ON MARCH’S CRITERION FOR TRANSIENCE ON ROTATIONALLY SYMMETRIC MANIFOLDS

Abstract

Abstract We show that March’s criterion for the existence of a bounded nonconstant harmonic function on a weak model (that is, $\mathbb {R}^n$ with a rotationally symmetric metric) is also a necessary and sufficient condition for the solvability of the Dirichlet problem at infinity on a family of metrics that generalise metrics with rotational symmetry on $\mathbb {R}^n$ . When the Dirichlet problem at infinity is not solvable, we prove some quantitative estimates on how fast a nonconstant harmonic function must grow.