10 problems

Let $D\subset\mathbb C^n$ be a bounded $\mathbb C$-convex domain, meaning that every non-empty intersection of $D$ with a complex line is contractible. Property $(\ast)$ is the equ…

Inverse-linear conjecture. If $h$ has nonnegative curvature, then the inverse-linear path from $h_0$ to $h_1=h$ consists entirely of metrics with nonnegative curvature.

Let $M$ be a Kobayashi hyperbolic complex manifold, meaning that its Kobayashi pseudodistance is a distance, and let the Wu metric be the invariant Hermitian metric on $M$. A Hermi…

Let $M=\mathsf{G}/\mathsf{H}$ be a homogeneous space, where $\mathsf{H}$ is a disconnected isotropy group. The identity components of $\mathsf{H}$ and $\mathsf{G}$ are denoted by…

Exhaustion conjecture. The domain $D$ can be exhausted by domains biholomorphic to convex domains.

Equality characterization conjecture. $F_{Omega}(w)=1$ if and only if there exists a balanced domain $Omega'$ (not necessarily convex) and a biholomorphic mapping $H:OmegatoOmega'$…

Invariant-metric conjecture. The topology of $X$ is metrizable by a $G$-invariant metric.

Let $G$ be a semisimple Lie group with property $(T)$, or let $\Gamma$ be a lattice in such a group, acting smoothly on a compact manifold $M$ as in the source. The measurable-to-s…

Let $\Gamma$ be a group with property $(T)$ acting smoothly on a compact manifold $M$ and preserving volume. Let $d(\Gamma)$ be the smallest dimension in which $\Gamma$ has an infi…

Let $\Omega_n$ be the spectral unit ball, let $A\in\Omega_n$, and let $B\in\mathcal M_n$. Write $k_{\Omega_n}(A;B)$ for the Kobayashi–Royden pseudometric and $C_A$ for the cone of…