Massin K ahler Geometry - Princeton University · Massin K ahler Geometry Claude LeBrun Stony Brook University 31st Annual Geometry Festival Princeton University, April 8, 2016 1

Mass in

Kahler Geometry

Claude LeBrunStony Brook University

31st Annual Geometry FestivalPrinceton University, April 8, 2016

1

Joint work with

2

Joint work with

Hans-Joachim HeinUniversity of Maryland

3

Joint work with


e-print: arXiv:1507.08885 [math.DG]

4

Joint work with


e-print: arXiv:1507.08885 [math.DG]

To appear in Comm. Math. Phys.

5

Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn

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6

Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn

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n ≥ 3

7

Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that

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n ≥ 3

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~x

gjk = δjk + O(|x|1−n2−ε)

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~x


gjk,` = O(|x|−n2−ε), s ∈ L1

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~x


gjk,` = O(|x|−n2−ε), s ∈ L1

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gjk,` = O(|x|−n2−ε), s ∈ L1

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gjk,` = O(|x|−n2−ε), s ∈ L1

16


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gjk,` = O(|x|−n2−ε), s ∈ L1

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gjk,` = O(|x|−n2−ε), s ∈ L1

18


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gjk,` = O(|x|−n2−ε), s ∈ L1

19

Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each “end” component of is diffeo-morphic to Rn −Dn in such a manner that

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gjk,` = O(|x|−n2−ε), s ∈ L1

20

Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R

n −Dn)/Γi, where Γi ⊂ SO(n), such that

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23

Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ SO(n), such that

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gjk,` = O(|x|−n2−ε), s ∈ L1

28

Definition. The mass (at a given end) of anALE n-manifold is defined to be

m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

29


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

30


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

31


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

32


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

33


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where

34


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where

• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;

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35


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where


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36


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where


• ν is the outpointing Euclidean unit normal;and

37


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where



• αE is the volume (n− 1)-from induced by theEuclidean metric.

38


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Seems to depend on choice of coordinates!

39


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




40


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.

41


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





42


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





gjk,` = O(|x|−n2−ε), s ∈ L1

43


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





44


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





45


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Chrusciel-type fall-off:

gjk − δjk ∈ C1−τ , τ >

n− 2

2

46


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Bartnik-type fall-off:

gjk − δjk ∈ W2,q−τ , τ >

n− 2

2, q > n

47


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Bartnik-type fall-off =⇒

gjk − δjk ∈ C1,α−τ , τ >

n− 2

2.

48


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





49

Motivation:

50

Motivation:

When n = 3, ADM mass in general relativity.

51

Motivation:


Reads off “apparent mass” from strength of thegravitational field far from an isolated source.

52

Motivation:



53

Motivation:



54

Motivation:



In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild

55

Motivation:




g = −(

1− 2m%n−2

)dt2+

(1− 2m

%n−2

)−1

d%2+%2hSn−1

56

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1

57

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1

Scalar-flat-Kahler Burns metric on C2⊂ C2 × CP1

58

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1

Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1

59

Motivation:



60

Motivation:



In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild

g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1


61

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1


62

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1

Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:

ω =i

2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2

63

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1


ω =i

2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2

64

Motivation:




g =

(1− 2m

%n−2

)−1

d%2+%2hSn−1


ω =i

2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2

also has mass m .

65

Positive Mass Conjecture:

66


Any AE manifold with s ≥ 0 has m ≥ 0.

67



68



69



Physical intuition:

70



Physical intuition:

Local matter density ≥ 0 =⇒ total mass ≥ 0.

71



Physical intuition:

Local matter density ≥ 0 =⇒ total mass ≥ 0.

72



73



Schoen-Yau 1979:

74



Schoen-Yau 1979:

Proved in dimension n ≤ 7.

75



Schoen-Yau 1979:


Witten 1981:

76



Schoen-Yau 1979:


Witten 1981:

Proved for spin manifolds (implicitly, for any n.)

77



Schoen-Yau 1979:


Witten 1981:

Proved for spin manifolds (implicitly, for any n).

78



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:

79



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:

Conjectured true in ALE case, too.

80



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:


L 1986:

81



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:


L 1986:

ALE counter-examples.

82



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:


L 1986:


Scalar-flat Kahler metrics

83



Schoen-Yau 1979:


Witten 1981:


Hawking-Pope 1978:


L 1986:


Scalar-flat Kahler metrics

on line bundles L→ CP1 of Chern-class ≤ −3.

84

Mass of ALE Kahler manifolds?

85


Scalar-flat Kahler case?

86



Lemma. Any ALE Kahler manifold has onlyone end.

87




88




89




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90




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n = 2m ≥ 4

91




Several different proofs are known.

92





Several are anlaytic:

93






each end is pseudo-convex at infinity.

94







Another is more topological:

95







Another is more topological:

intersection form on H2 of compactification.

96




97




Upshot:

98




Upshot:

Mass of an ALE Kahler manifold is unambiguous.

99




Upshot:

Mass of an ALE Kahler manifold is unambiguous.

Does not depend on the choice of an end!

100

We begin with the scalar-flat Kahler case.

101


Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.

102



103



104



105



106



That is, m(M, g, J) is completely determined by

107




108




• the smooth manifold M ,

109





• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and

110






• the Kahler class [ω] ∈ H2(M) of the metric.

111






• the Kahler class [ω] ∈ H2(M) of the metric.

In fact, we will see that there is an explicit formulafor the mass in terms of these data!

112

The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.

113


114


For example, the mass is zero for an ALE Ricci-flatKahler manifold. Arezzo asked whether, conversely,an ALE scalar-flat Kahler manifold with zero massmust be Ricci-flat. The answer is, No!

115



116



117



118



119



120



121



122



123



124



Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.

125




126




127




128




129




130




(Discovered independently by Rollin, Singer, & Suvaina,using different methods.)

131

Explicit formula depends on a topological fact:

132


Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map

133


Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map

134


Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map

135



136



H2c (M)→ H2

dR(M)

is an isomorphism.

137



H2c (M)→ H2

dR(M)

is an isomorphism.

138



H2c (M)→ H2

dR(M)

is an isomorphism.

Here

Hpc (M) :=

ker d : Epc (M)→ Ep+1c (M)

dEp−1c (M)

139



H2c (M)→ H2

dR(M)

is an isomorphism.

Here

Hpc (M) :=

ker d : Epc (M)→ Ep+1c (M)

dEp−1c (M)

where

Epc (M) := {Smooth, compactly supported p-forms on M}.

140



H2c (M)→ H2

dR(M)

is an isomorphism.

141



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:

142



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:

Compactify M as M = M ∪ (Sn−1/Γ).

143



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:


Then Hpc (M) = Hp(M,∂M).

144



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:



Exact sequence of pair:

145



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:




H1(∂M)→ H2(M,∂M)→ H2(M)→ H2(∂M)

146



H2c (M)→ H2

dR(M)

is an isomorphism.

If just one end:




H1(Sn−1/Γ)→ H2c (M)→ H2(M)→ H2(Sn−1/Γ)

147



H2c (M)→ H2

dR(M)

is an isomorphism.

148



H2c (M)→ H2

dR(M)

is an isomorphism.

Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use

149



H2c (M)→ H2

dR(M)

is an isomorphism.


150



H2c (M)→ H2

dR(M)

is an isomorphism.


151



H2c (M)→ H2

dR(M)

is an isomorphism.


♣ : H2dR(M)→ H2

c (M)

152



H2c (M)→ H2

dR(M)

is an isomorphism.


♣ : H2dR(M)→ H2

c (M)

to denote the inverse of the natural map

153



H2c (M)→ H2

dR(M)

is an isomorphism.


♣ : H2dR(M)→ H2

c (M)


H2c (M)→ H2

dR(M)

induced by the inclusion of compactly supportedsmooth forms into all forms.

154



H2c (M)→ H2

dR(M)

is an isomorphism.


♣ : H2dR(M)→ H2

c (M)


H2c (M)→ H2

dR(M)

induced by the inclusion of compactly supportedsmooth forms into all forms.

155

We can now state our mass formula:

Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by

156


Theorem C. Any ALE Kahler manifold (M, g, J)ofcomplex dimension m has mass given by

157



158



159



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

160



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

161



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

162



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

163



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

164



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where

165



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where

• s = scalar curvature;

166



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where


• dµ = metric volume form;

167



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where



• c1 = c1(M,J) ∈ H2(M) is first Chern class;

168



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where




• [ω] ∈ H2(M) is Kahler class of (g, J); and

169



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where




• [ω] ∈ H2(M) is Kahler class of (g, J); and

• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).

170

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

171

4πm(2m−1)(m−1)!

m(M, g) = − 4π(m−1)!

〈♣(c1), [ω]m−1〉+∫Msgdµg

172

For a compact Kahler manifold (M2m, g, J),

∫Msgdµg =

4π

(m− 1)!〈c1, [ω]m−1〉

173

For a compact Kahler manifold (M2m, g, J),

0 = − 4π

(m− 1)!〈c1, [ω]m−1〉 +

∫Msgdµg

174

For an ALE Kahler manifold (M2m, g, J),

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

175

For an ALE Kahler manifold (M2m, g, J),

4πm(2m−1)(m−1)!

m(M, g) = − 4π(m−1)!

〈♣(c1), [ω]m−1〉+∫Msgdµg

So the mass is a “boundary correction” to the topo-logical formula for the total scalar curvature.

176



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

177


Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

.

178


Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

.

So Theorem A is an immediate consequence!

179

Rough Idea of Proof:

180


Special Case: Suppose

181



•m = 2, n = 4;

182



•m = 2, n = 4;

• Scalar flat: s ≡ 0; and

183



•m = 2, n = 4;


• Complex structure J standard at infinity:

184



•m = 2, n = 4;



(M −K, J) ≈bih (C2 −B4)/Γ.

185



•m = 2, n = 4;



(M −K, J) ≈bih (C2 −B4)/Γ.

Since g is Kahler, the complex coordinates

186



•m = 2, n = 4;



(M −K, J) ≈bih (C2 −B4)/Γ.


(z1, z2) = (x1 + ix2, x3 + ix4)

are harmonic. So xj are harmonic, too, and

187



•m = 2, n = 4;



(M −K, J) ≈bih (C2 −B4)/Γ.


(z1, z2) = (x1 + ix2, x3 + ix4)


188



•m = 2, n = 4;



(M −K, J) ≈bih (C2 −B4)/Γ.


(z1, z2) = (x1 + ix2, x3 + ix4)


gjk(gj`,k − gjk,`

)ν`αE = −?d log

(√det g

)+O(%−3−ε).

189

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

190

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

191

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

ρ = dθ

is Ricci form, and

192

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

ρ = dθ

is Ricci form, and

−?d log(√

det g)

= 2 θ ∧ ω.

193

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

ρ = dθ

is Ricci form, and

−?d log(√

det g)

= 2 θ ∧ ω.Thus

m(M, g) = − lim%→∞

1

6π2

∫S%/Γ

θ ∧ ω

194

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

ρ = dθ

is Ricci form, and

−?d log(√

det g)

= 2 θ ∧ ω.Thus

m(M, g) = − lim%→∞

1

6π2

∫S%/Γ

θ ∧ ω

However, since s = 0,

d(θ ∧ ω) = ρ ∧ ω =s

4ω2 = 0.

195

m(M, g) = − lim%→∞

1

12π2

∫S%/Γ

? d(

log√

det g)

Now set θ = i2(∂ − ∂)

(log√

det g), so that

ρ = dθ

is Ricci form, and

−?d log(√

det g)

= 2 θ ∧ ω.Thus

m(M, g) = − 1

6π2

∫S%/Γ

θ ∧ ω

196

Let f : M → R be smooth cut-off function:

197

Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.

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1

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“end”

1

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M%

%

1

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radius

M%

%

1

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205


206


Set

ψ := ρ− d(fθ)

207


Set

ψ := ρ− d(fθ)

Compactly supported, because dθ = ρ near infinity.

208


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

209


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 =

∫Mψ ∧ ω

210


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 =

∫M%

ψ ∧ ω

where M% defined by radius ≤ %.

211


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 =

∫M%

ψ ∧ ω

212


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 =

∫M%

[ρ− d(fθ)] ∧ ω

213


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫M%

d(fθ ∧ ω)

because scalar-flat =⇒ ρ ∧ ω = 0.

214


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫M%

d(fθ ∧ ω)

215


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫∂M%

fθ ∧ ω

by Stokes’ theorem.

216


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫∂M%

θ ∧ ω


217


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫S%/Γ

θ ∧ ω


218


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫S%/Γ

θ ∧ ω


219


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫S%/Γ

θ ∧ ω


So

m(M, g) = − 1

6π2

∫S%/Γ

θ ∧ ω

220


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫S%/Γ

θ ∧ ω


So

m(M, g) = − 1

3π〈♣(c1), ω〉

221


Set

ψ := ρ− d(fθ)

[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)

〈2π♣(c1), ω〉 = −∫S%/Γ

θ ∧ ω


So

m(M, g) = − 1

3π〈♣(c1), ω〉

as claimed.

222

We assumed:

223

We assumed:

•m = 2;

• s ≡ 0; and

• Complex structure J standard at infinity.

224

General case:

225

General case:

•General m ≥ 2: straightforward. . .

226

General case:


227

General case:


• s 6= 0, compensate by adding∫s dµ. . .

228

General case:



• If m > 2, J is always standard at infinity.

229

General case:




• If m = 2 and AE, J is still standard at infinity.

230

General case:





• If m = 2 and ALE, J can be non-standard at∞.

231

General case:






The last point is serious.

232

General case:






Example: Eguchi-Hanson.

233

General case:







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235

General case:







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236

General case:







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237

General case:






One argument proceeds by osculation:

238

General case:






One argument proceeds by osculation:

J = J0 + O(%−3), OJ = O(%−4)

in suitable asymptotic coordinates adapted to g.

239

To understand J at infinity:

240


Let M∞ be universal over of end M∞.

241



Cap off M∞ by adding CPm−1 at infinity.

242




Added hypersurface CPm−1 has normal bundleO(1).

243





Belongs to m-dimensional family of hypersurfaces.

244






Moduli space carries O projective structure

245







with many totally geodesic hypersurfaces.

246








So flat if m ≥ 3.

247








So flat if m ≥ 3.

When m = 2, Cotton tensor may be non-zero,

248








So flat if m ≥ 3.

When m = 2, Cotton tensor may be non-zero,

but ‘‘flatter” than might naively expect.

249


250


AE case:

Compactify M itself by adding CPm−1 at infinity.

251


AE case:


Linear system of CPm−1 gives holomorphic map

M → CPmwhich is biholomorphism near CPm−1.

252


AE case:




Thus obtain holomorphic map

Φ : M → Cm

which is biholomorphism near infinity.

253


AE case:




Thus obtain holomorphic map

Φ : M → Cm

which is biholomorphism near infinity.

This has some interesting consequences. . .

254

Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:

255


256


257


258


AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.

259


AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.

Moreover, m = 0 ⇐⇒

260


AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.

Moreover, m = 0⇐⇒ (M, g) is Euclidean space.

261


AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.

Moreover, m = 0⇐⇒ (M, g) is Euclidean space.

Proof actually shows something stronger!

262

Theorem E (Penrose Inequality). Let (M, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum

∑jnjDj of com-

pact complex hypersurfaces with positive integercoefficients, with the property that

⋃jDj 6= ∅

whenever (M,J) 6= Cm. In terms of this divi-sor, we then have

263

Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum

∑jnjDj of com-


⋃jDj 6= ∅


264


∑jnjDj of com-


⋃jDj 6= ∅


265


∑jnjDj of com-


⋃jDj 6= ∅


266


∑jnjDj of com-


⋃jDj 6= ∅


267


∑jnjDj of com-


⋃jDj 6= ∅


268


∑jnjDj of com-


⋃jDj 6= ∅


269


∑jnjDj of com-


⋃jDj 6= ∅


270


∑jnjDj of com-


⋃jDj 6= ∅


271


∑jnjDj of com-


⋃jDj 6= ∅


m(M, g) ≥ (m− 1)!

(2m− 1)πm−1

∑njVol (Dj)

272


∑jnjDj of com-


⋃jDj 6= ∅


m(M, g) ≥ (m− 1)!

(2m− 1)πm−1

∑njVol (Dj)

273


∑jnjDj of com-


⋃jDj 6= ∅


m(M, g) ≥ (m− 1)!

(2m− 1)πm−1

∑njVol (Dj)

274


∑jnjDj of com-


⋃jDj 6= ∅


m(M, g) ≥ (m− 1)!

(2m− 1)πm−1

∑njVol (Dj)

with = ⇐⇒

275


∑jnjDj of com-


⋃jDj 6= ∅


m(M, g) ≥ (m− 1)!

(2m− 1)πm−1

∑njVol (Dj)

with = ⇐⇒ (M, g, J) is scalar-flat Kahler.

276

This follows from existence of a holomorphic map

Φ : M → Cm

which is a biholomorphism near infinity.

277


Φ : M → Cm


Indeed, we then have a holomorphic section

ϕ = Φ∗dz1 ∧ · · · ∧ dzm

278


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm

of the canonical line bundle which vanishes exactlyat the critical points of Φ.

279


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm


280


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm


The zero set of ϕ, counted with multiplicities, givesus a canonical divisor

281


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm



D =∑

njDj

282


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm



D =∑

njDj

and

−〈♣(c1),ωm−1

(m− 1)!〉 =

∑njVol (Dj)

283


Φ : M → Cm



ϕ = Φ∗dz1 ∧ · · · ∧ dzm



D =∑

njDj

and

−〈♣(c1),ωm−1

(m− 1)!〉 =

∑njVol (Dj)

so the mass formula implies the claim.

284

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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285

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

286

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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287

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

288

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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Documents

Massin K ahler Geometry - Princeton University · Massin K ahler Geometry Claude LeBrun Stony Brook University 31st Annual Geometry Festival Princeton University, April 8, 2016 1