Mathematics

Harmonic functions on cones of model manifolds

Yu. V. Goncharov, A. G. Losev, A. V. Svetlov

Volgograd State University

Abstract: The paper deals with harmonic functions on cones of model manifolds. $M$ is called a cone of model manifold, if $M=B\cup D$, where $B$ is a non-empty precompact set and $D$ is isometric to the product $[r_0,+\infty)\times \Omega$ ($r_0>0$, $\Omega$ is a compact Riemannian manifold with non-empty smooth boundary) with the metric
$$ds^2=dr^2+g^2(r)d\theta^2.$$
Here $g(r)$ is a positive smooth on $[r_0,+\infty)$ function, and $d\theta$ is a metric on $\Omega$. Note if $\Omega$ is a compact Riemannian manifold with no boundary, we have just a definition of model manifold.
Let's
$$H_0(M)=\{u: \Delta u=0, u|_{\partial M}=0\},$$
and
$$J=\int_{r_0}^\infty g^{1-n}(t)\left(\int_{r_0}^t g^{n-3}(\xi)d\xi\right)dt,$$
where $r_0={\rm const}>0,\ n=\dim M$.
The main results of the paper are following.
Theorem 1. Let's manifold $M$ has $J=\infty$. Then any bounded function $u\in H_0(M)$ is equal to zero identically.
Theorem 2. Let's manifold $M$ has $J=\infty$. Then for cone of positive harmonic functions from class $H_0(M)$ the dimension is equal to 1.

Keywords: Laplace–Beltrami equation, Liouville type theorems, model Riemannian manifolds, cones of model manifolds, dimension of solutions' space.

UDC: 517.95
BBK: 22.161.6