Harmonic functions on cones of model manifolds

The paper deals with harmonic functions on cones of model manifolds. ?? is called a cone of model manifold, if ?? = ?? ? ??, where ?? is a non-empty precompact set and ?? is isometric to the product [??0,+?) Ч ? (??0 > 0, ? is a compact Riemannian manifold with non-empty smooth boundary) with the metric ????2 = ????2 + ??2(??)????2. Here ??(??) is a positive smooth on [??0,+?) function, and ???? is a metric on ?. Note if ? is a compact Riemannian manifold with no boundary, we have just a definition of model manifold. Let's ??0(??) = {?? : ??? = 0, ??|???? = 0}, and ?? = ?? ? ??0 ??1???(??)(??? ?? ??0 ?????3(??)????)?????, where ??0 = const > 0, ?? = dim??. The main results of the paper are following. Theorem 1. Let's manifold ?? has ?? = ?. Then any bounded function ?? ? ??0(??) is equal to zero identically. Theorem 2. Let's manifold ?? has ?? = ?. Then for cone of positive harmonic functions from class ??0(??) the dimension is equal to 1.