Riemannian manifolds with bounded Dirichlet finite polyharmonic functions
Lung Ock Chung, Leo Sario, Cecilia Wang (1973)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Lung Ock Chung, Leo Sario, Cecilia Wang (1973)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Lung Ock Chung, Leo Sario, Cecilia Wang (1975)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Robert E. Greene, H. Wu (1975)
Annales de l'institut Fourier
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Let be a noncompact Riemannian manifold of dimension . Then there exists a proper embedding of into by harmonic functions on . It is easy to find harmonic functions which give an embedding. However, it is more difficult to achieve properness. The proof depends on the theorems of Lax-Malgrange and Aronszajn-Cordes in the theory of elliptic equations.
Choi, Gundon, Yun, Gabjin (2005)
International Journal of Mathematics and Mathematical Sciences
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Leo Sario (1974)
Annales de l'institut Fourier
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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 of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry -functions, and one might suspect that this is always the case. We shall show, however, that there do exist complete manifolds of any dimension...
Bent Fuglede (1978)
Annales de l'institut Fourier
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A harmonic morphism between Riemannian manifolds and is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim dim, since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where vanishes. Every non-constant harmonic morphism is shown to be...