Completeness and existence of bounded biharmonic functions on a riemannian manifold

Sario, Leo

Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 311-317.

A.S. Galbraith nous a communiqué la question suivante : est-ce que la complétion d’une variété implique, ou est impliquée par, la propriété que la classe $H^{2}B$ des fonctions bornées non harmoniques biharmoniques soit vide ? Parmi toutes les variétés considérées jusqu’ici dans la classification biharmonique, celles qui sont complètes ne portent pas de $H^{2}B$-fonctions et on peut suspecter qu’il en est toujours ainsi. Nous allons montrer cependant qu’il existe bien des variétés complètes de toute dimension qui portent des $H^{2}B$-fonctions. De plus, il existe des variétés complètes et des variétés incomplètes qui n’en portent pas et, de façon évidente, des variétés incomplètes qui en portent.

A.S. Galbraith has communicated to us the following intriguing problem: does the completeness of a manifold imply, or is it implied by, the emptiness of the class $H^{2}B$ of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry $H^{2}B$-functions, and one might suspect that this is always the case. We shall show, however, that there do exist complete manifolds of any dimension that carry $H^{2}B$-functions. Moreover, there exist both complete and incomplete manifolds not permitting these functions, and, trivially, incomplete manifolds possessing them.

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