Harmonic functions on cones of model manifolds

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

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2014), pp. 13-22.

Résumé

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.

Détail

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

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