skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries
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.  more » « less
Award ID(s):
1665412 2000050 1800549
PAR ID:
10234231
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Journal für die reine und angewandte Mathematik (Crelles Journal)
Volume:
2021
Issue:
770
ISSN:
0075-4102
Page Range / eLocation ID:
183 to 203
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, International Mathematics Research Notices)
    null (Ed.)
    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
  2. ; (, Journal of operator theory)
    null (Ed.)
    We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray–von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu’s free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples. 
    more » « less
  3. (, International Mathematics Research Notices)
    null (Ed.)
    Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow. 
    more » « less
  4. ; ; ; (, Bernoulli)
    We detail an approach to developing Stein’s method for bounding integral metrics on probability measures defined on a Riemannian manifold M. Our approach exploits the relationship between the generator of a diffusion on M having a target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for Rm, and moreover imply that the bounds for Rm remain valid when M is a flat manifold. 
    more » « less
  5. ; ; ; (, International Conference on Artificial Intelligence and Statistics)
    We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on Rd, while the quadratictime Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov’s theorem from large deviation theory and the weak metrizable properties of the MMD and KSD. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email