- DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 381
Applying the relative volume comparison theorem, which says-----,--VolB(x,s) < VolKB(s) < exp (c g-s)
Vol B (x,Vt) - VolKB(Vt) - Vtfor some Cg < oo depending only on n and K (this holds since s 2:: 1 2:: t),
we have2 { H(x,y,t)d(x,y)dμ(y)
jM-B(x,1)::; Csrn/^4 - exp --+Cg- ds
100
s ( 82 s )
1 4t 8t Vt
(26.141)where C10 < oo depends only on n and K.
REMARK 26.52. If we assume that t E (0, c:], then the corresponding
bound C10 tends to zero as E--+ 0.
Hence, by applying this and (26.135) to (26.134), we obtainIf (x, t) - u (x)I :'S C11,where Cn < oo depends only on n and K. This proves (26.133) and hence
completes the proof of Step 1.
Step 2. There exists Cn,K,1 E [1, oo) depending only on n and K such
that
(26.142) IV' f I (x, t) :'S Cn,K,1
for all x E M and t E (0, l].
First observe that(26.143)We have (see Exercise 2.20 in [45])
Hence
(26.144):t IV' f 12 = ~IV' f 12 - 2l\7\7f1^2 - 2 Re (\7 f, \7 J)
:'S ~IV' fl^2 + 2 (n - 1) K IV' fl^2.
We would like to apply the maximum principle to the above equation;
the issue is the noncompactness of M. Note that IV' JI (x, 0) = IV'ul (x) :'S 2.