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.
