3 problems

Let $\pi:(X,\alpha)\to(B,\beta)$ be a submersion, and consider the functional $\mathcal{N}$ on the space of relatively Kähler metrics that are cscK on each fibre. Properness conjec…

Properness and existence conjecture. For any $H\geq\frac12\operatorname{Ch}(X)$, every complete, connected $H$-surface embedded in $X$ with positive injectivity radius or finite to…

Let $F_A$ be the map associated with a matrix $A$, and let $S_{F_A}$ denote its set of non-proper values. Conjecture D. If a map $F_A(x)$ is non-proper, then $0\in S_{F_A}$. The te…