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Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
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Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
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Journal Article
Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
2023
Overview
Suppose (
M
,
g
) is a Riemannian manifold having dimension
n
, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity
C
(
X
) is an Euclidean cone over the cross-section
X
. Denote by
α
=
lim
r
→
∞
Vol
(
B
r
(
p
)
)
r
n
the asymptotic volume ratio. Let
h
k
=
h
k
(
M
)
be the dimension of the space of harmonic functions with polynomial growth of growth order at most
k
. In this paper, we prove an upper bound of
h
k
in terms of the counting function of eigenvalues of
X
. As a corollary, we obtain
lim
k
→
∞
k
1
-
n
h
k
=
2
α
(
n
-
1
)
!
ω
n
. These results are sharp, as they recover the corresponding well-known properties of
h
k
(
R
n
)
. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
