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Riemannian manifolds with connected Royden harmonic boundaries. (English) Zbl 0764.53032

Let \(M\) be a differentiable manifold \(M\) of dimension \(n\geq 2\) with local coordinates \(x=(x^ 1,\dots,x^ n)\). We assume that on \(M\) there exists a Riemannian metric \(g\) and it is not compact (because the case, where \(M\) is compact, is trivial). By the countability assumption on \(M\) we can find an exhaustion \(\{M_ K\}\), \(K\geq 0\) of \(M\), i.e. a sequence of relatively compact subregions \(M_ K\) of \(M\) with smooth relative boundaries \(\partial M_ K\) such that \(\overline M_ K\subset M_{K+1}\) \((K=1,2,\dots)\), \(M=\bigcup_{K\geq 0} M_ K\). The aim of this paper is to concern about a potential theoretic property of the open unit ball \(B^ n\). Therefore it is not considered a too general manifold. The main result of this paper can be expressed as follows: “For any dimension \(n\geq 3\), there exists a Riemannian metric \(ds\) on \(B^ n\) such that the Royden harmonic boundary \(\Delta(B^ n,ds)\) is disconnected so that \((B^ n,ds)\) does not possess the Virtanen property”.

MSC:

53C20 Global Riemannian geometry, including pinching
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