Margulis Lemma

24 VITALI KAPOVITCH AND BURKHARD WILKING
assume that by volume 3/4 of the points in B r/10 (q m ) are mapped by φ 1 to points
in B r/9 (p).
We choose such a point q ∈ B r/10 (q m ). In view of (17) we may assume in
addition that
∫ 1
By Lemma 3.7 this implies
0
Mx(‖∇·X‖ 3/2 ) 2/3 (φ t (q)) dt ≤ 4 3 C 6(L)rε i .
∫
(18) − dt r (1)(x, y) dµ(x) dµ(y) ≤ C 7 (L)r · ε i .
B r(q)×B r(q)
Note that by definition, dt r (1)(x, y) ≥ min{r, |d(φ 1 (x), φ 1 (y)−d(x, y)|}. Combining
this inequality with the knowledge that 3/4 of the points in B r/10 (q m ) end up in
B 1/9r (p), implies that we can find a subset B r/2 (q m ) ′ ⊂ B r/2 (q m ) with
(19)
(20)
vol(B r/2 (q m ) ′ ) ≥ (1 − C 8 (L)ε i r) vol(B r/2 (q m )) and
φ 1 (B r/2 (q m ) ′ ) ⊂ B r (p).
Set B r/2 (q m ) ′′ := B r/2 (q m ) ′ ∩ φ −1
1 (B r(p) ′ ). Then we get
φ 1 (B r/2 (q m ) ′′ ) ⊂ B r (p) ′
and
vol(B r/2 (q m ) ′′ ) ≥ vol(B r/2 (q m ) ′ ) − (vol B r (p) − vol B r (p) ′ )
(21)
by (19) and (11)
≥ (1 − C 8 (L)rε i ) vol B r/2 (q m ) − ¯C 1 rε i vol B r (p)
by (12)
≥ (1 − C 8 (L)rε i ) vol B r/2 (q m ) − 2 k+1 ¯C1 rε i vol B r/2 (q m )
≥ (1 − (C 8 (L) + 2 n ¯C1 )rε i ) vol B r/2 (q m ),
where we used that φ 1 is volume-preserving in the first inequality.
Using the induction assumption, we can prolong each integral curve from any
q ∈ B r/2 (q m ) ′′ to a point x ∈ B r (p) ′ by extending it by a previously constructed
integral curve from x to p ′ ∈ B Lρi (p) of a vector field Xold t (which depends on p′ ).
We set Xnew t = 2Xold 2t for 0 ≤ t ≤ 1/2 and Xt new = −2X for 1/2 ≤ t ≤ 1. Let c q (t)
be the integral curve of Xnew t with c q (1) = q and c q (0) = p ′ .
By the induction assumption and (17), we get
(22)
∫B r/2 (q m) ′′ ∫ 1
0
(Mx ‖∇·X t new‖ 3/2 ) 2/3 (c q (t)) dt dµ(q) ≤
≤ C 6 (L)(rε i ) vol B r/2 (q m ) + ¯C 2 (rε i ) vol B r (p)
(12)
≤ C 6 (L)(rε i ) vol B r/2 (q m ) + 2 n ¯C2 (rε i ) vol B r/2 (q m )
= ( C 9 (L) + 2 n ¯C2
)
(rεi ) vol B r/2 (q m ).
Recall that the balls B r/2 (q m ) cover B Lr (p). We put
B Lr (p) ′ := B Lr (p) ∩ ⋃ m
B r/2 (q m ) ′′ .