Einstein Metrics, Minimizing Sequences, and the. Differential Topology of Four-Manifolds. Claude LeBrun SUNY Stony Brook

Transcription

1 Einstein Metrics, Minimizing Sequences, and the Differential Topology of Four-Manifolds Claude LeBrun SUNY Stony Brook

2 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie

3 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R

4 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R the greatest blunder of my life! A Einstein, to G Gamow

5 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R the greatest blunder of my life! A Einstein, to G Gamow Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes JW von Goethe

6 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R the greatest blunder of my life! A Einstein, to G Gamow Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes JW von Goethe

7 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R the greatest blunder of my life! A Einstein, to G Gamow Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes JW von Goethe

8 Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature ie r = λg for some constant λ R the greatest blunder of my life! A Einstein, to G Gamow Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes JW von Goethe

9 Ricci curvature measures volume distortion by exponential map: p v exp(v)

10 Ricci curvature measures volume distortion by exponential map: p v exp(v) In geodesic normal coordinates metric volume measure is dµ g = [ ] r jk x j x k + O( x 3 ) dµ Euclidean, where r is the Ricci tensor r jk = R i jik

11 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric?

12 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know:

13 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann)

14 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann) When n = 3: Poincaré conjecture

15 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann) When n = 3: Poincaré conjecture Hamilton, Perelman, Yes!

16 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann) When n = 3: Poincaré conjecture Hamilton, Perelman, Yes! When n = 4: No! (Hitchin)

17 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann) When n = 3: Poincaré conjecture Hamilton, Perelman, Yes! When n = 4: No! (Hitchin) When n = 5: Yes?? (Boyer-Galicki-Kollár)

18 Question (Yamabe) Does every smooth compact 1-connected n-manifold admit an Einstein metric? What we know: When n = 2: Yes! (Riemann) When n = 3: Poincaré conjecture Hamilton, Perelman, Yes! When n = 4: No! (Hitchin) When n = 5: Yes?? (Boyer-Galicki-Kollár) When n 6, wide open Maybe???

19 Four Dimensions is Exceptional

20 Four Dimensions is Exceptional When n = 4, existence for Einstein depends delicately on smooth structure

21 Four Dimensions is Exceptional When n = 4, existence for Einstein depends delicately on smooth structure There are topological 4-manifolds which admit an Einstein metric for one smooth structure, but not for others

22 Four Dimensions is Exceptional When n = 4, existence for Einstein depends delicately on smooth structure There are topological 4-manifolds which admit an Einstein metric for one smooth structure, but not for others But seems related to geometrizations of 4-manifolds by decomposition into Einstein and collapsed pieces

23 Four Dimensions is Exceptional When n = 4, existence for Einstein depends delicately on smooth structure There are topological 4-manifolds which admit an Einstein metric for one smooth structure, but not for others But seems related to geometrizations of 4-manifolds by decomposition into Einstein and collapsed pieces By contrast, high-dimensional Einstein metrics too common, so have little to do with geometrization

24 Variational Problems If M smooth compact n-manifold, n 3, G M = { smooth metrics g on M}

25 Variational Problems If M smooth compact n-manifold, n 3, G M = { smooth metrics g on M} then Einstein metrics = critical points of normalized total scalar curvature functional

26 Variational Problems If M smooth compact n-manifold, n 3, G M = { smooth metrics g on M} then Einstein metrics = critical points of normalized total scalar curvature functional G M R g V (2 n)/n M s g dµ g where V = Vol(M, g) inserted to make scale-invariant

27 If g G M with s > 0, = any metric minimizing must be Einstein G M R g M s g n/2 dµ g

28 If g G M with s > 0, = any metric minimizing must be Einstein G M R g M s g n/2 dµ g If such Einstein minimizer exists, also minimizes g r n/2 g dµ g M

29 If g G M with s > 0, = any metric minimizing must be Einstein G M R g M s g n/2 dµ g If such Einstein minimizer exists, also minimizes g r n/2 g dµ g since M r g 2 = s2 g n + r 2 g s2 g n with Einstein

30 Two soft Invariants: I s (M) = inf g M s g n/2 dµ g

31 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g

32 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M)

33 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer

34 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer Some other goals of this talk:

35 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer Some other goals of this talk: compute these invariants for many 4-manifolds;

36 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer Some other goals of this talk: compute these invariants for many 4-manifolds; describe minimizing sequences for functionals;

37 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer Some other goals of this talk: compute these invariants for many 4-manifolds; describe minimizing sequences for functionals; show that above inequality often strict;

38 Two soft Invariants: I s (M) = inf g I r (M) = inf g M M s g n/2 dµ g r g n/2 dµ g which satisfy [ ] Ir(M) n n/4 Is(M) with = = Einstein minimizer Some other goals of this talk: compute these invariants for many 4-manifolds; describe minimizing sequences for functionals; show that above inequality often strict; provide context for Anderson s talk

39 What s so special about dimension 4? The Lie group SO(4) is not simple: so(4) = so(3) so(3)

40 What s so special about dimension 4? The Lie group SO(4) is not simple: so(4) = so(3) so(3) On oriented (M 4, g), = Λ 2 = Λ + Λ where Λ ± are (±1)-eigenspaces of : Λ 2 Λ 2, 2 = 1

41 What s so special about dimension 4? The Lie group SO(4) is not simple: so(4) = so(3) so(3) On oriented (M 4, g), = Λ 2 = Λ + Λ where Λ ± are (±1)-eigenspaces of : Λ 2 Λ 2, 2 = 1 Λ + self-dual 2-forms Λ anti-self-dual 2-forms

42 Riemann curvature of g R : Λ 2 Λ 2

43 Riemann curvature of g R : Λ 2 Λ 2 splits into 4 irreducible pieces: Λ + Λ Λ + W + + s 12 r Λ r W + s 12

44 Riemann curvature of g R : Λ 2 Λ 2 splits into 4 irreducible pieces: Λ + Λ Λ + W + + s 12 r Λ r W + s 12 where s = scalar curvature r = trace-free Ricci curvature W + = self-dual Weyl curvature W = anti-self-dual Weyl curvature

45 (M, g) compact oriented Riemannian 4-dimensional Gauss-Bonnet formula χ(m) = 1 8π 2 M ( ) s W W 2 r 2 dµ 2

46 (M, g) compact oriented Riemannian 4-dimensional Gauss-Bonnet formula χ(m) = 1 8π 2 M ( ) s W W 2 r 2 2 dµ for Euler-characteristic χ(m) = j ( 1) j b j (M)

47 4-dimensional Hirzebruch signature formula τ(m) = 1 12π 2 M ( W + 2 W 2) dµ

48 4-dimensional Hirzebruch signature formula τ(m) = 1 ( 12π 2 W + 2 W 2) dµ M for signature τ(m) = b + (M) b (M)

49 4-dimensional Hirzebruch signature formula τ(m) = 1 ( 12π 2 W + 2 W 2) dµ M for signature τ(m) = b + (M) b (M) Here b ± (M) = max dim subspaces H 2 (M, R) on which intersection pairing H 2 (M, R) H 2 (M, R) ( [ϕ], [ψ] ) is positive (resp negative) definite R ϕ ψ M

50 Associated square-norm H 2 (M, R) R [ϕ] [ϕ] 2 := M ϕ ϕ

51 Associated square-norm H 2 (M, R) R [ϕ] [ϕ] 2 := M ϕ ϕ

52 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if

53 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if they have the same Euler characteristic χ; they have the same signature τ; and both are spin, or both are non-spin

54 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if they have the same Euler characteristic χ; they have the same signature τ; and both are spin, or both are non-spin

55 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if they have the same Euler characteristic χ; they have the same signature τ; and both are spin, or both are non-spin

56 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if they have the same Euler characteristic χ; they have the same signature τ; and both are spin, or both are non-spin

57 Theorem (Freedman/Donaldson) Two smooth compact simply connected oriented 4-manifolds are orientedly homeomorphic if and only if they have the same Euler characteristic χ; they have the same signature τ; and both are spin, or both are non-spin Warning: Exotic differentiable structures! No diffeomorphism classification currently known! Typically, one homeotype many diffeotypes

58 Hitchin-Thorpe Inequality: (2χ ± 3τ)(M) = 1 4π 2 M ( ) s W ± 2 r 2 dµ g 2

59 Hitchin-Thorpe Inequality: (2χ ± 3τ)(M) = 1 4π 2 Einstein = 1 4π 2 M M ) W ± 2 r 2 2 ) ( s 2 ( s W ± 2 dµ g dµ g

60 Hitchin-Thorpe Inequality: (2χ ± 3τ)(M) = 1 4π 2 Einstein = 1 4π 2 M M ) W ± 2 r 2 2 ) ( s 2 ( s W ± 2 dµ g dµ g Theorem (Hitchin-Thorpe Inequality) If smooth compact oriented M 4 admits Einstein g, then and (2χ + 3τ)(M) 0 (2χ 3τ)(M) 0

61 Example Let CP 2 = reverse-oriented CP 2 Then jcp 2 #kcp 2 = CP 2 # #CP 2 }{{} j # CP } 2 # {{ #CP } 2, k

62 Connected sum:

63 Example Let CP 2 = reverse-oriented CP 2 Then jcp 2 #kcp 2 = CP 2 # #CP 2 }{{} j # CP } 2 # {{ #CP } 2, k has 2χ + 3τ = 4 + 5j k so Einstein metric if k 4 + 5j

64 Hitchin-Thorpe Inequality: (2χ ± 3τ)(M) = 1 4π 2 Einstein = 1 4π 2 M M ) W ± 2 r 2 2 ) ( s 2 ( s W ± 2 dµ g dµ g Theorem (Hitchin-Thorpe Inequality) If smooth compact oriented M 4 admits Einstein g, then and (2χ + 3τ)(M) 0 (2χ 3τ)(M) 0

65 Hitchin-Thorpe Inequality: (2χ ± 3τ)(M) = 1 4π 2 Einstein = 1 4π 2 M M ) W ± 2 r 2 2 ) ( s 2 ( s W ± 2 dµ g dµ g Theorem (Hitchin-Thorpe Inequality) If smooth compact oriented M 4 admits Einstein g, then and (2χ + 3τ)(M) 0 (2χ 3τ)(M) 0 Both inequalities strict unless finitely covered by flat T 4, Calabi-Yau K3, or Calabi-Yau K3

66 K3 = Kummer-Kähler-Kodaira manifold

67 K3 = Kummer-Kähler-Kodaira manifold Simply connected complex surface with c 1 = 0

68 K3 = Kummer-Kähler-Kodaira manifold Simply connected complex surface with c 1 = 0 Only one deformation type In particular, only one diffeotype

69 K3 = Kummer-Kähler-Kodaira manifold Simply connected complex surface with c 1 = 0 Only one deformation type In particular, only one diffeotype Spin manifold, b + = 3, b = 19

70 K3 = Kummer-Kähler-Kodaira manifold Simply connected complex surface with c 1 = 0 Only one deformation type In particular, only one diffeotype Spin manifold, b + = 3, b = 19 Theorem (Yau) K3 admits Ricci-flat metrics

71 Kummer construction of K3:

72 Kummer construction of K3: Begin with T 4 /Z 2 : T 4 T 2 T 2

73 Kummer construction of K3: Begin with T 4 /Z 2 : T 4 T 2 T 2 Replace R 4 /Z 2 neighborhood of each singular point with copy of T S 2

74 Approximate Calabi-Yau metric: Replace flat metric on R 4 /Z 2 T 4 T 2 T 2 with Eguchi-Hanson metric on T S 2 : g EH,ɛ = dϱ2 ( [ 1 ɛϱ 4 +ϱ2 θ θ ɛϱ 4] ) θ 2 3 (Page, Kobayashi-Todorov, LeBrun-Singer)

75 Examples of Einstein 4-Manifolds

76 Examples of Einstein 4-Manifolds Richest source: Kähler geometry

77 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding

78 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding

79 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding j : M CP k such that c 1 (M) is negative multiple of j c 1 (CP k )

80 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding j : M CP k such that c 1 (M) is negative multiple of j c 1 (CP k )

81 Corollary For any l 5, the degree l surface t l + u l + v l + w l = 0 in CP 3 admits s < 0 Kähler-Einstein metric

82 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding j : M CP k such that c 1 (M) is negative multiple of j c 1 (CP k ) Remark This happens c 1 (M) is a Kähler class Short-hand: c 1 (M) < 0

83 Examples of Einstein 4-Manifolds Richest source: Kähler geometry Theorem (Aubin/Yau) Compact complex manifold (M 2m, J) admits compatible Kähler-Einstein metric with s < 0 holomorphic embedding j : M CP k such that c 1 (M) is negative multiple of j c 1 (CP k ) Remark This happens c 1 (M) is a Kähler class Short-hand: c 1 (M) < 0 Remark When m = 2, such M are necessarily minimal complex surfaces of general type

84 Blowing up: If N is a complex surface, may replace p N with CP 1 to obtain blow-up M N#CP 2 in which new CP 1 has self-intersection 1

85 Blowing up: If N is a complex surface, may replace p N with CP 1 to obtain blow-up M N#CP 2 in which new CP 1 has self-intersection 1 A complex surface X is called minimal if it is not the blow-up of another complex surface

86 Blowing up: If N is a complex surface, may replace p N with CP 1 to obtain blow-up M N#CP 2 in which new CP 1 has self-intersection 1 A complex surface X is called minimal if it is not the blow-up of another complex surface Any complex surface M can be obtained from a minimal surface X by blowing up a finite number of times: M X#kCP 2 One says that X is minimal model of M

87 Compact complex surface (M 4, J) general type if dim Γ(M, O(K l )) al 2, l 0, where K = Λ 2,0 is canonical line bundle

88 Compact complex surface (M 4, J) general type if dim Γ(M, O(K l )) al 2, l 0, where K = Λ 2,0 is canonical line bundle If l 5, then Γ(M, O(K l )) gives holomorphic map f l : M CP N which just collapses each CP 1 with self-intersection 1 or 2 to a point Image X = f l (M) called pluricanonical model of M

89 Compact complex surface (M 4, J) general type if dim Γ(M, O(K l )) al 2, l 0, where K = Λ 2,0 is canonical line bundle If l 5, then Γ(M, O(K l )) gives holomorphic map f l : M CP N which just collapses each CP 1 with self-intersection 1 or 2 to a point Image X = f l (M) called pluricanonical model of M Pluricanonical model X is a complex orbifold with c 1 < 0 and singularities C 2 /G, G SU(2)

90 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0

91 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0 Thus, any surface M of general type obtained from Kähler-Einstein orbifold X in two steps:

92 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0 Thus, any surface M of general type obtained from Kähler-Einstein orbifold X in two steps: 1 Replace each orbifold point with ( 2)-curves intersecting according to Dynkin diagram determined by G SU(2)

93 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0 Thus, any surface M of general type obtained from Kähler-Einstein orbifold X in two steps: 1 Replace each orbifold point with ( 2)-curves intersecting according to Dynkin diagram determined by G SU(2) Related geometry: gravitational instantons

94 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0 Thus, any surface M of general type obtained from Kähler-Einstein orbifold X in two steps: 1 Replace each orbifold point with ( 2)-curves intersecting according to Dynkin diagram determined by G SU(2) Related geometry: gravitational instantons 2 Replace some non-singular points with ( 1)-curves

95 Aubin-Yau proof = Corollary (R Kobayashi) The pluricanonical model X of any compact complex surface M of general type admits and orbifold Kähler-Einstein metric with s < 0 Thus, any surface M of general type obtained from Kähler-Einstein orbifold X in two steps: 1 Replace each orbifold point with ( 2)-curves intersecting according to Dynkin diagram determined by G SU(2) Related geometry: gravitational instantons 2 Replace some non-singular points with ( 1)-curves Related geometry: scalar-flat Kähler metrics

96 Seiberg-Wittten theory: generalized Kähler geometry of non-kähler 4-manifolds

97 Seiberg-Wittten theory: generalized Kähler geometry of non-kähler 4-manifolds Can t hope to generalize operator to this setting

98 Seiberg-Wittten theory: generalized Kähler geometry of non-kähler 4-manifolds Can t hope to generalize operator to this setting But + does generalize:

99 Seiberg-Wittten theory: generalized Kähler geometry of non-kähler 4-manifolds Can t hope to generalize operator to this setting But + does generalize: spin c Dirac operator, preferred connection on L

100 Spin c structures:

101 Spin c structures: in image of w 2 (T M) H 2 (M, Z 2 ) H 2 (M, Z) H 2 (M, Z 2 )

102 Spin c structures: w 2 (T M) H 2 (M, Z 2 ) in image of H 2 (M, Z) H 2 (M, Z 2 ) = Hermitian line bundles L M with c 1 (L) w 2 (T M) mod 2

103 Spin c structures: in image of w 2 (T M) H 2 (M, Z 2 ) H 2 (M, Z) H 2 (M, Z 2 ) = Hermitian line bundles with L M c 1 (L) w 2 (T M) mod 2 Given g on M, = rank-2 Hermitian vector bundles V ± M

104 Spin c structures: in image of w 2 (T M) H 2 (M, Z 2 ) H 2 (M, Z) H 2 (M, Z 2 ) = Hermitian line bundles with L M c 1 (L) w 2 (T M) mod 2 Given g on M, = rank-2 Hermitian vector bundles V ± M which formally satisfy V ± = S ± L 1/2,

105 Spin c structures: in image of w 2 (T M) H 2 (M, Z 2 ) H 2 (M, Z) H 2 (M, Z 2 ) = Hermitian line bundles with L M c 1 (L) w 2 (T M) mod 2 Given g on M, = rank-2 Hermitian vector bundles V ± M which formally satisfy V ± = S ± L 1/2, where S ± are the (locally defined) left- and righthanded spinor bundles of (M, g)

106 Every unitary connection A on L

107 Every unitary connection A on L induces spin c Dirac operator D A : Γ(V + ) Γ(V )

108 Every unitary connection A on L induces spin c Dirac operator D A : Γ(V + ) Γ(V ) Weitzenböck formula: Φ Γ(V + ), Φ, D A D A Φ = 1 2 Φ 2 + A Φ 2 + s 4 Φ 2

109 Every unitary connection A on L induces spin c Dirac operator D A : Γ(V + ) Γ(V ) Weitzenböck formula: Φ Γ(V + ), Φ, D A D A Φ = 1 2 Φ 2 + A Φ 2 + s 4 Φ 2 +2 if A +, σ(φ)

110 Every unitary connection A on L induces spin c Dirac operator D A : Γ(V + ) Γ(V ) Weitzenböck formula: Φ Γ(V + ), Φ, D A D A Φ = 1 2 Φ 2 + A Φ 2 + s 4 Φ 2 +2 if A +, σ(φ) where F A + = self-dual part curvature of A,

111 Every unitary connection A on L induces spin c Dirac operator D A : Γ(V + ) Γ(V ) Weitzenböck formula: Φ Γ(V + ), Φ, D A D A Φ = 1 2 Φ 2 + A Φ 2 + s 4 Φ 2 +2 if A +, σ(φ) where F A + = self-dual part curvature of A, and σ : V + Λ + is a natural real-quadratic map, σ(φ) = Φ 2

112 Witten: consider both Φ and A as unknowns,

113 Witten: consider both Φ and A as unknowns, subject to Seiberg-Witten equations D A Φ = 0 F + A = iσ(φ)

114 Witten: consider both Φ and A as unknowns, subject to Seiberg-Witten equations Non-linear, but elliptic D A Φ = 0 F + A = iσ(φ)

115 Witten: consider both Φ and A as unknowns, subject to Seiberg-Witten equations D A Φ = 0 F + A = iσ(φ) Non-linear, but elliptic once gauge-fixing d (A A 0 ) = 0 imposed to eliminate automorphisms of L M

116 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4

117 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4 = moduli space compact, finite-dimensional

118 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4 = moduli space compact, finite-dimensional Seiberg-Witten map of Banach spaces proper map finite-dimensional spaces

119 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4 = moduli space compact, finite-dimensional Seiberg-Witten map of Banach spaces proper map finite-dimensional spaces Degree: classical Seiberg-Witten invariant

120 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4 = moduli space compact, finite-dimensional Seiberg-Witten map of Banach spaces proper map finite-dimensional spaces Degree: classical Seiberg-Witten invariant Stable homotopy class: Bauer-Furuta invariant

121 Weitzenböck formula becomes 0 = 2 Φ A Φ 2 + s Φ 2 + Φ 4 = moduli space compact, finite-dimensional Seiberg-Witten map of Banach spaces proper map finite-dimensional spaces Degree: classical Seiberg-Witten invariant Stable homotopy class: Bauer-Furuta invariant When invariant is non-zero, solutions guaranteed

122 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then

123 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R)

124 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class

125 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class c 1 (L) = a

126 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class c 1 (L) = a such that the Seiberg-Witten equations

127 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class c 1 (L) = a such that the Seiberg-Witten equations D A Φ = 0 F + A = iσ(φ)

128 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class c 1 (L) = a such that the Seiberg-Witten equations have a solution (Φ, A) D A Φ = 0 F + A = iσ(φ)

129 Definition Let M be a smooth compact oriented 4-manifold with b + 2 Then a H 2 (M, R) is called a monopole class of M iff there exists a spin c structure on M with first Chern class c 1 (L) = a such that the Seiberg-Witten equations D A Φ = 0 F + A = iσ(φ) have a solution (Φ, A) for every metric g on M

130 Proposition Let M be a smooth compact oriented 4-manifold with b + 2 The collection C H 2 (M, R) of all monopole classes is finite, and is an oriented diffeomorphism invariant of M

131 Proposition Let M be a smooth compact oriented 4-manifold with b + 2 The collection C H 2 (M, R) of all monopole classes is finite, and is an oriented diffeomorphism invariant of M

132 Proposition Let M be a smooth compact oriented 4-manifold with b + 2 The collection C H 2 (M, R) of all monopole classes is finite, and is an oriented diffeomorphism invariant of M

133 Proposition Let M be a smooth compact oriented 4-manifold with b + 2 The collection C H 2 (M, R) of all monopole classes is finite, and is an oriented diffeomorphism invariant of M

134 Proposition Let M be a smooth compact oriented 4-manifold with b + 2 The collection C H 2 (M, R) of all monopole classes is finite, and is an oriented diffeomorphism invariant of M

135 Definition Let C = {monopole classes} H 2 (M) If C, let Hull(C) = convex hull of C and set β 2 (M) = max {v v v Hull(C)}

136 Definition Let C = {monopole classes} H 2 (M) If C, let Hull(C) = convex hull of C and set β 2 (M) = max {v v v Hull(C)}

137 Definition Let C = {monopole classes} H 2 (M) If C, let Hull(C) = convex hull of C and set β 2 (M) = max {v v v Hull(C)}

138 Definition Let C = {monopole classes} H 2 (M) If C, let Hull(C) = convex hull of C and set β 2 (M) = max {v v v Hull(C)} If C =, set β 2 (M) = 0

139 Example If X is a minimal complex surface with b + > 1, and if M = X#lCP 2 then classical Seiberg-Witten invariant allows one to show that β 2 (M) = c 1 2 (X)

140 Example If X is a minimal complex surface with b + > 1, and if M = X#lCP 2 then classical Seiberg-Witten invariant allows one to show that β 2 (M) = c 1 2 (X) Example If X, Y, Z are minimal complex surfaces with b 1 = 0 and b + 3 mod 4, and if M = X#Y #Z#lCP 2 Bauer-Furuta invariant allows one to show that β 2 (M) = c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z) Similarly for 2 or 4

141 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied:

142 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied:

143 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied:

144 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied: M s 2 dµ g 32π 2 β 2 (M) ( M s 2 6 W + ) dµg

145 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied: M s 2 dµ g 32π 2 β 2 (M) ( M s 2 6 W + ) dµg 72π 2 β 2 (M)

146 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied: M s 2 dµ g 32π 2 β 2 (M) ( M s 2 6 W + ) dµg 72π 2 β 2 (M) Moreover, if β 2 (M) 0, equality holds in either case iff (M, g) is a Kähler-Einstein manifold with s < 0

147 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g

148 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M)

149 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M) Hence: Theorem A Let M be a smooth compact oriented 4-manifold with b + (M) 2

150 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M) Hence: Theorem A Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g,

151 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M) Hence: Theorem A Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ 3τ)(M) 1 3 β2 (M)

152 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M) Hence: Theorem A Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ 3τ)(M) 1 3 β2 (M) with equality only if (M, g) is flat T 4 or complex hyperbolic CH 2 /Γ

153 First curvature estimate implies 1 4π 2 M ( s W 2 ) dµ g 1 4π 2 M s 2 24 dµ g 1 3 β2 (M) Hence: Theorem A Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ 3τ)(M) 1 3 β2 (M) with equality only if (M, g) is flat T 4 or complex hyperbolic CH 2 /Γ = Einstein metric on CH /Γ unique,

154 Second curvature estimate implies 1 4π 2 M ( s W + 2 ) dµ g 2 3 β2 (M)

155 Second curvature estimate implies 1 4π 2 M ( s W + 2 ) dµ g 2 3 β2 (M)

156 Second curvature estimate implies 1 4π 2 Hence: M ( s W + 2 ) dµ g 2 3 β2 (M) Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2

157 Second curvature estimate implies 1 4π 2 Hence: M ( s W + 2 ) dµ g 2 3 β2 (M) Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g,

158 Second curvature estimate implies 1 4π 2 Hence: M ( s W + 2 ) dµ g 2 3 β2 (M) Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ + 3τ)(M) 2 3 β2 (M)

159 Second curvature estimate implies 1 4π 2 Hence: M ( s W + 2 ) dµ g 2 3 β2 (M) Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ + 3τ)(M) 2 3 β2 (M) with equality only if both sides vanish,

160 Second curvature estimate implies 1 4π 2 Hence: M ( s W + 2 ) dµ g 2 3 β2 (M) Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ + 3τ)(M) 2 3 β2 (M) with equality only if both sides vanish, in which case g must be hyper-kähler, and Mmust be diffeomorphic to either K3 or T 4

161 Example Let N be double branched cover CP 2, ramified at a smooth octic: N B B CP 2 Aubin/Yau = N carries Einstein metric

162 Now let X be a triple cyclic cover CP 2, ramified at a smooth sextic X B B CP 2

163 Now let X be a triple cyclic cover CP 2, ramified at a smooth sextic X B B CP 2 and set M = X#CP 2

164 Now let X be a triple cyclic cover CP 2, ramified at a smooth sextic X B B CP 2 and set Then M = X#CP 2 β 2 (M) = c 2 1 (X) = 3 (2χ + 3τ)(M) c 2 1 (X) 1 = 2

165 Now let X be a triple cyclic cover CP 2, ramified at a smooth sextic X B B CP 2 and set Then M = X#CP 2 β 2 (M) = c 2 1 (X) = 3 (2χ + 3τ)(M) = c 2 1 (X) 1 = 2

166 Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ + 3τ)(M) 2 3 β2 (M) & equality only if M diffeomorphic to K3 or T 4

167 Theorem B Let M be a smooth compact oriented 4-manifold with b + (M) 2 If M admits an Einstein metric g, then (2χ + 3τ)(M) 2 3 β2 (M) & equality only if M diffeomorphic to K3 or T 4 In example: β 2 (M) = 3 (2χ + 3τ)(M) = 2

168 X is triple cover CP 2 ramified at sextic X B B CP 2 and set M = X#CP 2 Theorem B = no Einstein metric on M

169 But M and N are both simply connected & non-spin,

170 But M and N are both simply connected & non-spin, and both have c 1 2 = 2, h 2,0 = 3, so

171 But M and N are both simply connected & non-spin, and both have c 1 2 = 2, h 2,0 = 3, so χ = 46 τ = 30

172 But M and N are both simply connected & non-spin, and both have c 1 2 = 2, h 2,0 = 3, so χ = 46 τ = 30 Hence Freedman = M homeomorphic to N!

173 But M and N are both simply connected & non-spin, and both have c 1 2 = 2, h 2,0 = 3, so χ = 46 τ = 30 Hence Freedman = M homeomorphic to N! Moral: Existence depends on diffeotype! Same ideas lead to infinitely many other examples Typically get non-existence for infinitely many smooth structures on fixed topological manifold Existence: look in Kähler-Einstein catalog

174 Until now, discussed arbitrary Einstein metrics Instead, focus on Einstein metrics which minimize g s 2 gdµ g M Related to soft invariants I s (M) = inf g I r (M) = inf g M M s 2 gdµ g r 2 gdµ g which satisfy [ ] Ir(M) 1 4 I s(m) with = = Einstein minimizer

175 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied: M s 2 dµ g 32π 2 β 2 (M) [ ] r gdµ 2 g 8π 2 2β 2 (2χ + 3τ) (M) M

176 Theorem (Curvature Estimates) For any C 2 Riemannian metric g on any smooth compact oriented 4-manifold M with b + 2, the following curvature bounds are satisfied: M s 2 dµ g 32π 2 β 2 (M) [ ] r gdµ 2 g 8π 2 2β 2 (2χ + 3τ) M (M) M r gdµ 2 g = 8π 2 (2χ + 3τ)(M) ( ) s W + 2 dµ g M

177 Theorem Suppose M 4 diffeo to non-minimal compact complex surface with b + > 1 Then M does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M

178 Theorem Suppose M 4 diffeo to non-minimal compact complex surface with b + > 1 Then M does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M By hypothesis M = X#kCP 2 where X minimal and k > 0 One shows so that I s (M) = 32π 2 c 1 2 (X) I r (M) = 8π 2 [c 1 2 (X) + k] I r (M) > 1 4 I s(m)

179 Theorem Let X, Y and Z be simply connected minimal complex surfaces with b + 3 mod 4 Then M = X#Y #Z#kCP 2 does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M

180 Theorem Let X, Y and Z be simply connected minimal complex surfaces with b + 3 mod 4 Then M = X#Y #Z#kCP 2 does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M In fact, I s (M) = 32π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z)] I r (M) = 8π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z) k]

181 Theorem Let X, Y and Z be simply connected minimal complex surfaces with b + 3 mod 4 Then M = X#Y #Z#kCP 2 does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M In fact, I s (M) = 32π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z)] I r (M) = 8π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z) k] Similarly for # of 2 or 4 complex surfaces

182 Theorem Let X, Y and Z be simply connected minimal complex surfaces with b + 3 mod 4 Then M = X#Y #Z#kCP 2 does not admit a metric which minimizes either g s 2 gdµ g or r 2 gdµ g M M In fact, I s (M) = 32π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z)] I r (M) = 8π 2 [c 1 2 (X) + c 1 2 (Y ) + c 1 2 (Z) k] Similarly for # of 2 or 4 complex surfaces Mystery: More summands? b + 1 mod 4?

183 K-E K-E K-E When X, Y and Z general type, however, minimizing {g j } with Gromov-Hausdorff limit 3 Kähler-Einstein orbifolds touching at points

184 K-E K-E K-E points where curvature has accumulated

185 K-E K-E K-E Predictable amount of r accumulates on necks

186 SFASD K-E K-E K-E Rescaled limit of neck carries AE metric with Example: s = 0 W + = 0 g = (1 + 1 ϱ 2) g Euclidean

187 RFALE K-E K-E K-E Orbifold singularities: rescaled metric tends to gravitational instanton: Asymptotically Locally Euclidean metric with r = 0 W + = 0

188 SFASD K-E K-E K-E Bubbling off CP 2 s: Asymptotically Euclidean metric with s = 0 W + = 0

189 Basic example: Burns metric on CP 2 { }: g B,ɛ = dϱ2 ( [ 1 ɛϱ 2 + ϱ2 θ θ ɛϱ 2] ) θ 2 3 Conformal Greens rescaling of Fubini-Study

190 K-E K-E If one of X, Y and Z is elliptic, collapses in limit to orbifold Riemann surface

191 Typical example: E Σ E Σ