massin k ahler geometry - princeton university · massin k ahler geometry claude lebrun stony brook...
TRANSCRIPT
Mass in
Kahler Geometry
Claude LeBrunStony Brook University
31st Annual Geometry FestivalPrinceton University, April 8, 2016
1
Joint work with
Hans-Joachim HeinUniversity of Maryland
e-print: arXiv:1507.08885 [math.DG]
To appear in Comm. Math. Phys.
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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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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
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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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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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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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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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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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gjk = δjk + O(|x|1−n2−ε)
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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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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
13
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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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
14
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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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
15
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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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
16
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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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
17
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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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
18
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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gjk = δjk + O(|x|1−n2−ε)
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 = δjk + O(|x|1−n2−ε)
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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21
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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22
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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24
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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25
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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26
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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27
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 = δjk + O(|x|1−n2−ε)
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
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
30
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
31
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
32
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
33
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
where
34
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
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..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
35
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
36
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
37
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
38
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Seems to depend on choice of coordinates!
39
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
40
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
41
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
42
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
43
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
44
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
45
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Chrusciel-type fall-off:
gjk − δjk ∈ C1−τ , τ >
n− 2
2
46
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik-type fall-off:
gjk − δjk ∈ W2,q−τ , τ >
n− 2
2, q > n
47
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik-type fall-off =⇒
gjk − δjk ∈ C1,α−τ , τ >
n− 2
2.
48
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
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-from induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
49
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
52
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
53
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
54
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
55
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g = −(
1− 2m%n−2
)dt2+
(1− 2m
%n−2
)−1
d%2+%2hSn−1
56
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
57
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2⊂ C2 × CP1
58
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
59
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
60
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
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
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
61
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
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
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
62
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
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
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
63
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
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
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
64
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
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
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
also has mass m .
65
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
71
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
72
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
75
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
76
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n.)
77
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
78
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
79
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
80
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1986:
81
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1986:
ALE counter-examples.
82
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1986:
ALE counter-examples.
Scalar-flat Kahler metrics
83
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1986:
ALE counter-examples.
Scalar-flat Kahler metrics
on line bundles L→ CP1 of Chern-class ≤ −3.
84
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
87
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
88
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
89
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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..................................
..................................
..................................................
.....................
...............................
90
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
...............................................................................................................................................................................................................................................................................................................................
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..................................
..................................
..................................................
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n = 2m ≥ 4
91
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Several different proofs are known.
92
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Several different proofs are known.
Several are anlaytic:
93
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Several different proofs are known.
Several are anlaytic:
each end is pseudo-convex at infinity.
94
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Several different proofs are known.
Several are anlaytic:
each end is pseudo-convex at infinity.
Another is more topological:
95
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Several different proofs are known.
Several are anlaytic:
each end is pseudo-convex at infinity.
Another is more topological:
intersection form on H2 of compactification.
96
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
97
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Upshot:
98
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Upshot:
Mass of an ALE Kahler manifold is unambiguous.
99
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
102
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
103
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
104
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
105
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
106
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
107
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
108
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
109
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
110
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
111
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• 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
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.
114
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.
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
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.
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!
116
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.
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!
117
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.
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!
118
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.
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!
119
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.
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!
120
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.
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!
121
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.
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!
122
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.
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!
123
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.
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!
124
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
125
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
126
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
127
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
128
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
129
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
130
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.
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!
Theorem B. There are infinitely many topolog-ical types of ALE scalar-flat Kahler surfaces thathave zero mass, but are not Ricci-flat.
(Discovered independently by Rollin, Singer, & Suvaina,using different methods.)
131
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
133
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
134
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
135
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
136
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
137
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
138
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
139
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
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
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
141
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
142
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
Compactify M as M = M ∪ (Sn−1/Γ).
143
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
Compactify M as M = M ∪ (Sn−1/Γ).
Then Hpc (M) = Hp(M,∂M).
144
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
Compactify M as M = M ∪ (Sn−1/Γ).
Then Hpc (M) = Hp(M,∂M).
Exact sequence of pair:
145
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
Compactify M as M = M ∪ (Sn−1/Γ).
Then Hpc (M) = Hp(M,∂M).
Exact sequence of pair:
H1(∂M)→ H2(M,∂M)→ H2(M)→ H2(∂M)
146
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
If just one end:
Compactify M as M = M ∪ (Sn−1/Γ).
Then Hpc (M) = Hp(M,∂M).
Exact sequence of pair:
H1(Sn−1/Γ)→ H2c (M)→ H2(M)→ H2(Sn−1/Γ)
147
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
148
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
149
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
150
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
151
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
152
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
153
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
154
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
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
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)ofcomplex dimension m has mass given by
157
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
158
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
159
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
160
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
161
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
162
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
163
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
164
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
165
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
166
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
167
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
168
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
169
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).
170
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
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
177
We can now state our mass formula:
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
We can now state our mass formula:
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:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
184
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
185
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
186
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
187
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
188
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
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)
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:≡ 0 away from end,≡ 1 near infinity.
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198
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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199
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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200
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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201
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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202
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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203
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radius
M%
%
1
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204
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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radius
M%
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1
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205
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
Compactly supported, because dθ = ρ near infinity.
208
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
209
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫Mψ ∧ ω
210
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
ψ ∧ ω
where M% defined by radius ≤ %.
211
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
ψ ∧ ω
212
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
213
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫M%
d(fθ ∧ ω)
215
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫∂M%
fθ ∧ ω
by Stokes’ theorem.
216
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫∂M%
θ ∧ ω
by Stokes’ theorem.
217
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
218
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
219
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
220
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), ω〉
221
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), ω〉
as claimed.
222
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
229
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
230
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
231
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
The last point is serious.
232
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Example: Eguchi-Hanson.
233
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Example: Eguchi-Hanson.
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234
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Example: Eguchi-Hanson.
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235
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Example: Eguchi-Hanson.
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236
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Example: Eguchi-Hanson.
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237
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
238
General case:
•General m ≥ 2: straightforward. . .
• s 6= 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
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:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
242
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
243
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
244
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
Moduli space carries O projective structure
245
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
Moduli space carries O projective structure
with many totally geodesic hypersurfaces.
246
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
Moduli space carries O projective structure
with many totally geodesic hypersurfaces.
So flat if m ≥ 3.
247
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
Moduli space carries O projective structure
with many totally geodesic hypersurfaces.
So flat if m ≥ 3.
When m = 2, Cotton tensor may be non-zero,
248
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Belongs to m-dimensional family of hypersurfaces.
Moduli space carries O projective structure
with many totally geodesic hypersurfaces.
So flat if m ≥ 3.
When m = 2, Cotton tensor may be non-zero,
but ‘‘flatter” than might naively expect.
249
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
M → CPmwhich is biholomorphism near CPm−1.
252
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
M → CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
253
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
M → CPmwhich is biholomorphism near CPm−1.
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
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
256
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
257
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
258
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
259
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0 ⇐⇒
260
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
261
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
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-
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
264
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-
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
265
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-
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
266
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-
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
267
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-
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
268
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-
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
269
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-
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
270
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-
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
271
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-
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
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
272
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-
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
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
273
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-
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
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
274
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-
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
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒
275
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-
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
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
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
278
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
279
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
280
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
281
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
282
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
283
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
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
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
287
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................. .....................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................................
.....................
...............................
289