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Certain Liouville theorems for Riemannian manifolds of a special form. (English. Russian original) Zbl 0764.58035
Sov. Math. 35, No. 12, 15-23 (1991); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1991, No. 12(355), 15-24 (1991).
Consider a complete Riemannian manifold $$M$$ with boundary such that the complement of a certain compact set $$B$$ in $$M$$ consists of $$m$$ connected components $$D_ 1,\dots,D_ m$$ and $$D_ i$$, $$i=1,\dots,m$$, is isometric to $$(0,\infty)\times K_ i$$, where $$K_ i$$ is compact with boundary and the metric on $$D_ i$$ is $$ds^ 2=h^ 2_ i(r)dr^ 2+g^ 2_ i(r)d\theta^ 2_ i$$, $$r\in(0,\infty)$$, and $$d\theta^ 2_ i$$ is the metric on $$K_ i$$. Consider $$I_ i=\int^ \infty_{r_ 0} h_ i(t)g^{1-n}_ i(t)\left(\int^ t_{r_ 0} h_ i(\xi)g^{n-3}_ i(\xi)d\xi\right)dt$$, where $$r_ 0=\text{const }>0$$, $$n=\dim M$$.
Theorem 1. Assume that $$M$$ is hyperbolic and $$I_ i=\infty$$ for all $$i=1,\dots,m$$. Then the dimension of the space of bounded harmonic functions on $$M$$ equals to the number of hyperbolic domains $$D_ i$$. Theorem 2. If $$M$$ is as in Theorem 1 then the dimension of the cone of positive harmonic functions on $$M$$ is $$m$$. The author presents also a similar result for the dimension of the space of functions $$u$$ satisfying $$\Delta u=0$$ and $$u=\bar 0(v_ \ell)$$.
Reviewer: V.Oproiu (Iaşi)

##### MSC:
 58J99 Partial differential equations on manifolds; differential operators 35B35 Stability in context of PDEs 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions