7 problems

Existence conjecture. There exists a complex-valued harmonic morphism

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\ge 2$. For each point $p\in M$, a complex valued harmonic morphism is a harmonic morphism … defined on an…

Kähler-structure conjecture. Every globally defined harmonic morphism $\phi:\mathbb{R}^4\to\mathbb{C}$, whether submersive or not, is holomorphic with respect to a Kähler structure…

Gudmundsson's conjecture. For each point $p\in M$, there exists a complex-valued harmonic morphism

Inversion conjecture. The map $z:{\mathbb R}^{2p}\setminus\{0\}\to{\mathbb C}$ is a complex-valued $(p,p)$-harmonic morphism for every $a\in{\mathbb C}^{2p}$.

The min–max conjecture. The min–max is realized by harmonic morphisms, and

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\ge 2$. For each point $p\in M$, consider complex-valued harmonic morphisms defined on an open neighbourho…