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

TL;DR

This paper studies harmonic morphisms from 4-manifolds to surfaces, showing they are pseudo-holomorphic near critical points and that singular fibers are branched minimal surfaces in certain cases.

Contribution

It demonstrates that harmonic morphisms from 4-manifolds are locally pseudo-holomorphic with respect to almost Hermitian structures near critical points, and characterizes their singular fibers.

Findings

Harmonic morphisms are pseudo-holomorphic near isolated critical points.

Singular fibers are branched minimal surfaces in compact, boundaryless 4-manifolds.

Results apply to both local and global geometric 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.