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Semi-conformal polynomials and harmonic morphisms. (Polynômes semi-conformes et morphismes harmoniques.) (French) Zbl 0963.53036

This paper is concerned with harmonic morphisms and semi-conformal mappings. The authors first prove two propositions: every harmonic morphism from \(\mathbb{R}^m\) to \(\mathbb{R}^n\) is polynomial of degree \(\leq(m-2)/(n-2)\), and every polynomial semi-conformal mapping from \(\mathbb{R}^m\) to \(\mathbb{R}^n\) is harmonic. With the help of these results they give the precise behavior of a semi-conformal map \(F\) from a Riemannian manifold \((M^m,g)\) to another \((N^n,h)\) near the singular point \(x_0\): either all differentials of \(F\) at \(x_0\) are zero, or \(x_0\) is a singular point of order \(\leq(m-2)/(n-2)\).
The authors complete the Ou-Wood’s classification of homogeneous polynomial harmonic morphisms of degree 2 [Y. L. Ou, J. C. Wood, Algebras Groups Geom. 13, 41-53 (1996; Zbl 0872.58022)] by asserting that every polynomial harmonic morphism of degree 2 can be expressed as a sum of a homogeneous polynomial of degree 2 and a projection.
For polynomial semi-conformal mappings from \(\mathbb{R}^n\) to \(\mathbb{C}\), the authors find three cases in which the mapping can be expressed as a holomorphic mapping: the polynomials of degree 2 from \(\mathbb{R}^m\), the dimension \(m=3\), and the polynomials of degree 5 from \(\mathbb{R}^5\).
Reviewer: Zhu Fulin (Hubei)

MSC:

53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.

Citations:

Zbl 0872.58022
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