It is important and interesting to study harmonic functions on a Riemannian
manifold. In an earlier work of Li and Tam [21] it was demonstrated that the
dimensions of various spaces of bounded and positive harmonic functions are closely
related to the number of ends of a manifold. For the linear space consisting of all
harmonic functions of polynomial growth of degree at most d on a complete
Riemannian manifold Mn of dimension n, denoted by
[Hscr ]d(Mn), it was proved by Li and
Tam [20] that the dimension of the space [Hscr ]1(M)
always satisfies dim[Hscr ]1(M) [les ] dim[Hscr ]1(ℝn)
when M has non-negative Ricci curvature. They went on to ask as a
refinement of a conjecture of Yau [32] whether in general
dim [Hscr ]d(Mn) [les ] dim[Hscr ]d(ℝn)
for all d. Colding and Minicozzi made an important contribution to this question in
a sequence of papers [5–11] by showing among other things that
dim[Hscr ]d(M) is finite
when M has non-negative Ricci curvature. On the other hand, in a very remarkable
paper [16], Li produced an elegant and powerful argument to prove the following.
Recall that M satisfies a weak volume growth condition if, for some constant A
and ν,
formula here
for all x ∈ M and r [les ] R, where
Vx(r) is the volume of the geodesic ball
Bx(r) in M;
M has mean value property if there exists a constant B such that, for any non-
negative subharmonic function f on M,
formula here
for all p ∈ M and r > 0.