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Zeitschrift für Analysis und ihre Anwendungen

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Volume 6, Issue 4, 1987, pp. 331–339
DOI: 10.4171/ZAA/254

Published online: 1987-08-31

On Polyharmonic Riemannian Manifolds

Rainer Schimming[1] and Jan Kowolik[2]

(1) Ernst-Moritz-Arndt-Universität Greifswald, Germany
(2) University of Opole, Poland

A natural generalization of the harmonic manifolds is considered: a Riemannian manifold is called $k$-harmonic or polyharmonic if it admits a non-constant $k$-harmonic function depending only on the geodesic distance $r = r(x, y)$ or rather on Synge’s function $\sigma = \sigma (x, y)$, i.e. a solution $F$ of $\Delta^k F(\sigma) = 0$. Certain theorems are generalized from harmonic to polyharmonic manifolds.

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Schimming Rainer, Kowolik Jan: On Polyharmonic Riemannian Manifolds. Z. Anal. Anwend. 6 (1987), 331-339. doi: 10.4171/ZAA/254