- DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 377
Given r > 0, t E (-oo, oo ), and O E (0, 1), we define the following
parabolic regions in M x ( -oo, oo):Q-=.. B(O ' or) x ( t---r^3 + 4 0 2 ' t---r 3 - 4 0 2) '
Q'_ ~ B(O, or) x (t -r^2 , t -
3
~ o r^2 ),
Q+ ~ B(O, or) x (t -l: o r^2 , t).
We have the following.THEOREM 26.4 7 (Weak Harnack inequality). Let (Mn, g) be a Riemann-ian manifold satisfying the properties above. Then there exist Po > 0 and
A < oo such that for any weak supersolution to the heat equation u with
u(x, t) > 0 a.e. in B(O, r) x (t - r^2 , t), where 0 < r < R, we have
1
(26.126) ( VoltQ~) l, u!"' dμ)"' <; A'~!u.We refer the reader to the original articles by Moser [135] and Saloff-
Coste [165] for the proof, which uses Moser iteration (compare with §1 of this
chapter). There is an excellent exposition in the corresponding elliptic case
by Li [115]. Moreover, the result above also holds for uniformly parabolic
operators in divergence form. An important consequence of Theorem 26.4 7
is the Harnack estimate for the heat equation.
THEOREM 26.48 (Harnack estimate). Under the same assumptions asin Theorem 26.41, there exists A < oo such that for any weak solution u to
the heat equation with u(x, t) > 0 a.e. in B(O, r) x (t - r^2 , t), 0 < r < R,
we have
(26.127) supu::::; Ainf u.
Q- Q+
We remark that for classical solutions to the heat equation on a manifold
with Re~ 0, the Li-Yau differential Harnack estimate implies a sharp form
of the Harnack estimate above.
4. Distance-like functions on complete noncompact manifolds
In this section we discuss the existence of distance-like functions with
uniform bounds on all higher covariant derivatives on complete noncompact
manifolds with bounded curvature. Such functions are useful in constructing
barrier functions used in applying the maximum principle on noncompact
manifolds (see for example §2 of Chapter 12 in Part II).^9
The following is Theorem 1 in Tam [177].
(^9) We also used such a function in §2 of the present chapter.