8 problems

Growth equality conjecture. The estimates labeled … are equalities, and $\eta=2$ in the case $D_g=3$ and $\eta=1$ in the case $D_g=4$, which implies $\eta=R$ in these cases.

Totally ramified value-number conjecture. For algebraic minimal surfaces, there exists $2.5\leq\kappa<3$ such that $\nu_g\leq\kappa$.

Sharp upper bound conjecture. The sharp upper bound of $D_g$ is $2$.

Fujimoto conjecture. These polynomials are in general position: any $m$ polynomials chosen from them are linearly independent over $\mathbb{C}$. This conjecture is the algebraic in…

Let $\Sigma\subset\mathbb{S}^3$ be fixed, let $G_\phi$ denote the Gauss graph associated with a map $\phi$, and consider a minimizing sequence of $\operatorname{Area}(G_\phi)$. Let…

Flat point conjecture. If $X$ has at least one flat point, then its Gauss map omits at most $3$ values.

Let $\Sigma$ be an orientable, complete, non-flat, immersed minimal surface with finite total curvature in $\mathbb{R}^{3}$, and let its Gauss map be the map from $\Sigma$ to…

Nirenberg's conjecture. If the Gauss map $G$ omits a neighbourhood of a point of $\mathbb{S}^{2}$, then $M$ is a plane.