300 B. SOME RESULTS IN COMPARISON GEOMETRY
COROLLARY B.36. If (Mn,g) is a complete Riemannian manifold withsectional curvatures bounded above by K > 0, then for any p E Mn,
~ 7f
inj _g(O) 2: VK.2.3. A Laplacian comparison theorem for distance functions.
Given any smooth function f on (Mn,g), the Bochner-Weitzenbock for-
mula says11::::. IY' fl^2 = IY'Y' fl^2 + (\7 f, \7 (!::::.!))+Re (\7 f, \7 f).
This formula is of fundamental importance and is used in the proof of the
Li-Yau gradient estimate [91] for the first eigenfunction and the Li- Yau dif-
ferential Harnack inequality [92] for positive solutions of the heat equation,
among many other results. It is easily proved using Ricci calculus:
1 !:::. IV' !1^2 = 1 'Vi 'Vi (Y' jf'Vj f) = 'Vi (Y' j 'Vi f'Vj f)
= 'Vi'Vj'Vif'Vj f + 'Vi'Vjf'Vi'Vj f
= \7 j \7 i 'Vi f'Vj f + Rjk 'Vj f'Vk f + \7 i \7 j f'Vi'Vj f= (\7 f::::.f, \7 f) +Re (\7 f, \7 f) + IY'Y' fl^2.
We say r : Mn ---+ [O, oo) is a generalized distance function if(B.l) IY'rl2 = 1
at all points where r is smooth. Generalized distance functions have the
property that the integral curves of 'Vr are geodesics.LEMMA B.37. If r is a generalized distance function, then wherever r is
smooth,
Y''Vr ('Vr) = 0.PROOF. Differentiating (B.l) shows that for all i = 1, ... , n,
2..
0 ='Vi IY'rl ='Vi (Y'jr'VJr) = 2\i'Jr'Vj'Vir = 2 (Y''Vr 'Vr)i.COROLLARY B.38. Wherever r is smooth,IY'Y'rl2 2: _ 1_ (!::::.r)2.
n-1
PROOF. \7\i'r has a zero eigenvalue, because
(\7\i'r) ('Vr, 'Vr) = \7ir\7jr\7i'Vjr = (V'r, 'V'Vr Y'r) = 0.Hence the standard estimate becomes
(!::::.r)^2 =(tr 9 \7\7r)^2 :::; (n - 1) IY'Y'rl^2.
DD