Title: On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics
Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric. more »« less
Huang, Xiaojun; Xiao, Ming(
, 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.
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.
Xiao, Ming(
, 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.
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.
Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X .
Ebenfelt, Peter, Xiao, Ming, and Xu, Hang. On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics. Retrieved from https://par.nsf.gov/biblio/10267943. International Mathematics Research Notices . Web. doi:10.1093/imrn/rnab120.
Ebenfelt, Peter, Xiao, Ming, & Xu, Hang. On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics. International Mathematics Research Notices, (). Retrieved from https://par.nsf.gov/biblio/10267943. https://doi.org/10.1093/imrn/rnab120
Ebenfelt, Peter, Xiao, Ming, and Xu, Hang.
"On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics". International Mathematics Research Notices (). Country unknown/Code not available. https://doi.org/10.1093/imrn/rnab120.https://par.nsf.gov/biblio/10267943.
@article{osti_10267943,
place = {Country unknown/Code not available},
title = {On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics},
url = {https://par.nsf.gov/biblio/10267943},
DOI = {10.1093/imrn/rnab120},
abstractNote = {Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric.},
journal = {International Mathematics Research Notices},
author = {Ebenfelt, Peter and Xiao, Ming and Xu, Hang},
editor = {null}
}
Warning: Leaving National Science Foundation Website
You are now leaving the National Science Foundation website to go to a non-government website.
Website:
NSF takes no responsibility for and exercises no control over the views expressed or the accuracy of
the information contained on this site. Also be aware that NSF's privacy policy does not apply to this site.
