Spherical harmonics generating bounded biharmonics
Authors:
Bradley Beaver, Leo Sario and Cecilia Wang
Journal:
Proc. Amer. Math. Soc. 84 (1982), 485-491
MSC:
Primary 31C12; Secondary 31A30, 31B30
DOI:
https://doi.org/10.1090/S0002-9939-1982-0643735-7
MathSciNet review:
643735
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold
. On the punctured Euclidean
-space
,
is void for
, whereas for
, it is generated by certain fundamental spherical harmonics. It is also known that
remains void on the Riemannian manifold
,
, obtained by endowing
with the non-Euclidean metric
,
.
The purpose of the present paper is to show that the fundamental spherical harmonics continue generating , despite the distorting metric
. An analogous result holds for
.
- [1] C. Müller, Spherical harmonics, Lecture Notes in Math., vol. 17, Springer-Verlag, Berlin-Heidelberg-New York, 1966. MR 0199449 (33:7593)
- [2] L. Sario and C. Wang, Generators of the space of bounded biharmonic functions, Math. Z. 127 (1972), 273-280. MR 0320349 (47:8888)
- [3]
-, Riemannian manifolds of dimension
without bounded biharmonic functions, J. London Math. Soc. (2) 7 (1974), 635-644. MR 0425834 (54:13784)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1982-0643735-7
Article copyright:
© Copyright 1982
American Mathematical Society