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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)

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