49 problems

A domain in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a connected open set. A circle domain is a domain whose complementary components are points or close…

Let $(M,\omega)$ be a compact $n$-dimensional Kähler manifold with positive holomorphic bisectional curvature. Frankel's conjecture. Then $M$ is biholomorphic to $\mathbb{CP}^{n}$.…

Yau's uniformization conjecture. $M$ is biholomorphic to $\mathbb{C}^n$.

Yau's conjecture. Such a simply connected Kähler manifold $M^n$ is biholomorphic to $\mathbb{C}\mathbb{P}^n$ if it is compact, and to $\mathbb{C}^n$ if it is noncompact.

Let $M$ be a non-compact Kähler manifold with positive bisectional curvature. Yau's uniformization conjecture. The manifold $M$ is biholomorphic to the complex Euclidean space. Thi…

Let $Omega$ be a normal Stein space with a compact, smooth, strongly pseudoconvex boundary. Let $Omega{operatorname{Reg}}$ denote its regular part, and let the Bergman metric be…

Let $X$ be a complete non-compact Kähler manifold of complex dimension $n$. For a Kähler metric, write $BK>0$ when its bisectional curvature is positive: for all real unit tangent…

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold with positive pseudohermitian bisectional curvature. Let $\mathbf{H}_n=\mathbb{C}^n\times\mathbb{R}$ be the standard…

Let $M$ be a complete noncompact $n$-dimensional Kähler manifold with positive holomorphic bisectional curvature. Yau's third uniformization conjecture. The manifold $M$ is biholom…

Yau's uniformization conjecture. $M$ is biholomorphic to $\mathbb{C}^n$.

Yau's conjecture. $M$ is biholomorphic to $\mathbb{C}^n$.

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…

Let $\Sigma_g$ be a closed orientable surface of genus $g\geq 1$, and let $\tau$ be a graph on $\Sigma_g$ that lifts to a triangulation of the universal cover. Let…

Greene–Wu–Yau conjecture. $M$ is biholomorphic to a complex Euclidean space.

Let $M$ be a complete noncompact Kähler manifold, meaning a noncompact complex manifold equipped with a complete Kähler metric. Its holomorphic bisectional curvature is positive wh…

Uniformization conjecture. $M$ is biholomorphic to

Yang–Mills flow uniformization conjecture. From such an initial connection, the Yang–Mills flow converges to the flat connection corresponding to a Riemannian geometry with constan…

Let $N_{\chi,n,b}$ denote the number of extremal disc configurations with Euler characteristic $\chi$, $n$ cusps and $b$ boundary components, and let $\Sigma_{\chi,n,b}$ denote the…

Let $X$ be a complete non-compact Kähler manifold of complex dimension $n$. Assume that $X$ admits a Kähler metric with positive sectional curvature, which is stronger than the pos…

Sasakian uniformization conjecture. Then $M$ is CR biholomorphic to the standard Heisenberg group $\mathbb{H}_{n}$.

Yau's uniformization conjecture. If $M$ has these properties, then $M$ is biholomorphic to the standard $m$-dimensional complex Euclidean space

Let $M$ be a complete noncompact Kähler manifold with positive holomorphic bisectional curvature. Yau's uniformization conjecture. The manifold $M$ is biholomorphic to…

Weak quasiconformal uniformization conjecture. Every metric sphere admits a weakly quasiconformal parametrization from $\hat{\mathbb{C}}$.

Countable invariant uniformization conjecture. Let $E$ be a Borel equivalence relation on a Polish space $X$. Then the following are equivalent: (a) $E$ is reducible to countable;…

Let $(M,g)$ be a compact Hermitian manifold with Griffiths non-negative Chern curvature and positive first Chern--Ricci curvature. A rational homogeneous manifold is a quotient…