Margulis Lemma

16 VITALI KAPOVITCH AND BURKHARD WILKING
Assume that Xi
t is divergence free on B r i+100(p i ) and that B ri+100(p i ) is compact.
Put
u s,i (x) := ( Mx ‖∇·Xi s ‖ α) 1/α
and suppose
∫ 1
0
u t,i (c i (t)) = ε i → 0.
Let f i = φ i1 be the flow of X t i evaluated at time 1. Then for all λ i → ∞
f i : (λ i M i , c(0)) → (λ i M i , c(1))
has the zooming in property. Moreover, for any lift ˜fi : ˜Mi → ˜M i of f i to the
universal cover ˜Mi of M i and for any lift ˜p i ∈ ˜M of c(0) = p i the sequence
˜f i : (λ i ˜Mi , ˜p i ) → (λ i ˜Mi , ˜f i (˜p i )) has the zooming in property as well.
The proposition remains valid if the assumption on Xi
t being divergence free
is removed. However, the proof is easier in this case and we do not have any
applications of the more general case. We will need the following
Lemma 3.7. There exists (explicit) C = C(n) such that the following holds. Suppose
(M n , g) has Ric ≥ −1 and X t is a vector field with compact support, which
depends on time (piecewise constant). Let c(t) be the integral curve of X t with
c(0) = p 0 ∈ M and assume that X t is divergence free on B 10 (c(t)) for all t ∈ [0, 1].
Let φ t be the flow of X t . Define the distortion function dt r (t)(p, q) of the flow
on scale r by the formula
{
}
dt r (t)(p, q) := min r, max ∣ d(p, q) − d(φ τ (p), φ τ (q))| .
0≤τ≤t
Put ε := ∫ 1
0 Mx(‖∇·X t ‖)(c(t)) dt. Then for any r ≤ 1/10 we have
∫
−
dt r (1)(p, q) dµ(p)dµ(q) ≤ Cr · ε
B r(p 0)×B r(p 0)
and there exists B r (p 0 ) ′ ⊂ B r (p 0 ) such that
vol(B r(p 0) ′ )
vol(B r(p 0))
≥ (1 − Cε) and φ t (B r (p 0 ) ′ ) ⊂ B 2r (c(t)) for all t ∈ [0, 1].
Proof. We prove the statement for a constant in time vector field X t . The general
case is completely analogous except for additional notational problems.
Notice that all estimates are trivial if ε ≥ 2 C
. Therefore it suffices to prove the
statement with a universal constant C(n) for all ε ≤ ε 0 . We put ε 0 = 1/2C and
determine C in the process. We again proceed by induction on the size of r.
Notice that the differential of φ s at c(0) is bilipschitz with bilipschitz constant
e R s
0 ‖∇·X‖(c(t))dt ≤ 1 + 2ε. Thus the Lemma holds for very small r.
Suppose the result holds for some r/10 ≤ 1/100. It suffices to prove that it
then holds for r. By induction assumption we know that for any t there exists
B r/10 (c(t)) ′ ⊂ B r/10 (c(t)) such that for any s ∈ [−t, 1 − t] we have
and
vol(B r/10 (c(t)) ′ ) ≥ (1 − Cε) vol(B r/10 (c(t))) ≥ 1 2 vol(B r/10(c(t)))
φ s (B r/10 (c(t)) ′ ) ⊂ B r/5 (c(t + s)),