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1. Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

   https://doi.org/10.1515/crelle-2020-0012  

   Huang, Xiaojun; Xiao, Ming (January 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We give an affirmative solution to a conjecture of Cheng proposed in 1979which asserts that the Bergman metric of a smoothly bounded stronglypseudoconvex domain in {\mathbb{C}^{n},n\geq 2} , is Kähler–Einsteinif and only if the domain is biholomorphic to the ball. We establisha version of the classical Kerner theorem for Stein spaces withisolated singularities which has an immediate application toconstruct a hyperbolic metric over a Stein space with a sphericalboundary. 

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2. Bergman metrics with constant holomorphic sectional curvatures

   https://doi.org/10.1515/crelle-2025-0013  

   Huang, Xiaojun; Li, Song-Ying; Treuer, John N (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in ${\mathbb{C}}^2$\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat. 

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3. The Kodaira dimension of complex hyperbolic manifolds with cusps

   https://doi.org/10.1112/S0010437X1700762X  

   Bakker, Benjamin; Tsimerman, Jacob (March 2018, Compositio Mathematica)

   We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. 

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4. Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

   https://doi.org/10.1515/crelle-2022-0029  

   Xiao, Ming (June 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = ( F 1 , … , F m ) {F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹 n {\mathbb{B}^{n}} , n ≥ 2 {n\geq 2} , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m {\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ω i ′ {\Omega_{i}^{\prime}} s are irreducible factors of Ω. We prove every non-constant component F i {F_{i}} must map generic boundary points of 𝔹 n {\mathbb{B}^{n}} to the boundary of Ω i {\Omega_{i}} . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to aproduct of unit balls and Lie balls. 

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5. Null, recursively starlike-equivalent decompositions shrink

   https://doi.org/10.1017/S0017089522000337  

   Meier, Jeffrey; Orson, Patrick; Ray, Arunima (May 2023, Glasgow Mathematical Journal)

   Abstract A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $$\mathbb{R}^n$$ for some n , sending E to a starlike set. A subset $$E\subset X$$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $$\{E_i\}_{i=0}^{N+1}$$ such that $$E_{i}/E_{i+1}\subset X/E_{i+1}$$ is starlike-equivalent for each i and $$E_{N+1}$$ is a point. A decomposition $$\mathcal{D}$$ of a metric space X is said to be recursively starlike-equivalent, if there exists $$N\geq 0$$ such that each element of $$\mathcal{D}$$ is recursively starlike-equivalent of filtration length N . We prove that any null, recursively starlike-equivalent decomposition $$\mathcal{D}$$ of a compact metric space X shrinks, that is, the quotient map $$X\to X/\mathcal{D}$$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture. 

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