Rigidity of Certain Harmonic Mappings.

Toshikazu Sunada

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A harmonic morphism $f : M \to N$ between Riemannian manifolds $M$ and $N$ is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim $M \geq$ dim $N$ , 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 $d f$ vanishes. Every non-constant harmonic morphism is shown to be...

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Let $M$ be a noncompact Riemannian manifold of dimension $n$ . Then there exists a proper embedding of $M$ into $R^{2 n + 1}$ by harmonic functions on $M$ . It is easy to find $2 n + 1$ 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.