On the existence of Evans potentials |
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Authors: | Wolfhard Hansen Ivan Netuka |
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Institution: | 1. Fakult?t für Mathematik, Universit?t Bielefeld, Postfach 100131, 33501, Bielefeld, Germany 2. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic
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Abstract: | It is shown that, for every noncompact parabolic Riemannian manifold $X$ and every nonpolar compact $K$ in $X$ , there exists a positive harmonic function on $X\setminus K$ which tends to $\infty $ at infinity. (This is trivial for $\mathbb{R }$ , easy for $\mathbb{R }^2$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space $X$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set $K$ , there is a symmetric (positive) Green function for $X\setminus K$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $\left0,\infty \right)\times \{0\}, \left0,\infty \right)\times \{1\}$ , and the line segments $\{n\}\times 0,1], n=0,1,2,\dots $ ). |
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