form u"('i; zi) = P "i u"(zi) +Q"iu"('i; zi); (1.2.5)where P "i u"(zi) = 12� 2�Z0 u"('i; zi)d'i:Let �"i : H1(G"i )! H1([0; li℄) that is de�ned by the formula�"iu"(zi) = 8><>:p"P "i (zi); zi 2 [Æ"; li � Æ"℄;p"P "i (Æ"); zi 2 [0; Æ");p"P "i (li � Æ"); zi 2 (li � Æ"; li℄:
Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 151
A. KhrabustovskyiWe have the following estimates:
ddzi�"iu"
2L2[0;li℄ = li�Æ"ZÆ" 0� ��zi 12� 2�Z0 u"('i; zi)p"d'i1A2 dzi� 12� li�Æ"ZÆ" 2�Z0 � ��ziu"('i; zi)�2 "d'idzi � (2�)�1kr"u"k20"; (1.2.6)k�"iu"k2L2[0;li℄ � Æ" h(�"iu"(Æ"))2 + (�"iu"(li � Æ"))2i+ (2�)�1ku"k2L2(G"i ): (1.2.7)Further, (�"iu"(Æ"))2 � 20�(�"iu"(zi))2 + li li�Æ"ZÆ" ���� ��zi�"iu"('i; zi)����2 dzi1A :Integrating this estimate on zi from Æ" to li � Æ" one has(�"iu"(Æ"))2 � C k�"iu"k2L2[Æ";li�Æ"℄ +
ddzi�"iu"
2L2[Æ";li�Æ"℄! (1.2.8)and, similarly,(�"iu"(li � Æ"))2 � C k�"iu"k2L2[Æ";li�Æ"℄ +
ddzi�"iu"
2L2[Æ";li�Æ"℄! : (1.2.9)It follows from (1.2.6)-(1.2.9) that �iu" is bounded in H1([0; li℄), and thereforefor i = 1; 2; 3 there exists a subsequen
e (still denoted by ") su
h that�"iu" !"!0 ui 2 H1([0; li℄) weakly in H1([0; li℄) and strongly in L2([0; li℄):(1.2.10)The following lemma says that u" is vanishingly small in B".Lemma 1.1. Let u" 2 H1(M "). Thenku"k2L2(B")� C�"2kr"u"k2L2(B") + "kr"u"k2L2([iG"i ) + "2j ln "j�kr"u"k2L2(
") + ku"k2L2(
")��:P r o o f. At �rst we note that u"
an be extended to the whole ball B" in su
ha way that kr"u"kL2(B") � Ckr"u"kL2(B") (see [9, p. 118, Ex. 4.10℄). Let us �x i152 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin Handlesfrom f1; 2; 3g. We introdu
e the spheri
al
oordinates ' 2 [0; 2�℄; � 2 [0; �℄ on B"su
h that the points of S"i have the
oordinates ' 2 [0; 2�℄; � = ar
sin("=b") =: �".So, we extend u" to B" and haveu"('; �) = u"('; �") + �Z�" �u"('; )� d :Further, 2�Z0 ���"Z�" (u"('; �))2b"2 sin �d�d'� C"20� ���"Z�" ��u"('; )� �2 sin d d' � ���"Z�" (sin )�1d + 2�Z0 (u"('; �"))2 d'1A :(1.2.11)Sin
e C1 � �" � C2, the �rst term is estimated by C"2kr"u"k2L2(B"). Nowwe estimate the se
ond term. Representing the
orresponding integral in the
ylindri
al
oordinates, one hasu"('; �") � u"('i; li) = u"('i; 0) + liZ0 �u"('i; zi)�zi dzi:Let D" and R" be the balls in
with the radii d" and r" (r" > d"). Then forany u 2 H1(R" nD") the following estimate is valid (see [1℄):kuk2L2(�D") � Cd" �j ln d"j � kruk2L2(R"nD") + 1(r")2 kuk2L2(R"nD")� : (1.2.12)Using (1.2.12), we have"2 2�Z0 (u"('i; li))2 d'i � C"2 2640� 2�Z0 u"('i; 0)d'i1A2 + 2�Z0 liZ0 ��u"('i; zi)�zi �2 dzi375� Ch"2j ln "j�ku"k2L2(
") + kr"u"k2L2(
")�+ "kr"u"k2L2(G"i )i:We denote D�"i = f('; �) 2 B" : � 2 [� � �"; �"℄g. It follows from (1.2.11) thatfor i = 1; 2; 3: ku"k2L2(B"n(D"i[D�"i ))� C�"2kr"u"k2L2(B") + "kr"u"k2L2(G"i ) + "2j ln "j�kr"u"k2L2(
") + ku"k2L2(
")��:Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 153
A. KhrabustovskyiThe lemma is proved sin
e Si=1;2;3 [B" n (D"i [D�"i )℄ = B".We return to the proof of Theorem 1.3. We denote u := (u0; u1; u2; u3). Letus prove that u = A0f , what is equal to the ful�lment of the following
onditions:I: ui(0) = 0; i = 1; 2; 3; (1.2.13)II: u1(l1) = u2(l2) = u1(l3); (1.2.14)III: (ru0;rw)H0 + (u0; w)H0 = (f0; w)H0 ; 8w 2 H1(
); (1.2.15)IV: 3Xi=1 liZ0 (ui(z))0(wi(z))0dz + 3Xi=1 liZ0 ui(z)wi(z)dz= 3Xi=1 liZ0 fi(z)wi(z)dz; 8wi 2 H1[0; li℄: (1.2.16)Let us verify the ful�lment of these
onditions.I. Using the tra
e theorem, for i = 1; 2; 3 we haveui(0) = lim"!0�"iu"(0) = lim"!0�"iu"(Æ") = p" lim"!0 �u"i ; (1.2.17)where �u"i is the mean value of u" over �D"i . It follows from (1.2.12), (1.2.17) that(1.2.13) is valid.II. For i; j = 1; 2; 3, one hasjui(li)� uj(lj)j = lim"!0p"jû"i � û"jj; (1.2.18)where û"i is the average value of u" over the
ir
le S"i .We denote v"(~x) := u"(~x)� U ", where U " is the average value of u" over B",and by v̂"i we denote the average value of v" over the
ir
le S"i .Using the inequality of the type (1.2.12) and Poin
are inequality, one hasjv̂"i j2 � C �����ln tan �"2 ���� � kr"v"k2L2(B") + 1(b"i )2 kv"k2L2(B")�� C ����ln tan �"2 ���� � kr"v"k2L2(B"): (1.2.19)Using (1.2.19), we havejû"i � û"j j = jv̂"i � v̂"j j � jv̂"i j+ jv̂"j j � Cs����ln tan �"2 ���� � kr"v"kL2(B"): (1.2.20)
154 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesSin
e ��ln tan �"2 �� < C, it follows from (1.2.18), (1.2.20) that (1.2.14) is ful�lled.III. Clearly, it is su�
ient to prove (1.2.15) for w su
h that9Æ > 0 8i = 1 : : : 3 : �g(supp(w); x"i ) � Æ;where �g is the distan
e on
generated by the metri
g (be
ause the set of thesew is dense in H1(
)). Then for these w and for su�
iently small " supp(w) �
".Let w"(~x) 2 L2(M "): w"(~x) = w(~x) in
" and w" = 0 in M " n
". Clearly,w" 2 H2(M ") for " being small enough.We have0 = lim"!0�(r"u";r"w")H" + (u"; w")H" � (f "; w")H"�= lim"!0�(r�"u";rw)L2(
) + (�"u"; w)L2(
) � (f;w)L2(
)�= (ru0;rw)H0 + (u0; w)H0 � (f0; w)H0 ;and (1.2.15) is valid.IV. It is su�
ient to prove (1:2:16) for su
h wi that9Æ > 0 8z 2 [0; Æ℄ : wi(z) = 0 and 8z 2 [li � Æ; li℄ : wi(z) = wi(li);be
ause the set of these wi is dense in the set of test fun
tions mentioned above.For su�
iently small " Æ" � Æ, and therefore these wi satisfy the
onditions:8z 2 [0; Æ"℄ : wi(z) = 0 and 8z 2 [li � Æ"; li℄ : wi(z) = wi(li).At �rst, let us estimate the reminder Q"u" on G"i . Using Poin
are inequality,we have li�Æ"ZÆ" 2�Z0 (Q"iu"('i; zi))2d'idzi � C li�Æ"ZÆ" 2�Z0 ��Q"iu"�'i �2 d'idzi= C li�Æ"ZÆ" 2�Z0 ��u"�'i�2 d'dzi � C"kr"u"k2L2(G"i ): (1.2.21)Using the above and the representation (1.2.5), we have3Xi=1 liZ0 (ui(zi))0(wi(zi))0dzi = 3Xi=1 lim"!0 12� 2�Z0 li�Æ"ZÆ" ��"iu"�zi �w"i�zi dzid'i= 3Xi=1 lim"!0 12� 24 2�Z0 li�Æ"ZÆ" �u"�zi � ��zi � wip"� "dzid'i + 2�Z0 li�Æ"ZÆ" p"Q"iu" � �2wi�z2i dzid'i35 :Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 155
A. KhrabustovskyiIn view of (1.2.21) the se
ond integral tends to zero. Therefore we have3Xi=1 liZ0 (ui(z))0(wi(z))0dz = 12� lim"!0(r"u";rw")H" ;where w" 2 H1(M ")w"(x) = 8><>:wi(zi)"�1=2; ~x = (zi; 'i) 2 Gi;0; ~x 2
";wi(li � Æ")"�1=2; ~x 2 B":In the same way using Lemma 1.1, we obtain3Xi=1 0� liZ0 ui(z)wi(z)dz � liZ0 fi(z)wi(z)dz1A = 12� lim"!0�(u"; w")H" � (f "; w")H"�:The last two equalities imply the
ondition (1.2.16).Thus we prove that u = A0f .It is easy to see that (1.2.2) follows from (1.2.4), (1.2.10), (1.2.21), andLemma 1.1. The
ondition C3 is ful�lled.4. It remains to verify the ful�lment of the
ondition C4. Let f " 2 H" besu
h that sup kf "kH" < 1. We denote u" = A"f ". It is
lear that the normsku"k2L2(M")+kr"u"k2L2(M") are uniformly bounded with respe
t to ". In the sameway as in item 3 one
an prove that there exists a subsequen
e (still denoted by ")su
h that the following limits existw0 = lim"!0�"0u" 2 H1(
) strongly in L2(
); (1.2.22)wi = lim"!0�"iu" 2 H1[0; li℄; i = 1; 2; 3 strongly in L2[0; li℄: (1.2.23)By means Lemma 1.1 we haveku"k2L2(B") ! 0; "! 0: (1.2.24)The ful�lment of the
ondition C4 (with w = (w0; w1; w2; w3)) follows easilyfrom (1.2.21)�(1.2.24).Theorem 1.3 is proved.We
ontinue the proves of Theorems 1.1 and 1.2. Let �"1 � �"2 � �"3 � : : : ��"k !k!1 0 be the eigenvalues of A" written with a
ount of their multipli
ityand let f "1 ; f "2 : : : be the
orresponding eigenve
tors normalized by the
ondition(f "i ; f "j )H" = Æij . Let �1 � �2 � �3 � : : : � �k !k!1 0 be the eigenvalues of A.156 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesIt is proved in [7℄ that the
onditions C1�C4 imply�"k ! �k; "! 0; k = 1; 2; 3 : : :and, moreover, if �k � �k+1 = �k+2 = : : : = �k+m > �k+m+1, then for any w 2N(�k+1) there exists a linear
ombination �f " of the eigenfun
tions f "k+1 : : : f "k+msu
h that k �f " �R"wkH" ! 0; "! 0:Sin
e �"k = 1�"k � 1, �k = 1�k � 1, u"k = f "k (and so N(�k) = N(�k)), it followsthat Theorems 1.1 and 1.2 are proved.2. Riemannian Manifold of In
reasing Genus2.1. Setting of the Problem and Main ResultLet
be a two-dimensional
ompa
t Riemannian manifold without boundaryand with a metri
g. By �
we denote the
orresponding Lapla
e�Beltramioperator. Let D"i , i = 1 : : : N(") = 3N1(") be a system of the balls in
with
enters x"i 2
and radii d". We
onsider the following domain with holes:
" =
nN(")[i=1 D"i :Let G"i , i = 1 : : : N("), be a set of tubesG"i = f~x = ('i; zi) : 'i 2 [0; 2�℄; zi 2 [0; 1℄g:We suppose that C"i = f~x = ('i; zi) 2 G"i : zi = 0g � �G"iis di�eomorphi
to �D"i . Using this di�eomorphism, we glue G"i , i = 1 : : : N(") to
". By S"i we denote the "ends" of G"iS"i = f~x = ('i; zi) 2 G"i : zi = 1g � �G"i :We divide the set f1 : : : N(")g into subsets, ea
h
onsisting of three elements.For any three indexes i; j; k we introdu
e the number Aijk, and set Aijk = 1 ifi; j; k belong to the same subset, and we set Aijk = 0 otherwise. If Aijk = 1, wesay that the
orresponding holes D"i ;D"j ;D"k are
onne
ted.
Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 157
A. KhrabustovskyiFor any i; j; k : Aijk = 1 we
onsider the sphere B"ijk � R3 with the radius b".Let D"i ;D"j ;D"k be the geodesi
balls on B"ijk with the radii b" ar
sin�d"b"�. It is
lear that the radii of the
ir
les �D"i ; �D"j ; �D"k are equal to d". LetB"ijk = B"ijkn(D"i [D"j [ D"k):One
an see that �D"i ; �D"j ; �D"k are di�eomorphi
to S"i ; S"j ; S"k, respe
tively. Us-ing these di�eomorphisms, we glue B"ijk to G"i [ G"j [ G"k. Thus we obtain themanifold (see Fig. 2)M " =
" [ 24 [i;j;k:Aijk=1(B"ijk [G"i [G"j [G"k)35 :We denote the points of the manifold by ~x. Clearly,M "
an be
overed by a systemof
harts, and suitable lo
al
oordinates fx1; x2g
an be introdu
ed. It is supposedthat M " is equipped with the metri
g" that
oin
ides with the metri
g on
"and with the metri
s indu
ed from R3 on B"ijk. On G"i the metri
is de�ned bythe formula for the square of the element of length:ds2 = q"i dz2i + (d")2d'2i ; q"i > 0:By g"�� , we denote the
omponents of metri
tensor in lo
al
oordinates.
Fig. 2: Manifold M ".We denote r"i = minj �g(x"i ; x"j), where �g is a distan
e on
generated bymetri
g. It is supposed that the following properties are valid:(i) j ln d"j�1 � C(r"i )2, r"i = O("), 0 < C1 � "2N(") � C2, "! 0;(ii) q"i � q" ! 0; "! 0, i.e., the lengthes of the
ylinders G"i tend to zero;(iii) (b")2�j ln d"j+ ��ln tan �"2 ��+ pq"d" �! 0; "! 0; �" = ar
sin d"b" :158 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesLet �" be a Lapla
e�Beltrami operator onM ". Let 0 = �"1 < �"2 � �"3 � : : : ��"k !k!11 be the eigenvalues of ��" written with a
ount of their multipli
ity,and u"1; u"2; u"3 : : : be the
orresponding eigenve
tors normalized by the
ondition(u"i ; u"j)H" = Æij.To des
ribe the behavior of �"k as "! 0 we introdu
e the notations:R"i = f~x 2
" : d" � �g(~x; x"i ) � r"i =2g; bC"i = f~x 2
" : �g(~x; x"i ) = r"i =2g;�"ijk = G"i [G"j [G"k [B"ijk; b�"ijk = R"i [R"j [R"k [ �"ijk:For i; j; k : Aijk = 1, we
onsider the problem�"v = 0 in b�"ijk; v = 1 on bC"i and v = 0 on bC"j [ bC"k: (2.1.1)The solution of (2.1.1) we denote by v"ijk. It is
lear that v"ijk = v"ikj .For i; j; k : Aijk = 1, we denoteW "ijk = � Zb�"ijk (r"v"ijk;r"v"jik)d~x;(here (r"u;r"v) := 2X�;�=1 g��" �u�x� �v�x� ), otherwise we set W "ijk = 0 (i.e., W "ijk =�(r"v"ijk;r"v"jik)L2(b�"ijk)).We introdu
e the generalized fun
tionW "(x; y) = Xi;j;k=1:::N(")W "ijkÆ(x� x"i )Æ(y � x"j) 2 D0(
�
):The limit(iv) 9 lim"!0W "(x; y) =W (x; y) 2 L1(
�
) - positive symmetri
fun
tion,is supposed to exist.We denote H" := L2(M "), H0 := L2(
).Theorem 2.1. For any k = 1; 2; 3 : : :�"k ! �k; "! 0;where 0 = �1 < �2 � �3 � : : : are the eigenvalues of the operator L : L2(
) !L2(
): [Lu℄(x) = ��
u(x) + Z
W (x; y)(u(x) � u(y))dy:
Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 159
A. KhrabustovskyiTheorem 2.2. Let R" : H0 !H":[R"f ℄(~x) = 8<:f(~x); ~x 2
";0; ~x 2 Si;j;k:Aijk=1�"ijk:Then the eigenfun
tions of ��"
onverge in the sense (1.1.1) to the eigen-fun
tions of the operator L.2.2. Proof of Theorems 2.1 and 2.2We denote by A" and A the operators inverse to ��"+I and L+I, respe
tively.Analogously as in the previous se
tion, Theorems 2.1 and 2.2 follow fromTheorem 2.3. The
onditions C1�C4 are ful�lled.P r o o f. The
onditions C1�C2 are trivial. Let us
he
k the
ondition C3.Let f 2 H. We denote u" = A"R"f , f " = R"f , u0 = A0f . Noti
e that thefollowing estimates are valid:ku"kL2(M") � kf "kL2(M"); kr"u"k2L2(M") � 2kf "kL2(M") � ku"kL2(M"): (2.2.1)It is well known that u" minimizes the fun
tionalJ"[u"℄ = ZM" �jr"u"j2 + (u")2 � 2f "u"� d~x (2.2.2)in the
lass of fun
tions H1(M "), while u0 minimizes the fun
tionalJ0[u℄ = Z
�jruj2 + u2 � 2fu�dx+ Z
Z
12W (x; y) (u(x)� u(y))2 dxdy (2.2.3)in the
lass of fun
tions H1(
). The
onverse assertions are also true.In order to prove that u"
onverges to u0, we
onsider the following abstra
ts
heme.Let H" be a Hilbert spa
e depending on the parameter " > 0, (u"; v")"; ku"k"be a s
alar produ
t and norm in this spa
e, and F " be the
ontinuous linear fun
-tionals in H" whi
h are uniformly bounded with respe
t to ". Let H be a Hilbertspa
e with the s
alar produ
t (u; v) and the norm kuk, and F be a
ontinuouslinear fun
tional in H.Consider the following two problems of minimization:ku"k2" + F "[u"℄! inf; u" 2 H"; (2.2.4)kuk2 + F [u℄! inf; u 2 H: (2.2.5)160 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesLet u" and u0 be the minimizants of the problems (2.2.4) and (2.2.5). The fol-lowing theorem is proved in [3℄.Theorem 2.4. Let M be a dense subset of H, let �" : H" ! H, and P " :M ! H" be the operators satisfying the following
onditions:(a) k�"w"k � Ckw"k;8w" 2 H";(b1) �"P "w ! w weakly in H as "! 0;8w 2M ;(b2) lim"!0 kP "wk" = kwk;8w 2M ;(b3) for any sequen
e
" 2 H" su
h that �"
" !
weakly as " ! 0, for anyw 2M one has lim"!0 j(P "w;
")"j � Ckwkk
k;(
) for any sequen
e
" 2 H", su
h that �"
" !
weakly, as "! 0, one haslim"!0F "[
"℄ = F [
℄:Then �"u" !"!0 u0 weakly in H:Noti
e that Theorem 2.4 holds true if the
onditions (b3) and (
) hold onlyfor su
h sequen
es
" that the norms k
"k" are uniformly bounded with respe
tto " be
ause in the proof of Theorem 2.4 the
onditions (b3) and (
) are used onlywith these sequen
es.Now we apply our abstra
t s
heme. Let H" be the Hilbert spa
e H1(M ") ofthe fun
tions on M " with the s
alar produ
t(u"; v")" = ZM" [(r"u";r"v") + u"v"℄ d~x;and let F " be a linear fun
tional de�ned by the formulaF "[u"℄ = ZM" �2f "u"d~x:Let H be the Hilbert spa
e H1(
) with the s
alar produ
t(u; v) = Z
[(ru;rv) + uv℄ dx+ Z
Z
12W (x; y)(u(x) � u(y))(v(x) � v(y))dxdy;and f be a linear fun
tional on it de�ned by the formulaF [u℄ = Z
�2fudx:Obviously, the fun
tionals F " are uniformly bounded with respe
t to ".Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 161
A. KhrabustovskyiNow we introdu
e the operators �" and P " satisfying the
onditions (a)�(
)of Theorem 2.4.The existen
e of the operator �" : H1(M ")! H1(
) that has the propertieskr�"u"k2L2(
) + k�"u"k2L2(
) � C�kru"k2L2(
") + ku"k2L2(
")�; (2.2.6)�"u"(~x) = u"(~x); ~x 2
" (2.2.7)is proved in [9, p. 118, Ex. 4.10℄).Clearly, (a) follows from (2.2.6).We introdu
e the operator P ". Let '(r) be a twi
e
ontinuously di�erentiablenon-negative fun
tion on the half-line [0;1) equal to 1 for r 2 [0; 1=4℄ and equalto 0 for r � 1=2. We set'"i (x) = '��g(x; x"i )r"i � ; '"0i(x) = '��g(x; x"i )d"0 � ;where d"0i = exp(�j ln d"j1=2).Let M = C2(
), M is dense in H1(
) and let w 2M . We de�ne the operatorP " by the equality[P "w℄(~x) = 8>>>><>>>>:w(~x) + (w"i � w(~x))'"0i(~x) +�(v"ijk(~x)� 1)w"i+v"jik(~x)w"j + v"kij(~x)w"k�'"i (~x); ~x 2 R"i ; j; k : Aijk = 1;v"ijk(~x)w"i + v"jik(~x)w"j + v"kij(~x)w"k; ~x 2 �"ijk;where w"i = w(x"i ).To see that the
onditions (b1)�(b3) hold, we use the following estimates ofthe solution v"ijk of (2.1.1).Lemma 2.1 Let R"0q = f~x 2
" : d"0q � �g(x; x"q) � r"q=2g. Then fori; j; k : Aijk = 1 and q 2 fi; j; kg:jD�(v"ijk(~x)� Æiq)j � C ����D�(ln�g(x"q; ~x))lnd" ���� ; ~x 2 R"0q j�j = 0; 1:The p r o o f of the lemma is
arried out in the same way as that ofLemma 2.4 in [9, p. 44℄ using the inequality 0 � v"ijk � 1 whi
h follows from themaximum prin
iple.
162 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesLemma 2.2 Let u" 2 H1(M "). Then for any i; j; k : Aijk = 1ku"k2B"ijk � C �(b")2 ����ln tan �"2 ���� � kr"u"k2L2(B"ijk) + pq"(b")2d" kr"u"k2L2(G"i[G"j[G"k)+(b")2�j lnd"j � kr"u"k2L2(R"i[R"j[R"k) + 1r"i 2 ku"k2L2(R"i[R"j[R"k)�� ;ku"k2G"i � C �q"kr"u"k2L2(Gi) + d"pq"�j lnd"j � kr"u"k2L2(R"i ) + 1r"i 2 ku"k2L2(R"i )�� :The p r o o f of this lemma is
arried out in the same way as the proof ofLemma 1.1.We verify that the
ondition (b2) holds. We denote bR"i = f~x 2
" : r"i =4 ��g(~x; x"i ) � r"i =2g. Let w 2M . ThenkP "wk2" = Z
" �jrwj2 + w2�dx+ Xi<j<k:Aijk=1 Zb�"ijk �w"i 2jrv"ijkj2 + w"j2jrv"jikj2 + w"k2jrv"kijj2+2w"iw"j (rv"ijk;rv"jik) + 2w"jw"k(rv"jik;rv"kij) + 2w"iw"k(rv"ijk;rv"kij)�d~x+ Æ("):(2.2.8)Here Æ(") are the remaining integrals estimated as follow?:jÆ(")j � C(w) Xi;j;k:Aijk=1 �J"ijk +E"ijk + I"ijk + Y "ijk + (d"0)2� ;where J"ijk = ZbR"i[ bR"j[ bR"k �jr"v"ijkj2 + 1r"i 2 jv"ijkj2� d~x;E"ijk = ZR"0i[R"0j[R"0k �jr"v"ijkj+ 1r"i jv"ijkj� d~x;I"ijk = ZR"i jv"ijk � 1j2d~x+ ZR"j[R"k jv"ijkj2d~x; Y "ijk = Z�"ijk jv"ijkj2d~x:?The sum Pi<j<k:Aijk=1 means that any three indexes fi; j; kg appear only ones in this sum.Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 163
A. KhrabustovskyiUsing Lemma 2.1 and maximum prin
iple for v"ijk, we haveJ"ijk � Cj lnd"j�2(1 + j ln r"i j2); (2.2.9)E"ijk � Cj lnd"j�1�r"i j ln r"i j+ d"0j lnd"0j�; (2.2.10)I"ijk � C�(d"0)2(1 + j lnd"j�1) + (r"i ln r"i = lnd")2�; (2.2.11)Y "ijk � C � j�"ijkj � C(d"pq" + (b")2); (2.2.12)Using (i)�(iii), we
on
lude thatÆ(") ! 0; "! 0: (2.2.13)We denote V "ijk = Rb�"ijk jr"v"ijkj2d~x, where i; j; k : Aijk=1. Sin
e v"ijk + v"jik +v"kij = 1 for any i; j; k : Aijk = 1, we have V "ijk = W "ijk +W "ikj. Therefore (2.2.8)
an be rewritten in the formkP "wk2 =Z
�jrwj2 +w2�dx+ Xi;j;k=1:::N(")W "ijk�(w(x"i ))2 � w(x"i )w(x"j)�+ Æ(")= Z
�jrwj2 + w2�dx+ 12 Xi;j;k=1:::N(")W "ijk�w(x"i )� w(x"j)�2 + Æ("): (2.2.14)It is easy to see that (b2) follows from (iv), (2.2.13) and (2.2.14).We verify the
ondition (b1). Let w 2 M . In view of the
onditions (a) and(b2), the norms k�"P "wk" are uniformly bounded with respe
t to ", and in thesame way as in (b2) one
an prove that �"P "w ! w strongly in L2(
). Thus the
ondition (b1) also holds.We verify the
ondition (b3). Let w 2 M , and the sequen
e
" 2 H" is su
hthat the norms k
"k" are uniformly bounded with respe
t to ", and �"
" !
weakly in H as "! 0. Integrating by parts, we have(P "w;
")" = (��
w + w;�"
")L2(
) + Æ("); (2.2.15)where Æ(") are the remaining integrals. Using Lemma 2.1, in the same way as in(b2) we obtain the estimatelim"!0 jÆ(")j � C lim"!08><>:0�N(")Xi=1 (w"i )2jR"i j1A1=2 k�"
"kL2(
)9>=>; :Sin
e �"
"
onverges weakly to
in H, then �"
"
onverges strongly to
inL2(
), and therefore we havelim"!0 jÆ(")j � CkwkL2(M") � k
kL2(
) � Ckwk � k
k: (2.2.16)164 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesIt follows from (2.2.15)�(2.2.16) that (b3) holds.Further, we verify that the
ondition (
) holds. Let the sequen
e
" 2 H" besu
h that �"
" !
weakly in H. Then �"
" !
strongly in L2(
). We havejF "[
"℄� F [
℄j = ������ Z
" f � (�"
" �
)d~x������+ ������� Z
n
" f
d~x�������! 0; "! 0;and so the
ondition (
) holds.Thus all the
onditions of Theorem 2.4 hold. Hen
e �"u" ! u0 weakly in H.Therefore, by the embedding theorem, �"u" ! u0 strongly in L2(
): Finally, wehave kA"R"f �R"A0fk2H" = ku"k2L2([�"ijk) + k�"u" � u0k2L2(
"):In view of Lemma 2.2, (i�iii) and (2.2.1) ku"k2L2([�"ijk) ! 0; " ! 0. Thus C3 isproved.And �nally, we verify the ful�lment of the
ondition C4. Let f " 2 H" besu
h that sup kf "kH" < 1. Let u" = A"f ". In view of (2.2.1), �"u" is weakly
ompa
t in H1(
) and so there exists the subsequen
e "0 and w 2 H1(
) su
hthat �"u" ! w strongly in L2(
): This and Lemma 2.2 imply C4.Theorem 2.3 and therefore Theorems 2.1 and 2.2 are proved.2.3. ExampleWe
onsider an example of the manifoldM " and
al
ulate the fun
tionW (x; y)expli
itly.Let
ontain the subset K, whi
h is a �at square with the side equal to l.Let " > 0 and let n" = �1"�1=3.We divide K into the squares K"�, � = 1 : : : n"2 with the side length l=n".Within ea
h squareK"� we
ut out n"4 holesD"i with the radius d" = exp ��n"6=l6�and su
h that their
enters form a periodi
latti
e with the period ln"3 . It is
learthat j lnd"j�1 = l4(r"i )2. The total number of D"i is equal to N(") = n"6.For ea
h hole D"i we denote the number of square K"�
ontaining this hole by�(i). Sin
e the number of holes within the square K"� is equal to n"2 � n"2, we
an assign to ea
h hole D"i � K"� the pair (�(i);
(i)), �(i);
(i) 2 f1 : : : n"2g. So,ea
h hole D"i is
hara
terized by (�(i); �(i);
(i)).If �(i) = �(j) =
(k); �(j) = �(k) =
(i); �(k) = �(i) =
(j) and only inthis
ase, then we join the boundaries of the holes D"i ;D"k;D"j by means of themanifold �"ijk = G"i [G"j [G"k [B"ijk.Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 165
A. KhrabustovskyiWe set q"i = [q � j lnd"j � d"℄2; q > 0and
hoose su
h b" that (iii) is valid andln�tan �"2 � = lnd" ! 0; "! 0; �" = ar
sin d"b"(for example d" � Cb").In order to
al
ulate W (x; y) we �nd a suitable approximation for the solutionv"ijk to (2.1.1). Namely, we represent it in the form v"ijk = bv"ijk + w"ijk, where
bv"ijk(~x) =
8>>>>>>>>>>><>>>>>>>>>>>:
a"i ln j~x� x"i j+ b"i ; ~x 2 R"i ;A"i z +B"i ; ~x = (zi; 'i) 2 G"i ;a"j ln j~x� x"jj+ b"j; ~x 2 R"j ;A"jz +B"j ; ~x = (zj ; 'j) 2 G"j;a"k ln j~x� x"kj+ b"k; ~x 2 R"k;A"kz +B"k; ~x = (zk; 'k) 2 G"k;C"ijk; ~x 2 B"ijk:We
hose the
onstants a"i ; b"i : : : A"k; B"k; C"ijk su
h that:1) bv"ijk is a harmoni
fun
tion in G"i [R"i , G"j [R"j , G"k [R"k,2) bv"ijk = 1 on bC"i , bv"ijk = 0 on bC"j [ bC"k,3) bv"ijkjS"i = bv"ijkjS"j = bv"ijkjS"k =M , where M is a
onstant,4) �bv"ijk�~n jS"i + �bv"ijk�~n jS"j + �bv"ijk�~n jS"k = 0, ~n is the outward (or inward) normal?.As a result, we obtaina"i = 2j ln d"j�13(1 + q) (1 + o(1)) = �2a"j = �2a"k;A"i = �a"ipq"id" ; A"j = �a"jpq"jd" ; A"k = �a"kpq"kd" ;b"i = 1� a"i ln(r"i =2); b"j = �a"j ln(r"j=2); b"k = �a"k ln(r"k=2);B"i = a"i lnd" + b"i ; B"j = a"j lnd" + b"j ; B"k = a"k lnd" + b"k:?Here the normal derivatives are taken in an arbitrary point of S"i . It is easy to see that the
onditions 1)�3) guarantee that �bv"ijk�~n are
onstant on S"i , as on S"j and S"k). The
ondition 4)determines the
onstant M from the
ondition 3).
166 Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2
On the Spe
trum of Riemannian Manifolds with Atta
hed Thin HandlesDire
t
al
ulations show thatkr"bv"ijkk2L2(b�"ijk) = 4�3(1 + q) j lnd"j�1(1 + �o(1)) ! 0; "! 0; (2.3.1)(r"bv"ijk;r"bv"jik)L2(b�"ijk) = � 2�3(1 + q) j ln d"j�1(1 + �o(1)); "! 0: (2.3.2)Now we prove that w"ijk gives vanishingly small
ontribution toW "ijk. Sin
e v"ijkminimizes the fun
tional I"[v℄ = kr"vk20" in the
lass of fun
tions from H1(b�"ijk)equal to 1 on bS"i and equal to 0 on bS"j [ bS"k, then kr"v"ijkk2L2(M") � kr"bv"ijkk2L2(M")and therefore kr"w"ijkk2L2(M") � 2 ��(r"w"ijk;r"bv"ijk)L2(M")�� :Using the properties of the fun
tion bv"ijk, we obtainkr"w"ijkk20" � 4�d" ����� A"ipq"i w"i + A"jpq"j w"j + A"ipq"j w"k����� = 4�ja"iw"i + a"jw"j + a"kw"kj= 4�ja"j(w"j � w"i ) + a"k(w"k � w"i )j; (2.3.3)where w"i ; w"j ; w"k are the average values of w"ijk in S"i ; S"j ; S"k, respe
tively.The following estimate is validjw"i � w"j j+ jw"i � w"kj � Crj ln tan �"2 j � kr"v"ijkk0"� Crj ln tan �"2 j � kr"bv"ijkk0" � Crj ln tan �"2 j=j ln d"j !"!0 0: (2.3.4)The proof is similar to that of (1.2.20).It follows from (2.3.3), (2.3.4) and from the form of the
oe�
ients a"i ; a"j ; a"kthat kr"w"k20" = �o(j lnd"j�1): (2.3.5)We have W "ijk = ��(bv"ijk; bv"jik)L2(b�"ijk) + (bv"ijk; w"jik)L2(b�"ijk)+ (w"ijk; bv"jik)L2(b�"ijk) + (w"ijk; w"jik)L2(b�"ijk)�: (2.3.6)It follows from (2.3.1), (2.3.2), (2.3.5), (2.3.6) thatW "ijk � �(bv"ijk; bv"jik)L2(b�"ijk) � 2�3(1 + q) j lnd"j�1:Journal of Mathemati
al Physi
s, Analysis, Geometry, 2009, vol. 5, No. 2 167
A. KhrabustovskyiLet w(x; y) 2 C1(
). ThenhW "; wi = Xi;j;k:Aijk=1 2�3(1 + q)w(x"i ; x"j)j ln d"j�1:By the
onstru
tion of the manifold M ", for any three squares K"�;K"� ;K"
thereare only three holes D"i��
;D"j��
;D"k��
su
h thatD"i��
� K"�; D"j��
� K"�; D"k��
� K"
; Ai��
j��
k��
= 1:Therefore the sum above
an be easily rewritten as follows:hW "; wi = 2�3(1 + q) n2(")X�;�;
=1w(xi��
; xj��
)j ln d"j�1= 2�3(1 + q) n2(")X�;�;
=1w(xi��
; xj��
)jK"�j� jK"� j� jK"
j !"!0 ZK ZK ZK 2�3(1 + q)w(x; y)dxdydz:Thus W (x; y) = �K(x)�K(y)ZK 2�3(1 + q)dz = 2�l23(1 + q)�K(x)�K(y);where �K is the
hara
teristi
fun
tion of K.A
knowledgements. The author is grateful to Prof. E.Ya. Khruslov forsetting the problem and his attention paid to this work. The work is partiallysupported by the Grant for Young S
ientists of National A
ademy of S
ien
esof Ukraine (No. 20207). Referen
es[1℄ C. Anne, Spe
tre du Lapla
ien et E
rasement d'Anses. � Ann. S
i. E
ole. NormSup. (4) 20 (1987), 271�280.[2℄ C. Anne and B. Colbois, Spe
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ien Agissant sur les p-FormesDi�e'rentielles et e'Crasement d'Anses. � Math. Ann. 303(3) (1995), 545�573.[3℄ L. Boutet de Monvel and E.Ya. Khruslov, Averaging of the Di�usion Equation onRiemannian Manifolds of Complex Mi
rostru
ture. � Tr. Mos
ow Mat. Obsh
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ow Math. So
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tra of Manifolds with Small Handle. � Comment.Math. Helv. 56(1) (1981), 83�102.168 Journal of Mathemati
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On the Spe
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Spe
trum of Manifolds ofIn
reasing Topologi
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