5 problems

Flatness conjecture. Then $M^4$ is flat.

Let $(M^4,J,g,\mathcal{X}^1,\mathcal{X}^2)$ be a scalar-flat toric Kähler 4-manifold with reduction $\Sigma^2$, and suppose that $\Sigma^2$ is not the strip. Assign boundary values…

Let $(M^4,J,g,\mathcal{X}^1,\mathcal{X}^2)$ be a scalar-flat toric Kähler 4-manifold, with reduction $\Sigma^2$ and momentum coordinates $(\varphi^1,\varphi^2)$. Polygonal reductio…

Quarter-plane rigidity conjecture. Theorem should hold with the phrase “up to homothety” removed.

Half-plane flatness conjecture. Theorem should continue to hold if $s\ge0$.