Special Geometry, Black Holes and Instantons - Fachbereich Mathematik
CURVATURE AND INJECTIVITY RADIUS ESTIMATES …Gravitational instantons for n= 4. Before describing...
Transcript of CURVATURE AND INJECTIVITY RADIUS ESTIMATES …Gravitational instantons for n= 4. Before describing...
CURVATURE AND INJECTIVITY
RADIUS ESTIMATES FOR EINSTEIN
4-MANIFOLDS
Jeff Cheeger
Courant Institute
April 10, 2018
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS1 / 36
Introduction.
We will review work with Gang Tian from 2005 on possibly
collapsed Einstein 4-manifolds M4, with say |RicM4| ≤ 3.
Key is an ε-regularity theorem yielding a uniform curvature
bound on B 12(p), if the L2-norm of the curvature on B1(p)
is at most ε, where ε > 0 is independent of the collapsing.
Dimension 4 enters the proof only via the positivity of the
Gauss-Bonnet form.
However, in [Ge,Jiang 2017] it is noted that the ε-regularity
theorem doesn’t extend to higher dimensions.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS2 / 36
ε-regularity in the possibly collapsed case.
Let R denote the Riemann curvature tensor.
Theorem ([Ch,Ti 2005]) There exists c, ε > 0 such that if
M4 is Einstein, |RicM4| ≤ 3, Br(p) ⊂M4, r ≤ 1, and∫Br(p)
|R|2 ≤ ε ,(1)
then
supBr/2(p)
|R| ≤ cr−2 .(2)
Note that the integral in (1) is not normalized by volume.
That case is due to Anderson, Tian, Nakajima ∼ 1990.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS3 / 36
Remarks.
Remark. Br/2(p) may be collapsed with bounded
curvature but not homeomorphic to B1(0n) ⊂ Rn.
If not, by [Ch,Fukaya,Gromov 1992], there is a
neighborhood containing a ball of a definite radius which
looks like a tube around an infranil manifold.
Remark. In [Ge,Jiang 2017] the ε-regularity theorem is
extended to 4-dimensional gradient shrinking Ricci solitons.
Similar results are due independently to S. Huang.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS4 / 36
General dimensions and bounded Ricci curvature.
From now on, Mn will denote a complete riemannian
manifold.
For n 6= 4, when we assume |RicMn| ≤ n− 1, we say so.
Our main results for n = 4 have direct generalizations to
the case in which the Einstein condition is dropped.
In the conclusions, sectional curvature bounds are replaced
by C1,α bounds on the metric in harmonic coordinates, for
all α < 1, as well as L2,p curvature bounds for all p <∞.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS5 / 36
Collapse implies L2 concentration of curvature.
Theorem. ([Ch,Ti 2005]) There exists v > 0, β, c, such
that if M4 is complete, Einstein, with |RicM4| ≤ 3,∫M4
|R|2 ≤ C ,
and for all p,Vol(Bs(p))
s4≤ v ,
then there exist p1, . . . , pN , such that
N ≤ βC ,(3)
∫M4\(
⋃iBs(pi))
|R|2 ≤ c
(∑i
Vol(Bs(pi))
s4+ lim
r→∞
Vol(Br(p))
r4
).
(4)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS6 / 36
Indication of proof.
The balls Bs(pi) are those on which the curvature
concentrates.
For the proof, apply Chern-Gauss-Bonnet to the region
M4 \ (⋃i
Bs(pi)) .
More precisely, one must first straighten out the boundary
via the Equivariant Good Chopping Theorem; see below.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS7 / 36
The case RicM4 = ±3g.
Note that ifRicM4 = ±3g ,
and ∫M4
|R|2 ≤ C ,
then
Vol(M4) ≤ C
6.
In this case, the last term on the r.h.s. of (4) vanishes.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS8 / 36
Margulis lemma type result in dimension 4.
Theorem. There exists w(C) > 0 such that if
RicM4 = ±3g and ∫M4
|R|2 ≤ C .(5)
Then for some p ∈M4,
Vol(B1(p))
Vol(M4)≥ w(C) .(6)
Proof. If RicM4 = 3g, a stronger result follows from
Meyers’ theorem and the Bishop-Gromov inequality.
If RicM4 = −3g, and diam(M4) ≤ d(C), then the same
argument applies.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS9 / 36
The case diam(M 4) ≥ d(C).
By the Bishop-Gromov inequality and N ≤ βC, if
diam(M4) ≥ d(C), there exists ρ(C) > 0 with
Vol(M4 \⋃i
Bs(pi)) ≥ ρ(C) · Vol(M4) .(7)
Thus, the l.h.s. of (4) is ≥ 6ρ(C) · Vol(M4).
By (3), the r.h.s. of (4) is ≤ cβC · v.
Thus,6ρ(C) · Vol(M4) ≤ cβC · v .
Cross multiplying gives (6).
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS10 / 36
Exponential decay of volume.
Theorem. There exist β, γ > 0, c, such that if M4 is
complete Einstein satisfying, RicMn = −3 and∫M4
|R|2 ≤ C ,
then there exist p1, . . . , pN , with
N ≤ βC ,
such that for r ≥ 5,
Vol(M4 \⋃i
Br(pi)) ≤ c · C · e−γr .
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS11 / 36
Gravitational instantons for n = 4.
Before describing the proof of the ε-regularity theorem we
mention the following recent related results.
There is a classification of complete Ricci flat hyper-Kahler
4-manifolds with faster than quadratic curvature decay
given by Gao Chen and Xiuxiong Chen.
For this, see [Chen,Chen 2015 I, II], [Chen,Chen 2016 III].
Remark. [Ch,Ti 2005] should eventually lead to further
general classification results for Einstein 4-manifolds.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS12 / 36
Positivity and Chern-Gauss-Bonnet, n = 4.
Let ωn denote the volume form for some local orientation.
If n = 4 the Chern-Gauss-Bonnet form, Pχ, satisfies
Pχ =1
8π2· |R|2 · ωn .(8)
In particular, for M4 compact,
1
8π2
∫M4
|R|2 = χ(M4) .(9)
Remark. As mentioned above, the remaining ingredients
in the proof are valid in any dimension.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS13 / 36
The curvature radius.
Definition. The curvature radius, r|R|(p) > 0, is the
supremal s such that
supBs(p)
|R| ≤ s−2 .(10)
In particular,
|R(p)| ≤ (r|R|(p))−2 .(11)
It follows easily that either R ≡ 0 and r|R| ≡ ∞ or the
function r|R| is 1-Lipschitz and hence, it varies moderately
on its own scale.
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Explanations concerning the curvature radius.
On the ball Br|R|(p)(p) with rescaled metric (r|R|(p))−2 · g,
the curvature is bounded in absolute value by 1.
Note that the rescaled ball has radius 1.
The fact that r|R| varies moderately on its own scale
enables one to generalize global constructions from the
theory of bounded curvature to ones on the scale of r|R|.
In these constructions, structures on overlapping regions
have to be perturbed slightly in order to match.
This applies to F -structures and N -structures.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS15 / 36
Collapse with locally bounded curvature.
Definition. U is v-collapsed with locally bounded
curvature if for all p in U ,
Vol(Br|R|(p)(p)) ≤ v · (r|R|(p))n .(12)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS16 / 36
Existence of N -structures.
By [Ch,Fu,Gv 1992], if v ≤ t(n), there exists
V ⊃ U , which carries an N-structure of positive rank.
Thus, V is the disjoint union of nilmanifolds, O, of
positive dimension > 0 called orbits.
Consequently, V has vanishing Euler characteristic,
χ(V ) = 0 .
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS17 / 36
Good chopping.
Let Tr(K) denote the r-tubular neighborhood of K ⊂Mn.
Put Ar1,r2(K) := Tr2(K) \ Tr1(K).
Theorem. ([Ch,Gv 1990], [Ch,Ti 2005]) There exists a
submanifold with boundary, Zn, such that
T 13r(K) ⊂ Zn ⊂ T 2
3r(K) ,(13)
Vol(∂Z) ≤ r−1 · Vol(A 13r, 2
3r(K) ,(14)
|II∂Z(z)| ≤ c(n) · (r·r|R|)−1) .(15)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS18 / 36
Equivariant good chopping.
Let K ⊂Mn be closed.
If Tr(K) is ε(n) collapsed with locally bounded curvature,
Zn can be chosen to be a union of orbits of the associated
N -structure and so,
χ(Zn) = 0 .(16)
Below, we use the notation,
Aa,b(K) := Tb(K) \ Ta(K) .
II∂Zn denotes the second fundamental form of ∂Zn.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS19 / 36
For the Chern-Gauss-Bonnet boundary term.
Theorem. There exists c = c(n) <∞, and Zn with
T 13r(K) ⊂ Zn ⊂ T 2
4r(K) ,
|II∂Zn| ≤ c · (r−1 + (r|R|)−1) ,
and for all k1, k2 > 0,
∫∂Zn|II∂Z |k1 · |R|k2 ≤ c
r·∫A 1
3 r,23 r
(K)
(r−(k1+2k2) + (r|R|)
−(k1+2k2)).
(17)
The proof uses a covering argument and Lip r|R| ≤ 1.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS20 / 36
Chern-Gauss-Bonnet in the locally collapsed case.
∣∣ ∫∂Zn
TPχ∣∣ ≤ c(n)r−1 ·
∫A 1
3 r,23 r
(K)
(r−(n−1) + (r|R|)
−(n−1)) .(18)
Since χ(Zn) = 0, we get
∣∣ ∫∂Zn
Pχ∣∣ ≤ c(n)r−1 ·
∫A 1
3 r,2r (K)
(r−(n−1) + (r|R|)
−(n−1)) .(19)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS21 / 36
|R|Ln/2 sufficiently small with respect to volume.
Let p ∈Mn−1, simply connected curvature ≡ −1.
Theorem (Anderson) There exists τ(n) such that if Mn is
Einstein, |RicMn| ≤ n− 1 , and
Vol(Br(p))
Vol(Br(p))·∫Br(p)
|R|n2 ≤ τ(n) ,(20)
then
supBr/2(p)
|R| ≤ c · r−2 ·(
Vol(Br(p))
Vol(Br(p))·∫Br(p)
|R|n2
) 2n
.(21)
Indication of proof. The proof uses Moser iteration.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS22 / 36
The local scale ρ(p).
If (20) holds for r = 1 put ρ(p) := 1.
Otherwise, define ρ(p) to be the largest solution of
Vol(Bρ(p)(p))
Vol(Bρ(p)(p))·∫Bρ(p)(p)
|R|n2 := τ(n) .(22)
By (21), we have
1
2ρ(p) ≤ r|R|(p) .(23)
If ρ(p) = 1, then
supB1/2(p)
|R| ≤ 4 .(24)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS23 / 36
The local scale and the maximal function.
ρ(p) is not a priori 1-Lipschitz, so it can’t be used directly.
We address this issue via the maximal function.
Definition. For (X,µ) a metric measure space put∫−A
|f | dµ :=1
µ(A)
∫A
|f | dµ .
Define the maximal function for balls of radius ≤ r,
Mf (x, r) := sups≤r
∫−Bs(x)
|f | dµ .
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS24 / 36
Slightly nonstandard maximal function estimate.
Lemma. If for x ∈ W , s ≤ 4r,
µ(B2s(x)) ≤ 2κµ(Bs(x)) ,
then for all α < 1,
(∫W
−Mf (x, r)α d µ
) 1α
≤ c(κ, α) · 1
µ(W )·∫T6r(W )
|f | dµ .(25)
Indication of proof. Similar to that of more standard
maximal function estimates.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS25 / 36
Bounding the local scale from below.
SinceVol(Bs(p)) ≤ c(n)sn (s ≤ 1) ,
we get
ρ(p)−1 ≤ c ·max(M|R|n2(p, s)
1n , s−1) .(26)
Thus, with Bishop’s inequality,
(r|R|(p))−(n−1) ≤ c ·
(s−(n−1) + (M|R|
n2(p, s))
n−1n
)(27)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS26 / 36
Chern-Gauss-Bonnet boundary estimate.
By (17) and (27) for s ≤ r ≤ 1,
∣∣ ∫∂Zn
TPχ∣∣ ≤ c(n)r−1 ·
∫A 1
3 r,23 r
(K)
(s−(n−1) + (M|R|
n2)n−1n
).
(28)
Choosing s = r512
, we get from (25), (28),
(Vol(A0,r(K))−1 ·∣∣∫∂Zn
TPχ∣∣
≤ c(n)
r−(n−1) + r−1
1
Vol(A0,1(K)
∫A 1
4 r,34 r
(K)
|R|n2
n−1n
.
(29)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS27 / 36
ε-regularity in the collapsed case, n = 4.
Recall, Pχ = 18π2 · |R|2.
Also, t-collapse with locally bounded curvature, t = t(4),
implies the existence of an N -structure.
So, if T1(E) is t-collapsed with locally bounded from
equivariant good chopping, we have by ((16)), χ(Zn) = 0,
and by (29), we get for c independent of M4,
(30)
Vol(E)
Vol(A0,1(E))·∫−E
|R|2 ≤ c·
1 +
1
Vol(A0,1(E))
∫A 1
4 ,34(E)
|R|2 3
4
.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS28 / 36
xi+1 ≤ ai + bi · xαi+1, i = 2, · · · .Similarly, short argument (using Lip r|R| ≤ 1), shows∫A
2−(i−1),1−2−(i−1) (E)
|R|2
≤ c24i · Vol(A0,1(E)) ·
1 +
(1
Vol(A0,1(E))
∫A2−i,1−2−i (E)
|R|2) 3
4
(31)
The exponentially growing factor 24i arises because we are
applying equivariant good chopping in the narrow region
A0,2−i(A2−(i−1),1−2−(i−1)(E)) .
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS29 / 36
Lemma on sequences.
Lemma. Let 0 ≤ α < 1, for i = 2, 3, . . . ,, ai, bi, xi be
nonnegative real numbers satisfying
xi ≤ ai + bi · xαi+1 ,
limi→∞
xαi
i = 1 ,
max(ai, bi) ≤ c ·Ki (K ≥ 1) .
Then
x2 ≤ (2c)1+α+α2+··· ·K1+α+2α2+3α3+··· .(32)
Proof. It follows by (upward) induction from
xαi
i ≤ 2 max(aαi
i , bαi
i · xαi+1
i+1 ) .
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS30 / 36
The key estimate n = 4.
Theorem. There exist δ > 0, c > 0, such that if (1), (2)
hold and E ⊂M4 denotes a bounded open subset such that
T1(E) is t-collapsed with∫B1(p)
|R|2 < δ (for all p ∈ T1(E)) .(33)
Then ∫E
|R|2 ≤ c · Vol(A0,1(E)) .(34)
Proof. Combine (31) with the lemma on sequences.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS31 / 36
Proof of ε-regularity for n = 4.
Step 1. (Getting in range) There exists ε, c1 > 0 such that
if |RicM4| ≤ 3 and for r ≤ 1,∫Br(p)
|R|2 ≤ ε ,
then for p ∈M4−3,
Vol(Br(p))
Vol(Br(p))·∫Br(p)
|R|2 ≤ c1 .(35)
However, to apply Anderson’s ε-regularity theorem
directly, we would need c1 ≤ ε.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS32 / 36
The decay estimate.
Step 2. It suffices to show that there exist γ < 1, β > 0,
such that (35) implies
Vol(Bβ(p))
Vol(Bβ(p))·∫|R|2 ≤ γ ·
Vol(Br(p)
Vol(Br(p))·∫|R|2(36)
We argue by contradiction.
If not, after rescaling, by Anderson’s ε-regularity theorem,
we get a sequence of balls, {B1/n(pi)} such that
supA1/n.1(pi)
|R| → 0 .(37)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS33 / 36
Continued.
Also, using the volume cone implies metric cone theorem
and local bounded covering geometry, the rescalings by a
factor n of the annuli {A 1n, 2n(pi)} have finite normal
coverings converging to a flat noncollapsed annulus.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS34 / 36
Continued.
By a refinement of equivariant good chopping, we get
almost conical Z4 with B 1n(pi) ⊂ Z4
i ⊂ B 2n(pi), such that
χ(Z4i ) = 0 ,(38)
∫∂Z4
i
TPχ > 0 ,(39)
∫Z4i
Pχ > 0 .(40)
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS35 / 36
Proof of ε-regularity concluded.
By (38)–(40) we contradict Chern-Gauss-Bonnet.
This completes the proof of the ε-regularity theorem.
Note that above, we can and do assume 6≡ 0.
Also, the refined equivariant good chopping leading to the
positivity in (39) relies on local bounded covering geometry
and convergence to the flat case.
Note that terms in both (39) and (40) are strictly positive,
though they may be arbitrarily small.
Jeff CheegerCURVATURE AND INJECTIVITY RADIUS ESTIMATES FOR EINSTEIN 4-MANIFOLDS36 / 36