Spaces of Harmonic Functions

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 M n of dimension n , denoted by H d ( M n ), it was proved by Li and Tam [ 20 ] that the dimension of the space H 1 ( M ) always satisfies dim H 1 ( M ) ⩽ dim H 1 (R n ) 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 H d ( M n ) ⩽ dim H d (R n )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 H 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, 1.1V x ( R ) ⩽ A( R r ) ν V x ( r )for all x ∈ M and r ⩽ 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 , 1.2V p ( r ) f ( p ) ⩽ B ∫ B p ( r )f ( x ) d xfor all p ∈ M and r > 0.