Completeness and existence of bounded biharmonic functions on a riemannian manifold
Annales de l'Institut Fourier, Volume 24 (1974) no. 1, p. 311-317

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.

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.

@article{AIF_1974__24_1_311_0,
     author = {Sario, Leo},
     title = {Completeness and existence of bounded biharmonic functions on a riemannian manifold},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {24},
     number = {1},
     year = {1974},
     pages = {311-317},
     doi = {10.5802/aif.502},
     mrnumber = {353203},
     zbl = {0273.31010},
     mrnumber = {50 \#5688},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1974__24_1_311_0}
}
Sario, Leo. Completeness and existence of bounded biharmonic functions on a riemannian manifold. Annales de l'Institut Fourier, Volume 24 (1974) no. 1, pp. 311-317. doi : 10.5802/aif.502. http://www.numdam.org/item/AIF_1974__24_1_311_0/

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