Spectral applications of metric surgeries

Transcription

1 Spectral applications of metric surgeries Pierre Jammes Neuchâtel, june 2013

2 Introduction and motivations Examples of applications of metric surgeries Let (M n, g) be a closed riemannian manifold, and λ 1 (g) the rst positive eigenvalue of the Laplacian on M.

3 Introduction and motivations Examples of applications of metric surgeries Let (M n, g) be a closed riemannian manifold, and λ 1 (g) the rst positive eigenvalue of the Laplacian on M. The conformal class of g is the set of metrics dened by [g] = { h 2 g, h C (M), h > 0 }

4 Introduction and motivations Examples of applications of metric surgeries Let (M n, g) be a closed riemannian manifold, and λ 1 (g) the rst positive eigenvalue of the Laplacian on M. The conformal class of g is the set of metrics dened by [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) < + g [g] Vol(M)=1

5 Introduction and motivations Examples of applications of metric surgeries Let (M n, g) be a closed riemannian manifold, and λ 1 (g) the rst positive eigenvalue of the Laplacian on M. The conformal class of g is the set of metrics dened by [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) < + g [g] Vol(M)=1 ν(m) = inf g sup λ 1 (M, g) g [g] Vol(M)=1

6 Introduction and motivations Examples of applications of metric surgeries Let (M n, g) be a closed riemannian manifold, and λ 1 (g) the rst positive eigenvalue of the Laplacian on M. The conformal class of g is the set of metrics dened by [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) < + g [g] Vol(M)=1 ν(m) = inf g sup λ 1 (M, g) g [g] Vol(M)=1 Theorem ν(m n ) is uniformly bounded on manifold of dimension n.

7 Introduction and motivations Let M n be a closed spin manifold. The index theorem gives a lower bound on the dimension of the kernel of the Dirac operator. Dim KerD i(m)

8 Introduction and motivations Let M n be a closed spin manifold. The index theorem gives a lower bound on the dimension of the kernel of the Dirac operator. Dim KerD i(m) Theorem (Bär, Dahl, Ammann, Humbert) This inequality is an equality for a generic set of metrics. In particular, The Dirac operator is generically invertible if n = 3, 5, 6, 7 mod 8.

9 Introduction and motivations Proposition If the scalar curvature of (M n, g) and (M n, g ) is positive (n 3), then M#M carries a metric of positive scalar curvature.

10 Introduction and motivations Proposition If the scalar curvature of (M n, g) and (M n, g ) is positive (n 3), then M#M carries a metric of positive scalar curvature. Theorem (Gromov, Lawson, '80) Every closed simply-connected non spin manifold of dimension 5 carries a metric of positive scalar curvature.

11 Surgeries I : connected sum

12 Surgeries I : connected sum Remove a small ball on each manifold. Attach a handle S n 1 [0, 1].

13 Surgeries I : connected sum Remove a small ball on each manifold. Attach a handle S n 1 [0, 1].

14 Surgeries II : denition Let M n be a closed manifold, and S k M n an embedded sphere whose tubular neighborhood is dieomorphic to S k B n k. (S k B n k ) = S k S n k 1

15 Surgeries II : denition Let M n be a closed manifold, and S k M n an embedded sphere whose tubular neighborhood is dieomorphic to S k B n k. (S k B n k ) = S k S n k 1 = (B k+1 S n k 1 ).

16 Surgeries II : denition Let M n be a closed manifold, and S k M n an embedded sphere whose tubular neighborhood is dieomorphic to S k B n k. (S k B n k ) = S k S n k 1 = (B k+1 S n k 1 ). Denition The manifold obtained from M by a surgery along S k (k dimensional surgery) is M\(S k B n k ) S k S n k 1 (B k+1 S n k 1 ) n k is the codimension of the surgery.

17 Surgeries II : denition & examples Example I The connected sum is a surgery along a sphere S 0.

18 Surgeries II : denition & examples Example I The connected sum is a surgery along a sphere S 0. Example II 1-codimensional surgery

19 Surgeries II : denition & examples Example I The connected sum is a surgery along a sphere S 0. Example II 1-codimensional surgery Example III The sphere S 3 is the union of two copies of S 1 D 2. A surgery along a trivial knot in S 3 produces the manifold S 1 S 2.

20 Surgeries III : applications Theorem (Gromov, Lawson, '80) Let M n be a closed riemannian manifold with positive scalar curvature. If M is obtained from M by a surgery of codimension 3, then M carries a metric of positive scalar curvature.

21 Surgeries III : applications Theorem (Gromov, Lawson, '80) Let M n be a closed riemannian manifold with positive scalar curvature. If M is obtained from M by a surgery of codimension 3, then M carries a metric of positive scalar curvature.

22 Surgeries III : applications Theorem (Gromov, Lawson, '80) Let M n be a closed riemannian manifold with positive scalar curvature. If M is obtained from M by a surgery of codimension 3, then M carries a metric of positive scalar curvature. Theorem (Bär, Dahl, '02) If the Dirac operator D is invertible on (M, g), there is a metric g on M such thaht D g is invertible.

23 Surgeries III : applications Theorem (Gromov, Lawson, '80) Let M n be a closed riemannian manifold with positive scalar curvature. If M is obtained from M by a surgery of codimension 3, then M carries a metric of positive scalar curvature. Theorem (Bär, Dahl, '02) If the Dirac operator D is invertible on (M, g), there is a metric g on M such thaht D g is invertible. Theorem (Ammann, Dahl, Humbert, '09) If D is invertible on M and M is obtained from M by a surgery of codimension 2, then D is invertible on (M, g ).

24 Surgeries IV: cancellation

25 Surgeries IV: cancellation A k-dimensional surgery is cancelled by a (n k)-surgery. S k B n k B k+1 S n k 1

26 Surgeries IV: cancellation A k-dimensional surgery is cancelled by a (n k)-surgery. S k B n k B k+1 S n k 1

27 Surgeries IV: cancellation A k-dimensional surgery is cancelled by a (n k)-surgery. S k B n k B k+1 S n k 1 A k-surgery is cancelled by a surgery along a (k + 1)-sphere that intersects transversally the belt sphere of the k surgery in one point (Smale's cancellation lemma).

28 Surgeries IV: cancellation Cancellation is a method to avoid 1-codimensional surgeries. It fails in two cases :

29 Surgeries IV: cancellation Cancellation is a method to avoid 1-codimensional surgeries. It fails in two cases : connected sum

30 Surgeries IV: cancellation Cancellation is a method to avoid 1-codimensional surgeries. It fails in two cases : connected sum non oriented handle

31 Cobordism I : denition Denition Let M and N be two closed n-dimensional manifolds. A cobordism between M and N is a compact n + 1-manifold W whose boundary is M N. M are N are cobordant if such a cobordim exists.

32 Cobordism I : denition Denition Let M and N be two closed n-dimensional manifolds. A cobordism between M and N is a compact n + 1-manifold W whose boundary is M N. M are N are cobordant if such a cobordim exists. Examples S 1 S 1 is cobordant to S 1 T 2 is cobordant to S 2

33 Cobordism I : denition Remark Cobordism is a equivalence relation.

34 Cobordism I : denition Remark Cobordism is a equivalence relation. Questions 1. What can we say about the quotient set? 2. What can we say about a given equivalence class?

35 Cobordism II : cobordism ring Let Ω n be the set of cobordism classes of n-dimensional manifolds.

36 Cobordism II : cobordism ring Let Ω n be the set of cobordism classes of n-dimensional manifolds. If M is the boundary of W n+1, we write [M] = [ ]. Remark : we also have [M] = [S n ].

37 Cobordism II : cobordism ring Let Ω n be the set of cobordism classes of n-dimensional manifolds. If M is the boundary of W n+1, we write [M] = [ ]. Remark : we also have [M] = [S n ]. Ω n is an abelian group for the disjoint union [M] + [N] = [M N]

38 Cobordism II : cobordism ring Let Ω n be the set of cobordism classes of n-dimensional manifolds. If M is the boundary of W n+1, we write [M] = [ ]. Remark : we also have [M] = [S n ]. Ω n is an abelian group for the disjoint union [M] + [N] = [M N] The identity element of this group is [ ] [M] + [M] = [ ] M M

39 Cobordism II : cobordism ring Ω = n Ω n is a ring: [M] [N] = [M N].

40 Cobordism II : cobordism ring Ω = n Ω n is a ring: [M] [N] = [M N]. Suppose that [M] = [M ], [N] = [N ], W 1 = M M, W 2 = N N. (W 1 N) = (M N) (M N) [M N] = [M N] (M W 2 ) = (M N) (M N ) [M N] = [M N ]

41 Cobordism II : cobordism ring Ω = n Ω n is a ring: [M] [N] = [M N]. Suppose that [M] = [M ], [N] = [N ], W 1 = M M, W 2 = N N. (W 1 N) = (M N) (M N) [M N] = [M N] (M W 2 ) = (M N) (M N ) [M N] = [M N ] Proposition The mod 2 Euler characteristic χ(m) Z/2Z is a cobordism invariant.

42 Cobordism II : cobordism ring Ω = n Ω n is a ring: [M] [N] = [M N]. Suppose that [M] = [M ], [N] = [N ], W 1 = M M, W 2 = N N. (W 1 N) = (M N) (M N) [M N] = [M N] (M W 2 ) = (M N) (M N ) [M N] = [M N ] Proposition The mod 2 Euler characteristic χ(m) Z/2Z is a cobordism invariant. Proof : let W 2n+1 be a cobordism between M 2n and N 2n. We obtain a closed manifold W by gluing two copies of W along their boundaries. χ(w ) = 2χ(W ) χ( W ) χ( W ) = χ(m) + χ(n) = 0 mod 2.

43 Cobordism II : cobordism ring Theorem (R. Thom, 1954) For each n, Ω n is nite.

44 Cobordism II : cobordism ring Theorem (R. Thom, 1954) For each n, Ω n is nite. As a ring, Ω = Z/2[X i, i 1, i 2 j 1]. If i is even, X i is the class of the real projective space P i (R).

45 Cobordism II : cobordism ring Theorem (R. Thom, 1954) For each n, Ω n is nite. As a ring, Ω = Z/2[X i, i 1, i 2 j 1]. If i is even, X i is the class of the real projective space P i (R). (Dold, 1956) If i = 2 r (2s + 1) 1 is odd, X i is the class of P(2 r 1, s2 r ) where P(k, l) = (S k P l (C))/(x, z) ( x, z).

46 Cobordism II : cobordism ring Theorem (R. Thom, 1954) For each n, Ω n is nite. As a ring, Ω = Z/2[X i, i 1, i 2 j 1]. If i is even, X i is the class of the real projective space P i (R). (Dold, 1956) If i = 2 r (2s + 1) 1 is odd, X i is the class of P(2 r 1, s2 r ) where P(k, l) = (S k P l (C))/(x, z) ( x, z). Ω 1 0 Ω 2 Z/2 P 2 (R) Ω 3 0 Ω 4 (Z/2) 2 P 2 (R) P 2 (R), P 4 (R) Ω 5 Z/2 P(1, 2)

47 Cobordism III : cobordism & surgeries Let W n+1 = M [0, 1] be a trivial cobordism. If S k 1 M {1} is an embedded sphere with trivial normal bundle, we obtain a new cobordism W by attaching a handle B k B n+1 k along S k 1 : M W

48 Cobordism III : cobordism & surgeries Let W n+1 = M [0, 1] be a trivial cobordism. If S k 1 M {1} is an embedded sphere with trivial normal bundle, we obtain a new cobordism W by attaching a handle B k B n+1 k along S k 1 : M W W is called an elementary cobordism of index k. The new boundary is obtained from M by a surgery along S k 1.

49 Cobordism III : cobordism & surgeries Consequences [M] + [N] = [M#N]. If M is obtained from M by a nite number of surgeries, then [M] = [M ].

50 Cobordism III : cobordism & surgeries Consequences [M] + [N] = [M#N]. If M is obtained from M by a nite number of surgeries, then [M] = [M ]. Theorem (Smale, Wallace) If W is a cobordism, then W = W 1 W 2... W p, where each W i is an elementary cobordism. Moreover, we can assume that the indices of these cobordisms are increasing with i.

51 Cobordism III : cobordism & surgeries Proof Let W be a cobordism between M and N, and f : W [0, 1] a Morse function such that f 1 (0) = M and f 1 (1) = N. f Morse function all critical points of f are non degererates.

52 Cobordism III : cobordism & surgeries Proof Let W be a cobordism between M and N, and f : W [0, 1] a Morse function such that f 1 (0) = M and f 1 (1) = N. f Morse function all critical points of f are non degererates. Near a critical point, f (x) = f (0) + x x 2 k x 2 k+1 x 2 n.

53 Cobordism III : cobordism & surgeries Proof Let W be a cobordism between M and N, and f : W [0, 1] a Morse function such that f 1 (0) = M and f 1 (1) = N. f Morse function all critical points of f are non degererates. Near a critical point, f (x) = f (0) + x x 2 k x 2 k+1 x 2 n. W compact f has nitely many critical points.

54 Cobordism III : cobordism & surgeries Proof Let W be a cobordism between M and N, and f : W [0, 1] a Morse function such that f 1 (0) = M and f 1 (1) = N. f Morse function all critical points of f are non degererates. Near a critical point, f (x) = f (0) + x x 2 k x 2 k+1 x 2 n. W compact f has nitely many critical points. We may assume that the critical values of f are distincts, and 0, 1.

55 Cobordism III : cobordism & surgeries Proof Let W be a cobordism between M and N, and f : W [0, 1] a Morse function such that f 1 (0) = M and f 1 (1) = N. f Morse function all critical points of f are non degererates. Near a critical point, f (x) = f (0) + x x 2 k x 2 k+1 x 2 n. W compact f has nitely many critical points. We may assume that the critical values of f are distincts, and 0, 1. If there is no critical value in [a, b], then f 1 ([a, b]) is a trivial cobordism.

56 Cobordism III : cobordism & surgeries

57 Cobordism III : cobordism & surgeries B k B n+1 k each critical point corresponds to an elementary cobordism.

58 Cobordism IV : oriented cobordism All manifolds are supposed orientable and oriented. If M is an oriented manifold, M will denote the same manifold with the opposite orientation. If W is an oriented manifold with boundary, the orientation on W induces an orientation on M.

59 Cobordism IV : oriented cobordism All manifolds are supposed orientable and oriented. If M is an oriented manifold, M will denote the same manifold with the opposite orientation. If W is an oriented manifold with boundary, the orientation on W induces an orientation on M. Denition Two oriented manifolds M and N are (oriented-)cobordant if there is a cobordism W such that W = M ( N).

60 Cobordism IV : oriented cobordism All manifolds are supposed orientable and oriented. If M is an oriented manifold, M will denote the same manifold with the opposite orientation. If W is an oriented manifold with boundary, the orientation on W induces an orientation on M. Denition Two oriented manifolds M and N are (oriented-)cobordant if there is a cobordism W such that W = M ( N). Remark For a trivial cobordism M [0, 1], the orientation induced on M {0} and M {1} are opposite. [M] = [ M]

61 Cobordism IV : oriented cobordism Let Ω SO be the oriented cobordism ring. Theorem (R. Thom, 1954) For each n, Ω SO n Ω SO is nitely generated. Q = Q[Y 4i ], i 1 with Y 4i = [P 2i (C)]. dimension group Z Z/2 0 0 Z 2

62 Handle decomposition Every compact connected manifold M admits a handle decomposition, i. e. M is obtained by attaching handle to a ball (0-handle).

63 Handle decomposition Every compact connected manifold M admits a handle decomposition, i. e. M is obtained by attaching handle to a ball (0-handle). If M is closed, it admits a handle decomposition with only two balls (one 0-handle and one n-handle).

64 Handle decomposition Every compact connected manifold M admits a handle decomposition, i. e. M is obtained by attaching handle to a ball (0-handle). If M is closed, it admits a handle decomposition with only two balls (one 0-handle and one n-handle). If M has a boundary, it admits a handle decomposition with one ball (0-handle).

65 Handle decomposition Every compact connected manifold M admits a handle decomposition, i. e. M is obtained by attaching handle to a ball (0-handle). If M is closed, it admits a handle decomposition with only two balls (one 0-handle and one n-handle). If M has a boundary, it admits a handle decomposition with one ball (0-handle). Proof : two 0-handle + one 1-handle = one 0-handle

66 Handle decomposition Exercise Every compact surface with boundary admits a at metric. 2πχ(M) = K da + k dl M M

67 Conformal bounds for λ 1 Let (M, g) be a closed connected riemannian manifold. : C (M) C (M) 0 = λ 0 (M, g) < λ 1 (M, g) λ 2 (M, g)...

68 Conformal bounds for λ 1 Let (M, g) be a closed connected riemannian manifold. : C (M) C (M) 0 = λ 0 (M, g) < λ 1 (M, g) λ 2 (M, g)... λ 1 = inf M f =0 f 2 M M f 2

69 Conformal bounds for λ 1 Let (M, g) be a closed connected riemannian manifold. : C (M) C (M) 0 = λ 0 (M, g) < λ 1 (M, g) λ 2 (M, g)... λ 1 = inf M f =0 f 2 M M f 2 sup Vol(M,g)=1 λ 1 (M, g) = + (Colbois & Dodziuk)

70 Conformal bounds for λ 1 Let (M, g) be a closed connected riemannian manifold. : C (M) C (M) 0 = λ 0 (M, g) < λ 1 (M, g) λ 2 (M, g)... λ 1 = inf M f =0 f 2 M M f 2 sup Vol(M,g)=1 λ 1 (M, g) = + (Colbois & Dodziuk)

71 Conformal bounds for λ 1 [g] = { h 2 g, h C (M), h > 0 }

72 Conformal bounds for λ 1 [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) =? g [g] Vol(M)=1

73 Conformal bounds for λ 1 [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) =? g [g] Vol(M)=1 If ϕ : M n S k is a conformal immersion, we dene V c (ϕ) = sup γ Gk Vol(γ ϕ(m)), where G k is the group of conformal dieomorphism of S k (Möbius group).

74 Conformal bounds for λ 1 [g] = { h 2 g, h C (M), h > 0 } sup λ 1 (M, g) =? g [g] Vol(M)=1 If ϕ : M n S k is a conformal immersion, we dene V c (ϕ) = sup γ Gk Vol(γ ϕ(m)), where G k is the group of conformal dieomorphism of S k (Möbius group). Denition The conformal volume of M is the inmum of V c (ϕ) on all conformal immersion ϕ S k, for all k. V c (M, [g]) = inf ϕ V c(ϕ)

75 Conformal bounds for λ 1 Theorem (Li & Yau, El Sou & Ilias) λ 1 (M, g)vol(m) 2/n nv c (M, [g]) 2/n equality (M, g) admits a homothetic minimal immersion in a sphere.

76 Conformal bounds for λ 1 Theorem (Li & Yau, El Sou & Ilias) λ 1 (M, g)vol(m) 2/n nv c (M, [g]) 2/n equality (M, g) admits a homothetic minimal immersion in a sphere. Examples of manifold that admits a minimal immersion in the sphere : S n, P n (R), P n (C), P n (H),...

77 Conformal bounds for λ 1 S k R k+1 The coordinates x i of R k+1 satises i x 2 i = 1 on S k i dx 2 i = g can on R k+1 and S k.

78 Conformal bounds for λ 1 S k R k+1 The coordinates x i of R k+1 satises i x 2 i = 1 on S k i dx 2 i = g can on R k+1 and S k. Let ϕ : (M, g) S k be a conformal immersion (assume that Vol(M, g) = 1). The Idea is to use the coordinates as test functions. γ G k such that M γ ϕ idv g = 0 for all i.

79 Conformal bounds for λ 1 S k R k+1 The coordinates x i of R k+1 satises i x 2 i = 1 on S k i dx 2 i = g can on R k+1 and S k. Let ϕ : (M, g) S k be a conformal immersion (assume that Vol(M, g) = 1). The Idea is to use the coordinates as test functions. γ G k such that γ ϕ M idv g = 0 for all i. λ 1 ϕ 2 i dv g ϕ i 2 g dv g M M

80 Conformal bounds for λ 1 S k R k+1 The coordinates x i of R k+1 satises i x 2 i = 1 on S k i dx 2 i = g can on R k+1 and S k. Let ϕ : (M, g) S k be a conformal immersion (assume that Vol(M, g) = 1). The Idea is to use the coordinates as test functions. γ G k such that γ ϕ M idv g = 0 for all i. λ 1 ϕ 2 i dv g ϕ i 2 g dv g M M λ 1 ϕ 2 i dv g ϕ i 2 dv g M i M i

81 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M i

82 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M λ 1 i ( n/2 ϕ i g) 2 dv g M i 2/n

83 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M λ 1 i ( n/2 ϕ i g) 2 dv g M i Up to a rotation of S k, we have ϕ i 2 g = 0 for i > n. 2/n

84 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M λ 1 i ( n/2 ϕ i g) 2 dv g M i Up to a rotation of S k, we have ϕ i 2 g = 0 for i > n. For i n, ϕ i g is exactly the conformal factor between g = g M and g S k 2/n

85 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M λ 1 i ( n/2 ϕ i g) 2 dv g M i Up to a rotation of S k, we have ϕ i 2 g = 0 for i > n. For i n, ϕ i g is exactly the conformal factor between 2/n g = g M and g S k ( i ϕ i 2 g) n/2 dv g = n n/2 dv gs k

86 Conformal bounds for λ 1 λ 1 ϕ i 2 g dv g M λ 1 i ( n/2 ϕ i g) 2 dv g M i Up to a rotation of S k, we have ϕ i 2 g = 0 for i > n. For i n, ϕ i g is exactly the conformal factor between g = g M and g S k ( ϕ i i g) 2 n/2 dv g = n n/2 dv gs ( ) k 2/n λ 1 n = nvol(ϕ(m)) 2/n nv c (ϕ) 2/n dv gs k M 2/n

87 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n.

88 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n. ν(m n ) ν(s n ).

89 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n. ν(m n ) ν(s n ). ν(s 2 ) = 8π, ν(p 2 (R)) = 12π.

90 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n. ν(m n ) ν(s n ). ν(s 2 ) = 8π, ν(p 2 (R)) = 12π. ν(t 2 ) = ν(k 2 ) = 8π (Girouard, '09).

91 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n. ν(m n ) ν(s n ). ν(s 2 ) = 8π, ν(p 2 (R)) = 12π. ν(t 2 ) = ν(k 2 ) = 8π (Girouard, '09). Denition The Möbius volume of M is dened by V M (M) = inf V c(m, [g]) = inf sup Vol(γ ϕ(m)) g ϕ:m S k γ G k

92 Conformal bounds for λ 1 Denition The Friedlander-Nadirashvili invariant of a closed manifold M is dened by ν(m) = inf g sup g [g] λ 1 (M, g)vol(m) 2/n. ν(m n ) ν(s n ). ν(s 2 ) = 8π, ν(p 2 (R)) = 12π. ν(t 2 ) = ν(k 2 ) = 8π (Girouard, '09). Denition The Möbius volume of M is dened by V M (M) = inf V c(m, [g]) = inf sup Vol(γ ϕ(m)) g ϕ:m S k γ G k ν(m n ) nv M (M) 2/n

93 Conformal bounds for λ 1 Theorem There is a constant c(n) > 0 such that for all closed manifold M n, V M (M) c.

94 Conformal bounds for λ 1 Theorem There is a constant c(n) > 0 such that for all closed manifold M n, V M (M) c. Corollary ν(m n ) is uniformly bounded for a given n.

95 Conformal bounds for λ 1 Theorem There is a constant c(n) > 0 such that for all closed manifold M n, V M (M) c. Corollary ν(m n ) is uniformly bounded for a given n. Principle of the proof : to study the behavior of the Möbius volume when performing surgeries.

96 Conformal bounds for λ 1 Proof for n = 2 Lemma Let M be a compact surface. If M is obtained by adding a handle to M, then V M (M ) sup{v M (M), c} where c doesn't depend on M.

97 Conformal bounds for λ 1 Proof for n = 2 Lemma Let M be a compact surface. If M is obtained by adding a handle to M, then V M (M ) sup{v M (M), c} where c doesn't depend on M. Let ϕ : M S k such that V M (M) V (ϕ) V M (M) + ε and V c (ϕ) ε Vol(ϕ(M)) V c (ϕ).

98 Conformal bounds for λ 1 Proof for n = 2 Lemma Let M be a compact surface. If M is obtained by adding a handle to M, then V M (M ) sup{v M (M), c} where c doesn't depend on M. Let ϕ : M S k such that V M (M) V (ϕ) V M (M) + ε and V c (ϕ) ε Vol(ϕ(M)) V c (ϕ). Stereographic projection : S k R k { } 4 g S k = (1 + x 2 ) g 2 eucl

99 Conformal bounds for λ 1 By attaching a thin handle to ϕ(m), we obtain an immersion ϕ : M S k such that Vol(ϕ (M )) Vol(ϕ(M)) (a) (b)

100 Conformal bounds for λ 1 By attaching a thin handle to ϕ(m), we obtain an immersion ϕ : M S k such that Vol(ϕ (M )) Vol(ϕ(M)) (a) (b) For γ G k, what is Vol(γ ϕ (M ))?

101 Conformal bounds for λ 1 By attaching a thin handle to ϕ(m), we obtain an immersion ϕ : M S k such that Vol(ϕ (M )) Vol(ϕ(M)) (a) (b) For γ G k, what is Vol(γ ϕ (M ))? Up to an isometry of S k, γ is equivalent to a translation or a homothety.

102 Conformal bounds for λ 1 By attaching a thin handle to ϕ(m), we obtain an immersion ϕ : M S k such that Vol(ϕ (M )) Vol(ϕ(M)) (a) (b) For γ G k, what is Vol(γ ϕ (M ))? Up to an isometry of S k, γ is equivalent to a translation or a homothety. If γ is a translation, Vol(γ ϕ (M )) is close to or smaller than Vol(ϕ(M)).

103 Conformal bounds for λ 1 By attaching a thin handle to ϕ(m), we obtain an immersion ϕ : M S k such that Vol(ϕ (M )) Vol(ϕ(M)) (a) (b) For γ G k, what is Vol(γ ϕ (M ))? Up to an isometry of S k, γ is equivalent to a translation or a homothety. If γ is a translation, Vol(γ ϕ (M )) is close to or smaller than Vol(ϕ(M)). Same conclusion if the factor of the homothety is small or or not too large.

104 Conformal bounds for λ 1 Suppose that the factor of γ is large. The parts of M that are close to have a small area.

105 Conformal bounds for λ 1 Suppose that the factor of γ is large. The parts of M that are close to have a small area. M or (c)

106 Conformal bounds for λ 1 Proof for n 3 Lemma Let M n be a closed manifold. If M is obtained from M by a surgery of codimension 2, then V M (M ) sup{v M (M), c(n)}.

107 Conformal bounds for λ 1 Proof for n 3 Lemma Let M n be a closed manifold. If M is obtained from M by a surgery of codimension 2, then V M (M ) sup{v M (M), c(n)}. S k B n k (ε) B k+1 S n k 1 (ε) Codimension 2 n k 1 1

108 Conformal bounds for λ 1 Let M n and M n be cobordant manifolds.

109 Conformal bounds for λ 1 Let M n and M n be cobordant manifolds. Case 1 : M and M are orientable. Thanks to the cancellation lemma, M can be obtained from M by surgeries of codimension 2.

110 Conformal bounds for λ 1 Let M n and M n be cobordant manifolds. Case 1 : M and M are orientable. Thanks to the cancellation lemma, M can be obtained from M by surgeries of codimension 2. Case 2 : M and M are non orientable. γ 2 γ 1

111 Conformal bounds for λ 1 Let M n and M n be cobordant manifolds. Case 1 : M and M are orientable. Thanks to the cancellation lemma, M can be obtained from M by surgeries of codimension 2. Case 2 : M and M are non orientable. γ 2 γ 1 Since M is non orientable, we can nd a transversally orientable loop and apply the cancellation lemma.

112 Conformal bounds for λ 1 The Möbius volume is bounded on each class of oriented cobordism on all non orientable manifold of a given class of non oriented cobordism

113 Conformal bounds for λ 1 The Möbius volume is bounded on each class of oriented cobordism on all non orientable manifold of a given class of non oriented cobordism Final step : V M (M 1 M 2 ) = sup{v M (M 1 ), V M (M 2 )}

114 Conformal bounds for λ 1 The Möbius volume is bounded on each class of oriented cobordism on all non orientable manifold of a given class of non oriented cobordism Final step : V M (M 1 M 2 ) = sup{v M (M 1 ), V M (M 2 )} the Möbius volume is bounded on all classes of oriented cobordism

115 Conformal bounds for λ 1 : manifolds with boundary Theorem If M n is a compact manifold with boundary, then ν(m) = ν(s n ).

116 Conformal bounds for λ 1 : manifolds with boundary Theorem If M n is a compact manifold with boundary, then ν(m) = ν(s n ). Proved by G. Kokarev et N. Nadirashvili for oriented surfaces ('10) using complex analysis.

117 Conformal bounds for λ 1 : manifolds with boundary Theorem If M n is a compact manifold with boundary, then ν(m) = ν(s n ). Proved by G. Kokarev et N. Nadirashvili for oriented surfaces ('10) using complex analysis. Example Let M n R n be an euclidean domain. The stereographic projection induces a conformal immersion ϕ : M S n. For all γ G n, γ ϕ(m) is a domain of S n, hence Vol(γ ϕ(m)) Vol(S n ). V M (M) Vol(S n )

118 Conformal bounds for λ 1 : manifolds with boundary Every compact surface with boundary is obtained by attaching 1-handles to a disk.

119 Conformal bounds for λ 1 : manifolds with boundary Every compact surface with boundary is obtained by attaching 1-handles to a disk. It admits a at embedding in R 3, which is the union of thin bands.

120 Conformal bounds for λ 1 : manifolds with boundary Every compact surface with boundary is obtained by attaching 1-handles to a disk. It admits a at embedding in R 3, which is the union of thin bands. Let ϕ be such an embedding. The bands are supposed to have the same width ε.

121 Conformal bounds for λ 1 : manifolds with boundary Every compact surface with boundary is obtained by attaching 1-handles to a disk. It admits a at embedding in R 3, which is the union of thin bands. Let ϕ be such an embedding. The bands are supposed to have the same width ε. If ε is small, Vol(ϕ(M)) is small. If γ G 3 has not a large homothetic factor, Vol(γ ϕ(m)) is still small.

122 Conformal bounds for λ 1 : manifolds with boundary We consider γ G 3 with large factor. Near the origin, γ ϕ(m) looks like a domain D R 2 { } = S 2.

123 Conformal bounds for λ 1 : manifolds with boundary We consider γ G 3 with large factor. Near the origin, γ ϕ(m) looks like a domain D R 2 { } = S 2. If ε is small enough, the part D of γ ϕ(m) near has an area < Vol(S 2 \D).

124 Conformal bounds for λ 1 : manifolds with boundary We consider γ G 3 with large factor. Near the origin, γ ϕ(m) looks like a domain D R 2 { } = S 2. If ε is small enough, the part D of γ ϕ(m) near has an area < Vol(S 2 \D). g S 3 = 4 (1+ x 2 ) 2 g eucl If we increase the ratio of γ, the volume of D decreases as fast as Vol(S 2 \D).

125 Conformal bounds for λ 1 : manifolds with boundary We consider γ G 3 with large factor. Near the origin, γ ϕ(m) looks like a domain D R 2 { } = S 2. If ε is small enough, the part D of γ ϕ(m) near has an area < Vol(S 2 \D). g S 3 = 4 (1+ x 2 ) 2 g eucl If we increase the ratio of γ, the volume of D decreases as fast as Vol(S 2 \D). Vol(γ ϕ(m)) < Vol(S 2 ).

126 Conformal bounds for λ 1 : manifolds with boundary Proof for n 3 M admits a handle decomposition without n-handle. Induction on the number of handles.

127 Conformal bounds for λ 1 : manifolds with boundary Proof for n 3 M admits a handle decomposition without n-handle. Induction on the number of handles. True if M = B n.

128 Conformal bounds for λ 1 : manifolds with boundary Proof for n 3 M admits a handle decomposition without n-handle. Induction on the number of handles. True if M = B n. Let ϕ : M S k an almost extremal embedding. Vol(γ ϕ(m)) < Vol(S n ) for all γ G k.

129 Conformal bounds for λ 1 : manifolds with boundary Proof for n 3 M admits a handle decomposition without n-handle. Induction on the number of handles. True if M = B n. Let ϕ : M S k an almost extremal embedding. Vol(γ ϕ(m)) < Vol(S n ) for all γ G k. Let ϕ : M S k the embedding obtained by attaching a handle B i B n i (ε) on ϕ(m).

130 Conformal bounds for λ 1 : manifolds with boundary Proof for n 3 M admits a handle decomposition without n-handle. Induction on the number of handles. True if M = B n. Let ϕ : M S k an almost extremal embedding. Vol(γ ϕ(m)) < Vol(S n ) for all γ G k. Let ϕ : M S k the embedding obtained by attaching a handle B i B n i (ε) on ϕ(m). Control of the volume of Vol(γ ϕ (M )) in the same way as for dimension 2.

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132 Bibliography P. Jammes Première valeur propre du laplacien, volume conforme et chirurgies, Geom. Dedicata, 135, p. 2937, 2008 B. Ammann, M. Dahl and E. Humbert Surgery and harmonic spinors, Adv. Math., 220 (2), p , 2009 B. Ammann, M. Dahl and E. Humbert Smooth Yamabe invariant and surgery, J. Dier. Geom., 94, p. 158, 2013 P. Jammes Spectre et géométrie conforme des variétés compactes à bord, preprint