4 problems

Flatness conjecture. Every two-step nilpotent hypersymplectic Lie algebra is flat.

Torus-fibration conjecture. $M$ is finitely covered by a manifold diffeomorphic to a fiber bundle over an $m$-dimensional torus.

Lacunary-function conjecture. The harmonic metrics generated by the real and imaginary parts of any lacunary holomorphic function on $D$ are never subset metrics of the target flat…

Let $S$ be the underlying surface and let $\mathrm{Flat}(S)$ denote the space of flat metrics on $S$. For $\varphi\in\mathrm{Flat}(S)$, let $L_{\varphi}$ be its Liouville geodesic…