David Catlin

The Bergman Kernel and a Theorem of Tian

Given a domain Ω in ℂn, the Bergman kernel is the kernel of the projection operator from L 2 (Ω) to the Hardy space A 2 (Ω). When the boundary of Ω is strictly pseudoconvex and smooth, Fefferman [2] gave a complete description of the asymptotic behavior of K(z, z) as z approaches the boundary. This work was then extended by Boutet de Monvel and Sjostrand [1] who showed that, for the same domains, a similar asymptotic expansion for K(z, w) holds off the diagonal. Moreover, they showed that the Bergman kernel is a Fourier integral operator with a complex phase function.

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding.

Sufficient conditions for the extension ofC R structures

LetM be a smoothC R manifold of dimension 2n − 1 such that at each point, either the Levi form has at least 3 positive eigenvalues or it hasn − 1 negative eigenvalues. IfD is a smoothly bounded subdomain ofM, then there is a smoothly bounded integrable almost complex manifoldX of dimension 2n such thatM is contained in the boundary ofX and such that theC R structure thatM inherits as a subset ofX coincides with the original structure ofM.

Invariant metrics on pseudoconvex domains

Let Ω be a bounded domain in ℂn. Each of the metrics of Bergman, Caratheodory, and Kobayashi assigns a positive number to a given non-zero tangent vector X above a point z in Ω. This assignment is invariant in the sense that if f is a biholomorphism of Ω onto another bounded domain Ω′, then the metric applied to X equals the value of the metric on Ω′ applied to the tangent vector df(X) at the point f(z). Although it is very difficult to calculate the precise value of the above metrics in all but a few special cases, it is sometimes possible to compute a formula for the asymptotic behavior of the metric as the point z approaches the boundary of Ω. When Ω is a smoothly bounded strongly pseudoconvex domain, asymptotic formulas for the Bergman metric were obtained by Diederich [3] and later in much more precise form by Fefferman [4]. Formulas for the asymptotic behavior of the Caratheodory and Kobayashi metric on the same domains were obtained by Graham [5]. In this note we shall consider the case of pseudoconvex domains of finite type in ℂ2 Instead of determining an asymptotic formula for the above metrics, we obtain only a formula that expresses the approximate size of the metrics. In a sense which we shall make precise, these metrics are all equivalent for the given class of domains. We also obtain a formula for the approximate size of the Bergman kernel K(z,\( \bar{\text z} \)) of the domain Ω

PROGRAM OF THE CONFERENCE

12:00–14:00 Registration 14:00–14:15 Opening Chair: Michal Kř́ıžek 14:15–15:00 Pavel Kroupa, The star-formation histories of nearby galaxies raises questions on cosmology (ZOOM presentation) 15:00–15:30 Itzhak Goldman, Astrophysical bounds on mirror dark matter (ZOOM presentation) 15:30–16:00 Coffee Break Chair: Michal Kř́ıžek 16:00–16:30 Wolfgang Oehm, A possible constraint on the validity of general relativity for very strong gravitational fields 16:30–17:00 Alexei A. Starobinsky, New cosmological constraints on scalartensor gravity models and consequences for the H0 tension (ZOOM presentation) 17:00–17:30 Yurii V. Dumin, The problem of causality in the uncertaintymediated inflationary model (ZOOM presentation) 17:30–17:50 Yurii V. Dumin, The problem of flatness in various cosmological models (ZOOM presentation)