# On harmonic morphisms from $4$-manifolds to Riemann surfaces and local almost Hermitian structures

Abstract : We investigate the structure of a harmonic morphism $F$ from a Riemannian $4$-manifold $M^4$ to a $2$-surface $N^2$ near a critical point $m_0$. If $m_0$ is an isolated critical point or if $M^4$ is compact without boundary, we show that $F$ is pseudo-holomorphic w.r.t. an almost Hermitian structure defined in a neighbourhood of $m_0$. \\ If $M^4$ is compact without boundary, the singular fibres of $F$ are branched minimal surfaces.
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https://hal.archives-ouvertes.fr/hal-00844307
Contributor : Marina Ville <>
Submitted on : Sunday, July 14, 2013 - 4:38:42 PM
Last modification on : Thursday, December 5, 2019 - 1:26:33 AM
Long-term archiving on: Tuesday, October 15, 2013 - 4:08:58 AM

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• HAL Id : hal-00844307, version 1
• ARXIV : 1307.3903

### Citation

Ali Makki, Marina Ville. On harmonic morphisms from $4$-manifolds to Riemann surfaces and local almost Hermitian structures. 2013. ⟨hal-00844307⟩

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