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Sur le spectre du laplacien des fibrés en tores qui s'effondrent

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(13) G I 5-1$?^2)GF2 L7 . , O , . N ./,9* I=0 If5-<)<2=1 V-7i51$G 2ZD I 2@0[1+2 ]=7Z5 1 , , O ,98f.;8 01+2 1 a=7i5-1 365-7 J .. Q. ’. (M, gi ). ). jlk. &.

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(17) J. J [J. J. J J. λp,mp +1 (M, g) ≥ C1 (n,. (n,. e 6= 0  C2 ke(M )k22 ε2 dim H 2 (N, R) = 1 . ≤ λ1,1 (M, g) ≤ C3 ke(M )k22 ε2. C2 ke(M )k22 ε2 ≤ λp,k (M, g) ≤ C3 ke(M )k22 ε2 pour 1 ≤ k ≤ mp . 5-3-2Z?. =. mp = bp (N ) + bp−1 (N ) − bp (M ) J.  ™.

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(20) . . . . @. Z. . . -. . . . . . . ε. &. . . mp = 0 e 6= 0 ! 1. '}. e(M ). r. X. . c. '. dim H 2 (N, R) ≥ 1 . @q[p. .  $. . .  . . . . . . L„. . . . C. 4.  . . . . . . . . . . . . G. ∆pinv. . . . . . Γ  Ωp (M )G. . :. . . . . . . . . . ∆pinv. . . ∆p . G. Q. B. ∆pinv.

(21)  m6" ,. ..  . 6 !#"+&%“”u:w " .. *. n kRk. ,/./,. z. .. s. ∆. . * . . . 0 GF2=Iv? 1 I 5-1 Q 2^I  2 . , , O X , : 8 ; . 8 , ,9* d I 7 ? 2:<2:1 A I 3-2^I 2 2=I 02†I 2=I 01+2 1 a=7i51. 365-7 GF2tG <2:1QI 1 , \ 8:. , d * * ƒ * O d O O 2 0  *Y <R0[1 2TG 0[1+2< 7 02U < V 1+2 A 0[7 5 02 2 ∞ .O * . * . , . * * 51 7^<2hG-0 2=1If2=0[7hGF2R? 0[7 ]=07E2^‚2 I 2^I 0[1+2a 7=<2˜A+7 A+7Z2)G-0 ∞ 2^I O O , * .PO O * 8 ,N −2 5 A ^ 5? 2=1tI=07 G 1 53W5 2:07_A$7 A$7Z2 3 7 2 ∞ O€* .P, , . 5 7I 2^I 1365-7 51 2 J .. P". . O#O. {Œ| '. –. O. 2ZD ,. ,. I. .. a(n) a (n). 2. c(n). M. M. λ. kRk diam(M ) ≤ a α λ < a diam(M ) − ckRk. α. Œ 790/798MHM@\8;: %D67 14365SF2>G:W@\:f8M7 5W7 V 398M391W@\>GKL 547 ’ 0[ @E5 <X:4OEFM5j>L765 V 14OEF20/765f143654OE>GF2R2>L765 8 798 “791S1W@ F28 7 279ƒ‡02>L7Y@\F 0[@\1W@ V 1W@\02„M7 ! . .

(22) . . .

(23) I7@\8MBC5z@\KG:>L7 rD4Fo„[P F2@\8M027KG:S14‹7 @\1S KL39:48M3dOE rFMF25,K 539:SPQ7 FMHUOEKL8M7914HUOE148M7I5 5SF2H21 765 F28 MR2D6147936145zD9>L7XH27 @EHUMƒ,R214797 :S:47dF28M547IF21‡5S:S1S>A@ FMD9R[:SF2@E145477 H:S2KL7,OE8 5WOE7 > U0E2@\1S>GKLKLD939KG:4:43‡7I76HM5S@\:X8MH25 3 F24< 8 D6D OE@E825 8;5SF KGƒ‡ 02 >L7 7 %>L765 MoR2F 1436’ 5 Ÿ D6OE8M5SOEKLH2FM395Y1479368 5N54Bt79791414OEOE8;8M:N5IH2F23 8MM7z82KLD65NOE8MD6OE5S:Sƒ‡1SFMƒ,D 7 Ž 5S39FM>L3954ƒ,0/79798M8;5S: KLOE8_H P F28 HUK ™36OEƒ,dOE7 1S0202„2>GFMKL545 ƒ,78MOE>GFMKG8M5‡3 @\Bt79KG14147OE8MH25‡7 >CP>A„ @ ;M0™R2OE14:S7„MZ6547 5414KG79ƒ‡02021436>GK 54[79D 8r@\:4:S3 1SKL0[D67@\1 ;F2FM8 7 @\>G>LOE@E8MHU5Jƒ,ƒ,79:J79:SF2:S8T147X>LO 79V 8 @\1S3 KG%:SKL„2H2ƒ,798M7XD6147X365479OE> 8r: 76H25NOE02821S8MKG368M76D95NKG0[0[@\@\>L1765 >L7 0214OE021SKL39:4365 HUFT5S0/76D9:S147= rFM7X8MOEFM5 . . . . S1 . c. . . . Œ. .   . . .  %. . =. '. Š. . . . A ∈ SLn (Z). . A. Tn. . Tn. . . . X. :. qr%(. "nmWop. jhk O. d. 5. , G. <2=1I . O. 'h(>*-,#.. ,9* 1. G-0‘I ,. n ≥ 2  A ∈ SLn (Z) *. .;8. 0FI=`i2^I;A 5?2t?E5-7Z5? ,9* * *. 2. 7. ,. I. .. B ∈ Mn (R). ./, d O. *. 025WII , .. .. O. d. A = exp(B)  * 5M3W5 2:07RA$7 A$7E2 0 G = G(B) ⊂ 0 AQ2 N 8 7XAnUQ2l5-0 ]:7 M G2. 2 I  O. ,X8. ?. *. 0Q2. O. d0 5‘G <2=1I 1 GF2HI 1 1 ,5-0QJL– 2ZD I 2H0[1vV-7 . 8 . O *-,/. * 8Z* * U < < GLn+2 (R) 2 01†7 I 2Z5-0 Γ ⊂ G 2 d 02 Γ\G I N N ./,9* * . ,#. ,9* , 8E* * , n GF2 1 ? 1QI 7^0 ]:7Z2 T ]=7Z5 1 p : M → S A 57PI^0IXA2=1QI 1G-0)G < 7XA$U I=<2 (n, O * * O f 8 . , * . * * ƒ Ž . . O O d d GF2BA 0I 1 I^0!AA I 2 02 2^IR< 7 02^IhI=07 M I 1 U < V 1+2=I2 2 2^I AJ *-,/. ,9* , , * 8:. , d *O d 02 p I 01+2cI^0]=<2=7^I 1H7 2=<5-1n1 2:11+2A 07c0[1+2c< 7 0Q2lGF2c3 0[<2 1 I=07 O€* S 1 g5 7^I B. GF2. 2. .   J.  . J [J. J. dim Ker ∆1inv = d0 + 1 ,. G. ./,. I. *Ž* .X*. 0[7 . 0. .. 1+?. *. 2^I. .X*. 0 8 . f. 2c<. *. 0

(24) 1. . 7. ,. 02. 

(25). |K(M, g)| < a .  . I. J J. J $J. (n,. O*. ∆1inv. .. 5FG-<2. n − d0. 1+J ,#O. a > 0. , d. . 2. *. 13W5-7. ,. 15 *. ,. 2ZD. .. .. I. *. 2C01>2C?. .. 3W5. O. .. *. A$7E2^I1. *. 1C1n0. O#O. 2=I. . . O O * d 5-1 2 2 2 0Q2ŠA 0 7 [ * .O * ./,9* O#O 8 ,N G 1 5Š? 0 7f]=0[7E2cI 2Z? [ 11+2 2B3 7 2 * .X* . >A 0[7 0 J. 1QI. c(B, a) > 0. 5-1 2 g I=07 M λinv 1,i (M, g) < c .X*. 2:07^ILA$7. i. ,#O. ,. .. *. .. .. 0 a > 0 2ZD I 20[1+2T? 1I 5-1 2 c (B, a) > 0 O * ./,/* O#O 8 ,N d 2 2 0Q2cA 0[7 0 2Š< 7 02 g I^0[7 M 5H? 0[7f]:07E2If2Z? 1n1+2 2Š3 7 2 * 0 inv |K(M, g)| < a  15 λ1,d−d0 +1 (M, g) > c J (n, . O* . , O *. . .l,#O , . ,#. 2ZD I 2C0[1+2HI^0 2†GF2 d = n 2 d0 6= n l5 7^I G 2^I 1 A 2=1 R2 . O O O * * # , . , * 8 . * 8 . 8f. , d d < 7 02^I gε I^0[7 M 2 2 02 5

(26) ? 0[7f]=0[7E2)I 0[1 a 7=< <2=1 ] 7=1 22 * .X* . d 0 λinv 0 < i ≤ n − d0 J 1,i (M, gε ) → 0 0Q5-1$G ε → 0  A 0[7 (n, O€* . .X* .bO * * , d d = d0 = n 5 7I G = Rn  M 2^I 0[1 7E2l2 2^Iba 7^<2^I_U 5-7^< 1 02^I .. O O. *. *Ž*. 1. d 6= n B5. 7IcA .X*. *. .. 0[7. .. 2ZD5? .X*. . 0. *. 2=<2=1. 07. .. .LO. 8:.. *. 2^Ia. GF24? 0[7f]:07E2†2 * . . 00 ? 1QI 5 1 2 . inv L2. G. ,. ƒ:.. 5-<. , d. ,. 7=<2^I. k ≤ d − d0  .. 0. ,#O. 2ZD ,. 7E2@0[1 a . O#O 2 2. c (B, k) > 0 λ1,k+1 (M, gεk ) > c00. ε→0. 1n365-7 , *. ,. 5-1. .. 2^IfJ. ,#O#O 8:. , d 2†0[1+2la:5< 2 t GF2†< 7 0 2^I Q 8 . * 8 * . 7=< <2=1 ] 7 1 = I A 577Z5^AA  7 * d inv k 0Q2 A 0[7 ε 1,i , I J I. .. k.  gε. ε i ≤ k. λ (M, g ) → 0 k<n. M. I^0[7 . 2 01+2 d 0Q5-1$G. 02:S1S14KLOE7?02H214767Y5J>CPQ0/7 79F2OE: 8MMHU ;14F[79ƒ,@\8M79H 8rT7 : >A@ 0/Œ :4OEOEKG798;0™:S: :4OE7?>L‰O 5SV KG:SKL7IF[@\>Lƒ,7d:SKLOEOE0™8r8 79:S1S14HUƒ,7 K ™79 ;: Z9UFM147=Bt7 OEH21S>L:47TOE798Mƒ,8MD=OE79HUƒY8;F : R214‹D 7 @\@E1S5 H2KL79O 7T1 T0/@ >A“79@ :S76KGDXM:476R2>A5@ 147=V‹7636@\5SOE>L79:dƒ,F2H2143 75Ž HUKGƒ,798M5SKLOE8 239:SFMHUKL3I0[@\1 Œ OE>GR/OEKL5 79: Œ OEF21S:4OEKL5 HM@\8M5 Œ Œ ‘‹‘   >L7 D @E5 H P F28 7 OE8MHU14@=79ƒ,H23979ƒ,8r: OE8M0[54@\:S1 1W@\„M:SKLOEOEƒ,8 OEH2:S7=„M39‰ :SKL76‡5 ƒ,H27T79: >A@798 M3 R2;14KL7 H279 8M5SKGD6:S7dF[>L@\7 :SKLBCOE@\8 KG:fH2 ;7TFM7d>GKGHMƒ‡@\KG8M:475 Tqp.  .

(27) . Qm ‰q~% '. . . T. . . . . 1. Tqp. Qm ‰qt%. y. '. ‚. . . B’). . . . :. . . ™. f. . %. .

(28) E@ HUKA@\R[@\:SKL rFM7 Ÿ [KG> 8oP @ 0[@E5JH27d0/79:SKG:4765 E@\>L79F2145 0214OE0214765 >L7I5S0/76D9:S147IH27 8M:4797N8MD6H OE8M“54793614 ;5 FM79398M14D6O 7 [@\KGƒ‡>LOEƒ,145 36>LHU7 KA@\V :4147 OEF2HU0™F 7 14365SF276>G5S:W:@\:d82KG‰ >G0™OEX:476795S8r:f: rFM7 5SK[:4OEF2: P \ @ 2 F S : 4 1 I 7 [ 0 \ @ S 1 : [ 6 D d 7 S : M „ 6 3 E O 4 1 9 Z , ƒ Y 7 / 0 9 7 S 1 , ƒ 9 7 J : 2 H Y 7 2 H E O 2 8 M 8 9 7 J 1 2 F M 8 X 7 6 D E O M 8 U H G K S : L K E O T 8 M 8 6 3 6 D 6 7 4 5 W 5 \ @ G K 4 1 X 7 9 7 : Ž 5SF z5W@\8r:47 5SF21 0™OEF21 >CPQ7 UKL5S:4798MD67 H27j0/79:SKG:4765 ‹@\>L79F2145 0214OE02147650™OEF21>L765 BtOE1Sƒ,765 =. @q[p. Qm ‰. . qC%[‹. ‚. !. '. . &. H. ∆1inv. Q. . G. . . . .   mWqM%. xl"nmW". . B. 'R(+*. O. 0I. . *.. 2^IU ^ , A. ƒ. U. .. If2^I

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(38)

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(61) . F. . . iT. . . Λ∗ (n∗ )  .  . ∇af f . . . Ω∗ (N, E) . T HM. E. T. g Fb. hE. . . Ω(N, E) ,→ Ω(M ) Λ∗ (n∗ )  A0[1]. . A0[0]. E. . . . .  . . . Ω∗ (N, E) ,→ Ω∗ (M ). . W. A0[2] . T H M P . . _. E.

(62) Ž @ S 5 S : S 1 M F 9 D S : 2 F 4 1 , 7 2 H 7 M 2 R 4 1 z 3 @ , M 8 , 7 S 1 L K 9 7 z ƒ \ @ 2 8 2 8 L K 9 7  8 G K M 8 U H 2 F G K Y : 2 H E O M 8 , D < A > @ B E O L K d 5 2 F M 8 , 7 S 5 2 F ™ 0 9 7 1 DF26OE8 82>A8M@\7 02UK>AL@EOED98 KL79879: F28 025SF214O%1 HUF2KG:J54D @\>A@\KG147I8 5SF28M1OE:W@\8r2: 79: 0™791Sƒ,79>L: 7I0[5S@\0™1J76D9D6:SOE148M7d54HU36 ;F FM>A79@\8r02: >A@EH2D97IKL79H28 3 M82KG1 Ž E O S : : , ƒ E O r 8 S : 4 1 d 7 ; M F 7 M H \ @ M 8 5 6 D 9 7 S : 4 : 7 S 5 G K S : [ F \ @ S : L K E O 8 % L > 6 7 f 5 / 0 9 7 S : G K 4 : 6 7 5 ‹ \ @ L > 9 7 2 F 4 1 } 5 2 0 4 1 E O 2 0 4 1 6 7 5 U H F A > \ @ 2 0 A > @ D9KL798 54F21 54OE8;:JD679>G>L7XH27  O ‘‹ R U:S„M36OE14Z9ƒ,7z‰ Ÿ . . . . Ω(N, E) . ∆E.   . M. "nmWop. qr%n'/% s. .. R. O. R. .. g. M. .. 2^I. N. . . ∆pE . ˜. . . . , , . ,L* 5 v1+2R7 2=<5-1n1 2:1I^0[7 2 I ./, . . 2=1QI 2=0[7IlGF2h? 0[7f]:07E2^Il7E2^IXA2Z? 3W2=<2=1 I^0[7 2 I 07 = M 2 F 2^I b A 8:. , d O * 8 , , ƒ:. N < 7 0Q2 5t] 7=1+2I^0!A 7 2:07E24GF2^IG 5-< 7E2^IGF2^I ]=7E2^I@2 Fb  * * * . O N O* ,#O , . * . . I 2Z? 1$G2ga 7=<2ba 1$G5<2:1 5 2lGF2=I ]=7Z2=Ig5 7^I 2ZD I 2RGF2^Il? 1QI 5 1 2^I , 8 . d . O#O , d d 2 0 1+2lG AQ2=1+GF2=1 02hG2 2 2=I 0Q2I F ∞ * .X* .  A 07 0 jhk. 'R(n,. σ(∆pE ). .  ∆pE. ∆M. . E. 0[1. *. ]=7. 8. N. M. diam(F ). c. dim(M ). kR k. p ≤ dim(M ). F. 1. 1 *. *.. 07. .. Π a  a0 O* diam(F ) ≤ a0 5. 2 O. O. 5 5 2. .. 7^I. σ(∆pM ) ∩ [0, a · diam(F )−2 − c(kRM k∞ + kΠk2∞ + kT k2∞ )[ =. σ(∆pE ) ∩ [0, a · diam(F )−2 − c(kRM k∞ + kΠk2∞ + kT k2∞ )[.. 8 _ 6 7 S 5 ? : 2 H E O M 8 z D W 1 \ @ , ƒ 9 7 M 8 3 < A > @ 4 1 6 7 4 D M „ 9 7 4 1 W D M „ 7 H P 2 F  8 E O ™ 0 9 3 W 1 \ @ 4 : 9 7 2 F ? 1 G > G K ‡ ƒ G K 4 : 7 S 5 2 F 1 ƒ,765S839:I:SD68M1SKLOE7 ;8MFM0/5S7:W79@\F28;: :4D67zOE@\8MFŠ5S8 KLH2D67 39OE14F2797914:1N5X54H2390[7?@\8M>C14PQ7Y739ƒ,H2OE39798M0™8;HU:79148M>A79@XH ƒ,>G0[79KGƒ‡8r@E:5 KG:4H27d797=: H2>A7 @‡>Aƒ,@YH25439F23:SV1S0/KL39 ;798M14FMZ9D67=14OE7,825SF28MH217 OEUK8MLD‡OE8 5W79@ : >GH2KGƒ‡OE798M:fKGD=:4H27‡7 79>G8M>A>L@77 0™H27d791SD6ƒ,OE798M:J5SKL0[H2@E39145N79H21 7Y>CPQH2798M3 M547982ƒ=KG1JR2F2>L7 8 OE0™391W@\:479F21 %@EOH SOEKG8r: D676OE5S:jƒ‡>Cƒ,PQ76547 0[@ED67 H2765 5SP KLF2H20™367Y7914H2D6OE7 828MOE7 :SUK: LOE768M5S5: 79H2H2:d77 H24H2@\7=7 F V ;V 14FM7z3OE54:S‰ F2KL791 8r5SF2:4791 1I ;D6F279Kq:d79:0279148M3654547979ƒY1 “>CR279PQ76>L8;7?5S:Y0[0[@E>A@\@ D61 7 V >L1W7H2@E76VHU5 14F[OE02@\F2:S140™KLO;OE7HU8 F2KG:4‚ 5>jH2OE5476R2D 5 @\:SKL>A79@\8rKG14:‡765 @\>L79OEŽ FM14:S5XD91W>G@\KL>L8MHU7‡5KL7914B OE368M5S1S5 F2ƒz>G5S:W@\F2@\:S1 :?KLOEH28M75 D6OEƒ‡0[@ED9KG:43=5SF2K E@\8;:  O ‘‹ R M:S„M36OE14Z9ƒ,7 “Ÿ . . Ω(N, E) A0. . hE. . . . A0. M. . . . E. Z.  . . . GE. GL(E). . . . . ,X8:.;8 , , N 8 01>2Ž3657 7 = 2 <5-1n1 2:11+2 D   . O , . ./, * . 2 JP– 2ED I 201+2 AQ5-7 24? <BA 5F? 2 E E E , . ,X8:.;8 , , , ,9* . O O d d n 02ŠI 2^I 0[1+2Š3W5-7 7 2=<5-11 2=n 1 1+2GF2G <2:1QI 1 2 2 0 2 Q . O* O O \8 d , O * O 2 5 7I 5 ? W 5 II 2 G 0 3W5 2:1$ ? 2‘G-0 ? 0!A 2 ∞ GH , ,/. ./, . 1$G-0 A 5-7 5=AA 5-7 2:1 J E. jhk. qr%n'/%[ 'R(+*-,/.. . E. HE. . "nmWop. (A0 )∗ . (SE × HE )   SE. E. c. hE. W. (N, hN ). n > dim(N ) K(n, a, ε) ⊂ (S × H )/G n. a>0. ε>0 . O#O 2 2 (M , g) kRM k ≤ a d (M, N ) ≤ ε  0 (A , h ) g K. † q W : \ @ r 8 J : 2 H E O 2 8 M 8 6 3 d 7 2 F M 8 d 7 S 5 2 F G K 4 : d 7 2 H d 7 , ƒ 9 3 S : S 1 L K ; M F 6 7 5 ; 2 F K 7 ™ E O M 8 U H 4 1 7 S 5 2 F 1 W 5 = @ [ R E @ 4 5 7 F2OED6OE88M8M7 5S0™:S54791SOEF2F2FM:KG514Ž7J7 54U:F2F2SKG8z:41W7@\>AKG@\14 ;027ŠF2>A@EKfH2D9D67 KL79OE88>A@ “>G79KG54ƒ‡1 F2V KGKG7 :4:47 7 “79145?“F2H28MOE7 8;54:}F2>L0/7N7954140/D676OEH D982:SPQ398M14>L77 39UKƒ,76L5SOE79:q8_8;>A:4@d>G5TKG>Gƒ‡KGH2ƒ‡KG7 :4KG7 :47 ;HUF2FzKf0/5S0™79761Sƒ,D9:S79147J:‡HUH2F 7 >A@\02>A@ED9 KL798  5SF21 V , @ S 5 2 F G K 4 : 7 M 8 X 7 6 D E O 8 “ 9 7 1 = 7 [ 0 E @ 5 M 8 6 3 6 D 6 7 4 5 5 \ @ G K 4 1 9 7 , ƒ 9 7 r 8 : 8 0™79F2F28M:X7d0[Bt@\OE1X8MD97 :S27KL9OEƒ‡8 02 r>LF27zK @\O‹FŠ54D90[KG>G@\>L7X1W@ 79V 8r1W:S@\14027 „M7 79‰ : G‰,021479@ 8MHU021414779ƒ‡KLZ9147 ‹@\>L79F21J0214OE7902:X14D47 „MEOE@KL5SKGO‹1=54D90/KGOE>G>LF27911 79H28;7 :S147 79: 5W@\8M5D6OE8 “791 V 791 PQOE0/391W@\:479F21 8M7 D6OE8 “791 V 7NH2OE8MD 0[@E5 \82K2>A@ D9>A@E54547    OEF21J>CPQ39:SFMH27IHUF 540/76D9:S147d5SF21 >L765jBtOE8MD9:SKLOE8M5 2OE8 0™79F2: 547 14765 Ž :479KG8MHU147J@\F MR2143 ‹ ;F2KM765S:F28 MR2143 :S1SK ;KA@\>a798,HU14OEKG:47 143679>G>L7 5SF21 Œ 790™798MHM@\8r: . . Tqp. M. Qm ‰q. [(A0i , hE,i )]i∈N. 4. %$'#%[“'‡. γ. F. Qm ‰q%n'/%(. . E. . 1. . 2. α = 0 β = 1. . . ∆E. '. !. F. [(A0i , hE,i )] . 1 t (A0t , hE,t ) .  M (SE × HE )/GE. ∆∞ . . Tqp. . (gi )i∈N. l. . . !. 0. . . . N. .

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