Some Observations on Liouville’s Theorem on Surfaces and the Dirichlet Problem at Infinity

Abstract

In this paper we explore Liouville's theorem on Riemannian cones as defined below. We also study the Strong Liouville Property, that is, the property of a cone having spaces of harmonic functions of a fixed polynomial growth of finite dimension.

An Observation on the Dirichlet Problem at Infinity in Riemannian Cones

“…As mentioned before, the previous argument inspired the following result, whose proof follows along the same lines as shown above, and which shall be presented elsewhere [2].…”

“…The following computation, which was originally suggested in [4], and that we reproduce for the convenience of the reader, shows that the boundary data are satisfied in an L 2 -sense as described above. Again, using the orthogonality of the eigenfunctions of the Laplacian with different eigenvalues, we can estimate f (ω) − u (ω, r) ≤ K.…”

“…In order to state our next result, let us show a proof of Liouville's theorem devised by the author of this note and that has been generalised by the author and by J. E. Bravo. Before we proceed, we want to point out two facts: first, there is a simple proof of Liouville's theorem in the case of an entire function using Fourier series; and, second, that in the case of dimension two, the fact that harmonic functions remain harmonic after conformal deformation can be used to give another proof of Liouville's theorem as presented below (see Section 3 in [3]): however, this second proof cannot be generalised to higher dimensions. So let's start with the promised proof.…”

“…and this gives a bound from below for the slowest possible growth of a nonconstant harmonic function defined on the cone with metric dr 2 + φ (r) 2 g ω , g ω being an arbitrary metric on S n−1 . If g ω is the round metric, then it is well known that λ 1 = √ n − 1, and thus a bound from below for the slowest possible growth of a nonconstant harmonic function in this case would be, for 2) , which does give Liouville's theorem in its classical version, but not its sublinear version (with the remarkable exception of n = 3).…”

“…Theorem 2. Assume that (M, g), with g as in (4), is a smooth Riemannian manifold (thus, N = S n−1 ). Furthermore, assume that there is an R 0 such that φ (r) ≥ 0 if r > R 0 , and that (2) holds.…”