Completeness and existence of bounded biharmonic functions on a riemannian manifold

Leo Sario

Annales de l'institut Fourier (1974)

  • Volume: 24, Issue: 1, page 311-317
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Sario, Leo. "Completeness and existence of bounded biharmonic functions on a riemannian manifold." Annales de l'institut Fourier 24.1 (1974): 311-317. <http://eudml.org/doc/74166>.

@article{Sario1974,
abstract = {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 $H^2B$ of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry $H^2B$-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 $H^2B$-functions. Moreover, there exist both complete and incomplete manifolds not permitting these functions, and, trivially, incomplete manifolds possessing them.},
author = {Sario, Leo},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {311-317},
publisher = {Association des Annales de l'Institut Fourier},
title = {Completeness and existence of bounded biharmonic functions on a riemannian manifold},
url = {http://eudml.org/doc/74166},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Sario, Leo
TI - Completeness and existence of bounded biharmonic functions on a riemannian manifold
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 1
SP - 311
EP - 317
AB - 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 $H^2B$ of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry $H^2B$-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 $H^2B$-functions. Moreover, there exist both complete and incomplete manifolds not permitting these functions, and, trivially, incomplete manifolds possessing them.
LA - eng
UR - http://eudml.org/doc/74166
ER -

References

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  3. [3] Y.K. Kwon, L. Sario, B. Walsh, Behavior of biharmonic functions on Wiener's and Royden's compactifications, Ann. Inst. Fourier (Grenoble) 21 (1971), 217-226. Zbl0208.13703MR49 #5385
  4. [4] N. Mirsky, L. Sario, C. Wang, Bounded polyharmonic functions and the dimension of the manifold, J. Math. Kyoto Univ., 13 (1973), 529-535. Zbl0284.31007MR50 #2530
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  17. [17] L. Sario, C. Wang, The class of (p, q)-biharmonic functions, Pacific J. Math., 41 (1972), 799-808. Zbl0237.31013MR47 #5780
  18. [18] L. Sario, C. Wang, Counterexamples in the biharmonic classification of Riemannian 2-manifolds, Pacific J. Math. (to appear). Zbl0252.31011
  19. [19] L. Sario, C. Wang, Generators of the space of bounded biharmonic functions, Math. Z., 127 (1972), 273-280. Zbl0217.10503MR47 #8888
  20. [20] L. Sario, C. Wang, Quasiharmonic functions on the Poincaré N-ball, Rend. Mat. (to appear). Zbl0318.31011
  21. [21] L. Sario, C. Wang, Riemannian manifolds of dimension N ≥ 4 without bounded biharmonic functions, J. London Math. Soc. (to appear). Zbl0276.31007
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  25. [25] L. Sario, C. Wang, Parabolicity and existence of bounded biharmonic functions, Comm. Math. Helv. 47, (1972), 341-347. Zbl0247.31012MR50 #2532
  26. [26] L. Sario, C. Wang, Positive harmonic functions and biharmonic degeneracy, Bull. Amer. Math. Soc., 79 (1973), 182-187. Zbl0252.31010MR46 #9899
  27. [27] L. Sario, C. Wang, Parabolicity and existence of Dirichlet finite biharmonic functions, J. London Math. Soc. (to appear). Zbl0278.31009
  28. [28] L. Sario, C. Wang, Harmonic and biharmonic degeneracy, Kodai Math. Sem. Rep., 25 (1973), 392-396. Zbl0272.31005MR48 #12403
  29. [29] L. Sario, C. Wang, M. Range, Biharmonic projection and decomposition, Ann. Acad. Sci. Fenn. A.I., 494 (1971), 1-14. Zbl0219.31007MR57 #9962
  30. [30] C. Wang, L. Sario, Polyharmonic classification of Riemannian manifolds, Kyoto Math. J., 12 (1972), 129-140. Zbl0227.31008MR45 #8825

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