A polynomial-growth harmonic function of order at most 
on a complete Riemannian
manifold is a harmonic function 
for which
 |
for a fixed base point 
and some finite constant 
. Let 
be the dimension of this
vector space. On Euclidean
space 
, these are precisely the harmonic
polynomials of degree at most 
for a nonnegative integer 
, giving
 |
where the second term is zero for 
.
In dimension three this is 
. The harmonic
dimension comparison conjecture concerns these dimensions under nonnegative Ricci curvature.
