An Observation on the Dirichlet Problem at Infinity in Riemannian Cones

Abstract

In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.

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

“…Indeed, the proof is exactly the same as that given in [7] for the Dirichlet Problem at Infinity. First, we must show that the sum does converge in L 2sense for every r ≤ R. Here we need the fact that the ϕ m 's are nonnegative and nondecreasing: this is shown in [7] under the assumption that φ ′ ≥ 0. Therefore 0 ≤ ϕ m (r) /ϕ m (R) ≤ 1 for 0 ≤ r ≤ R and our assertion follows immediatly.…”

“…Since the ϕ m 's are non decreasing (by the arguments in [7]), it follows that x ′ m (t) ≥ 0 which is a basic but important observation. Let…”

“…The harmonic extension of h R to M R (O), and thus, by uniqueness, u on M R (O) (see Theorem 3 in [7]), is given by…”

“…Therefore 0 ≤ ϕ m (r) /ϕ m (R) ≤ 1 for 0 ≤ r ≤ R and our assertion follows immediatly. Next, we must show that that the boundary condition at ∂M R (O) is met as r → R, and for this we also ask the reader to consult [7]. From now on, we shall assume, without loss of generality, that c 0,0 = 0.…”