Vol. 50, No. 1, 1974

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Counterexamples in the biharmonic classification of Riemannian 2-manifolds

Leo Sario and Cecilia Wang

Vol. 50 (1974), No. 1, 159–162

Crucial counterexamples in the biharmonic classification theory of Riemannian 2-manifolds have been deduced from certain general principles. The present note is methodological in nature: the aim is to supplement the theory by showing that very simple counterexamples can be directly constructed. Whereas earlier work has been devoted to the class H2 of nonharmonic biharmonic functions, here the class W of all biharmonic functions is discussed. This is of interest, since the classes OWB and OWD of Riemannian manifolds without (nonconstant) bounded or Dirichlet finite biharmonic functions are strictly contained in the corresponding classes OH2B and OH2D, as is seen by endowing the unit disk with a suitable conformal metric. Moreover, for W-functions the biharmonic equation need not be reduced to the Poisson equation but can be dealt with directly.

Mathematical Subject Classification 2000
Primary: 31B30
Secondary: 53C20
Received: 19 July 1972
Published: 1 January 1974
Leo Sario
Cecilia Wang