- DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 383
we obtainfl f e-ar2(x)77 (x) [\7f12 (x, t) dμ (x) dt
lo 1B(0,2R)
1
11 ear^2 (x) 2
S 2C + -( -) 1\7 (e-ar2
(x)77 (x)) I f^2 (x, t) dμ (x) dt.
O B(0,2R) 77 X
Now since f (x, t) :::; r (x) + C and1\7 ( e-ar2(x)77(x))12 S 2e-2ar2(x) ( 4a2r2 (x) 772 (x) + [\777[2 (x))
(note that [\7r[ :::; 1 a.e.), we havefl f ear2(x) I 2 12
lo 1B(0,2R) 77 (x) \7 ( e-ar (x)77 (x)) f2 (x, t) dμ (x):::; f
1f 2 (r (x) + C)^2 e-ar
2
(x) (4a^2 r (x)^2 77 (x) + [\7^77 [2
(x)) dμ (x),
lo 1B(0,2R) 77 (x)
which, for a sufficiently large depending only on n and K, is bounded in-
dependent of the RE [1, oo) used in the definition of 77 (where we used the
volume comparison theorem).
Step 3. There exist constants Cn,K,2 < oo depending only on n and K
such that
[\7\7 f[ (x, 1) S Cn,K,2for all x EM.
Given any point x E M, we have that
(26.146) expx: B ( 0, p/VK) -+ B ( x, p/VK)
is a local diffeomorphism, where p > 0 is a universal constant and where
B ( 0, p/VK) C TxM is the ball ofradius p/VK centered at 0 with respect
to the inner product g (x). We consider the pulled-back metric
(26.147)on B (o,p/VK). Define
j: iJ (o,p/VK)-+ IR
by
J (V, t) ~ f (expx (V), t) - u (x).
Note that from Step 1,
If (O,t)I = [f(x,t)-u(x)[:::; C1