Siqi Fu

The Bergman kernel function: Explicit formulas and zeroes

We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.

Semi-classical analysis of Schrödinger operators and compactness in the ∂-Neumann problem

Abstract We study the asymptotic behavior, in a “semi-classical limit,” of the first eigenvalues (i.e., the groundstate energies) of a class of Schrodinger operators with magnetic fields and the relationship of this behavior with compactness in the ∂ -Neumann problem on Hartogs domains in C 2 .

Finite type conditions on reinhardt domains

We prove that, if ρ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in , then the variety type at ρ is identical to the regular type.

Correction to “Semi-classical analysis of Schrödinger operators and compactness in the ∂̄-Neumann problem”: [J. Math. Anal. Appl. 271 (2002) 267–282] ☆

We correct an inaccuracy in a proof in the above paper. Keep the notation from the paper (in particular regarding the L2 and Sobolev norms). The inaccuracy occurs on page 278, where it was incorrectly stated that the term ∫ 1 0 ‖u‖−1, Mr dr is estimated from above by ‖u‖−1, Ω. This term should be estimated as follows. The surfaces Mr can be thought of as pieces of the boundaries of domains Ωr which are sublevel sets of some smooth defining function for Ω. Denote by ∆−1 the inverse of the isomorphism ∆ : W 1 0 (Ω) → W−1(Ω). Then, u − ∆−1(∆u) is harmonic, and ∆−1 is compact as an operator from W−1(Ω) → W 3/4(Ω). For harmonic functions, the trace theorem (with loss of 1/2 derivative) holds for all Sobolev indices, whereas in general, it holds for indices greater than 1/2. Consequently, for all δ > 0, there is a constant Cδ such that

The Kobayashi metric in the normal direction and the mapping problem

Estimates of the Kobayashi metric in the normal direction are used to study the mapping problem in several complex variables.

On a Domain in with generic piecewise smooth levi-flat boundary and non-compact automorphism group

In this paper, we study the problem of classification of piecewise smooth domains in with non-compact automorphism groups. We prove that a simply-connected domain in with generic piecewise smooth Levi-flat boundary and non-compact automorphism group is biholomorphic to a bidisc.

Positivity of the \( \bar \partial \)-Neumann Laplacian

We study the \( \bar \partial \)-Neumann Laplacian from spectral theoretic perspectives. In particular, we show how pseudoconvexity of a bounded domain is characterized by positivity of the \( \bar \partial \)-Neumann Laplacian.

Transformation formulas for the Bergman kernels and projections of Reinhardt domains

Formulas that relate the Bergman kernel and projection of a bounded Reinhardt domain whose closure does not intersect the coordinate planes to those of its covering tube domain are obtained via the Poisson summation formula.

Geometry of Reinhardt domains of finite type in ℂ2

The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in ℂ2 is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ωm = {(z1,z2); ¦z1¦2 + ¦z2¦2m < 1 or a tube domainTm = {(z1,z2); Imz1 + (Imz2)2m < 1}. The Bergman metric for the tube domainTm is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domainTm at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kähler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in ℂ2.