Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Publication date: 15 September 2021

Abstract: 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

a l p h a = l i m_{r i g h t a r r o w i n f t y} f r a c m a t h r m V o l (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 a upper bound of

h_{k}

in terms of the counting function of eigenvalues of

X

. As a corollary, we obtain

l i m_{k i g h t a r r o w i n f t y} k^{1 - n} h_{k} = f r a c 2 a l p h a (n - 1)! o m e g a_{n}

. These results are sharp, as they recover the corresponding well-known properties of

h_{k} (m a t h b b R^{n})

. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.

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