SPACES OF HARMONIC FUNCTIONS

Abstract

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 V_x(r) is the volume of the geodesic ball B_x(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.