Lecture 24 - Einstein Manifolds VII - Epsilon regularity
May 6, 2010
1
Energy ratio improvement
R
n
1
We have called scale-invariant quantity VR B
| Rm | 2 the “energy ratio.” It is
Bp (r)
p (r)
convenient to modify this, and consider the quantity
Z
n
Volλ B(r)
| Rm | 2 .
Vol Bp (r) Bp (r)
Since we have Ric ≥ λg, this quantity is more useful in the use of relative volume comparison.
If we restrict ourselves to a ≤ 1 then these quantities are equivalent.
R
n
Lemma 1.1 Assume M n is an Einstein manifold, r ≤ 1, and Bp (r) | Rm | 2 < δ. Then
there exist numbers C < ∞, η > 0 so that either
Z
Z
Volλ B(r/2)
Volλ B(r/2)
2
| Rm | ≤ (1 − η)
| Rm |2
(1)
Vol Bp (r/2) Bp (r/2)
Vol Bp (r) Bp (r)
or else the annulus Bp (5r/8) − Bp (3r/8) has
| Rm | <
Vol Bp (r)
Vol Bp (r/2)
√
Cr−2 η
≥ (1 − η)
Volλ B(r)
.
Volλ B(r/2)
Volλ B(r/2)
Vol Bp (r)
Z
≥ (1 − η)
Volλ B(r/2)
≥ (1 − η)
Vol Bp (r)
Z
Pf
If (1) does not hold, then
Z
Volλ B(r/2)
| Rm |2
Vol Bp (r/2) Bp (r/2)
1
| Rm |2
Bp (r)
Bp (r/2)
| Rm |2 ,
and we have
Volλ B(r/2)
Vol Bp (r/2)
Z
| Rm |
λ
B(r/2)
≥ (1 − η) Vol
Vol Bp (r) . On the other hand
2
≤
Bp (r)−Bp (r/2)
!Z
1 Volλ B(r/2) Vol Bp (r)
− 1
| Rm |2 .
1 − η Volλ B(r) Vol Bp (r/2)
Bp (r/2)
Bishop-Gromov volume comparison gives
Z
Volλ B(r/2) Vol Bp (r)
Volλ B(r) Vol Bp (r/2)
| Rm |2 ≤
Bp (r)−Bp (r/2)
The Key Estimate now gives
Z
| Rm |2 ≤
Bp (r)−Bp (r/2)
η
1−η
Z
≤ 1, so therefore
| Rm |2 .
Bp (r/2)
η
Cr−4 (Vol Bp (r) − Vol Bp (r/2)) .
1−η
Now let q ∈ Bp (5r/8) − Bp (3r/8), so that Bq (r/8) ⊂ Bp (r) − Bp (r/2). We have
Z
Z
| Rm |2 ≤
| Rm |2
Bq (r/8)
Bp (r)−Bp (r/2)
≤
≤
≤
η
Cr−4 (Vol Bp (r) − Vol Bp (r/2))
1−η
η
Cr−4 Vol Bq (2r)
1−η
η
Volλ Bq (2r)
Cr−4 Vol Bq (r/8)
1−η
Volλ Bq (r/8)
so that
Volλ B(r/8)
Vol Bq (r/8)
Z
| Rm |2
≤
Bq (r/8)
η
Cr−4 Volλ Bq (2r).
1−η
With r ≤ 1 we have that there exists a C so that
Z
Volλ B(r/8)
| Rm |2
Vol Bq (r/8) Bq (r/8)
≤
η
C.
1−η
Now if η is chosen small enough that Cη/(1 − η) < 0 , then -regularity holds and we get
√
|Rmq | ≤ C r−2 η
Lemma 1.2 The second alternative in Lemma 1.1 does not hold.
2
Pf
The small curvature and almost-volume annulus together imply the existence of a
Cheeger-Colding function r̂ that has the following properties
4r̂2 = 8
|r̂ − r| ≤ Φ
Z
1
2
|∇r̂ − ∇r| ≤ Φ
|r̂−1 (a)| r̂−1 (a)
|∇r̂|
1 −
≤ C
−1 r̂ (a) ≤ Φ
|∂Bp (a)| Z
1
IIr̂−1 (a) − 1 gr̂−1 (a) ⊗ ∇r̂ ≤ Φ.
|r̂−1 (a)| r̂−1 (a) r̂
for some Φ = Φ(η) where limη→0 Φ = 0. We can pass to the universal covering space, where
the injectivity radius is bounded and we can take a limit. On this space the injectivity
radius is bounded, so it is possible to take a limit as η → 0. On the limit space the function
r̂ has ∇2 r = 1r̂ g, so the limit is a warped product with level sets of r̂ being space forms.
This gives C 1,α -convergence of r̂.
Therefore r̂−1 converges in the pointwise sense to a space form. Thus the annulus has
(almost) the metric structure of an annulus in a Euclidean cone.
Now consider again the Chern-Gauss-Bonnet theorem
Z
Z
χ(Bp (3r/4)) =
| Rm |2 +
T P χ.
∂Bp (3r/4)
The boundary term converges to the Euclidean boundary term, which is positive. Since the
left-hand side is negative due to the F-structure, we have a contradiction.
Theorem 1.3 (-regularity) If r ≤ 1 and
R
Bp (r)
| Rm |2 ≤ δ, then for some µ > 0
sup | Rm |2 ≤ Cr−2 .
Bp (µr)
Pf
The Key Estimate gives
Z
| Rm |2 ≤ Cr−4 |Vol Bp (r) − Vol Bp (r/2)|
Bp (r/2)
Volλ B(r/2)
Vol Bp (r/2)
Z
| Rm |2 ≤ C
Bp (r/2)
3
for some C. Lemmas 1.1 and 1.2 give
Volλ B(r2−k−1 )
Vol Bp (r2−k−1 )
Choosing k >
log(0 /C)
log(η)
Z
| Rm |2 ≤ Cη k .
Bp (r/2)
gives
Volλ B(r2−k−1 )
Vol Bp (r2−k−1 )
Z
| Rm |2 ≤ 0 ,
Bp (r/2)
whereupon standard -regularity goes through.
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