16 problems

Kähler–Einstein conjecture. If $g$ is Kähler–Einstein, then $(U,g)$ is biholomorphically isometric to a bounded symmetric domain.

Let $D^{\mathrm{I}}_{p,q}$ denote the irreducible type-I bounded symmetric domain, and let $F:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p',q'}$ be a proper holomorphic map, where…

Let $S$ be a compact Kähler–Einstein complex surface with strictly negative sectional curvature, and let $B^2$ denote the complex unit $2$-ball. Siu–Yang's uniformization conjectur…

Folklore conjecture. Every bounded convex domain that admits a finite volume quotient is biholomorphic to a bounded symmetric space.

Let $\Omega \subset \mathbb{C}^n$, $n \geq 1$, be a bounded domain. Write $R^\Omega$ for the curvature tensor of its Bergman metric, and let…

Let $\Omega$ be a bounded symmetric domain of rank $r$, with invariants $p$, $b$, $d$, and $q_\Omega$, and let $\Gamma_\Omega$ denote the associated gamma function. Let $m$ and…

Let $\mathbf{k}\in\mathbb{Z}_{++}^3$ and $\mathbf{l}\in\mathbb{Z}^3$ satisfy $l_1\geq 0$, $0\leq l_2\leq k_1-k_2$, and $0\leq l_3\leq k_2-k_3$. Let…

Wavelet characterization conjecture. The Bergman spaces $A^p_\alpha$ should admit a wavelet characterization throughout this entire range, without the restriction requiring…

A Cartan classical domain of type IV is a bounded symmetric domain in complex dimension $n$. Uniform estimates for the $\overline{\partial}$-equation mean that solutions can be cho…

Let $\Omega$ be a bounded symmetric domain, let $\Gamma\subset {\rm Aut}(\Omega)$ be a torsion-free arithmetic lattice, and let … be the uniformization map. For an algebraic subset…

Let $\Omega$ be a bounded symmetric domain in its Harish-Chandra realization, let $L^p_{\nu}=L^p(\Omega,v_{\nu})$ for $p\in(1,\infty)$, and let $A^p_{\nu}\subset L^p_{\nu}$ be the…

Clozel–Ullmo conjecture. The assumption that the image of the holomorphic isometry is invariant under those automorphisms should be removable; that is, every germ of a holomorphic…

Type-IV total geodesy conjecture. Then

Rank-2 type-I total geodesy conjecture. If

Rigidity conjecture. Any such map $H$ is a holomorphic isometry, and hence is equivalent to one of the maps in Theorem 1.

Conjecture. Such a manifold is a bounded symmetric domain of rank one. This means, in particular, that it is a product of complex hyperbolic spaces.