17 problems

Let $(M,h)$ be a simply-connected complete Kähler manifold satisfying … for two positive constants $A$ and $B$. Let $\mathfrak{K}$ denote the Kobayashi metric on $M$. Greene–Wu's c…

Let $\varphi:E\longrightarrow \mathcal E(p)$ be a $\kappa$-geodesic, with component parameters $\alpha_j$ and exponents $p_j$ as in Proposition 2. In particular, the preceding desc…

Abelian-variety entire-curve conjecture. Every non-constant holomorphic map

Pointed Brody conjecture. If the infinitesimal Kobayashi–Royden pseudometric is degenerate on $T_xX$, then there exists a non-constant holomorphic map

Let $c_X:X\to C(X)$ be an admissible representative of the core of $X$, and let $\Delta(c_X)$ be its base orbifold. Conjecture IV'H. One should have … In addition, if $(X/\Delta)$…

Let $X\in\mathcal C$ be a compact complex manifold with core $c_X:X\to C(X)$, and let $d_{C(X)}^*$ be the induced pseudometric on the core. Conjecture IVH. The pseudometric…

Let $X\in\mathcal C$, and call $X$ $H$-special when its Kobayashi pseudometric satisfies $d_X\equiv 0$. Conjecture IIIH. The manifold $X$ is special if and only if it is $H$-specia…

Let $M$ and $M'$ be Kobayashi hyperbolic complex manifolds. Let $f:(M,K_M)\to (M',K_{M'})$ be a Kobayashi isometry. If $M$ and $M'$ are not biholomorphic to a product of complex ma…

Let $M$ and $N$ be Kobayashi hyperbolic manifolds, and let $F:M\to N$ be an isometry with respect to the Kobayashi metric or distance. Assume that $M$ and $N$ are not Cartesian pro…

Let $X$ be a bounded domain in $\mathbb{C}^{N}$ equipped with the Kobayashi metric, and assume that it is a proper metric space. Let $\overline{X}\subset\mathbb{C}^{N}$ be the stan…

Let $d\geq 1$, let $C\subset\operatorname{\mathbb{R}}^d$ be a convex domain, and let … be its tube domain. Assume that $(\Omega,K_\Omega)$ is Gromov hyperbolic; equivalently, $C$ i…

Kobayashi's non-hyperbolicity conjecture. Every compact Calabi–Yau manifold $X$ is Kobayashi non-hyperbolic; equivalently, there exists a non-constant holomorphic map

Continuous extension conjecture. Every complex geodesic of $\Omega$ extends continuously to $\overline{\mathbb{D}}$.

Special-manifold characterization conjecture. The manifold $X$ is special if and only if both of the following properties, conjecturally equivalent to each other, hold: $d_X$ vanis…

Let $D$ be a smooth bounded pseudoconvex domain, let $d_D(z)$ denote the distance from $z$ to the boundary of $D$, and let $\b7$ denote the Kobayashi-Royden metric. Fu's conjecture…

Campana's core conjecture. The generic fiber of $c_X$ has a vanishing Kobayashi metric, and $C(X)$ is Brody hyperbolic modulo a proper subvariety.

Polynomial realization conjecture. If