arXiv subjects
Publications and source records attributed to Robert Xin Dong.
We prove that if the Carathéodory metric on a strictly pseudoconvex domain with a smooth boundary is locally Kähler near the boundary, then the domain is biholomorphic to a ball. We also establish a local rigidity theorem for domains with Bergman metrics of constant holomorphic sectional curvature, and highlight this relationship with the Lu constant.
By using the Bergman representative coordinate and Calabi's diastasis, we extend a theorem of Lu to bounded pseudoconvex domains whose Bergman metric is incomplete with constant holomorphic sectional curvature. We characterize such domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. We also provide a multidimensional generalization of Carathéodory's theorem on the continuous extension of the biholomorphisms up to the closures. In particular, sufficient conditions are given, in terms of the Bergman kernel, for the boundary of a biholomorphic ball to be a topological sphere.
We obtain uniform estimates for the canonical solution to $\bar\partial u=f$ on the Cartesian product of bounded planar domains with $C^2$ boundaries, when $f$ is continuous up to the boundary. This generalizes Landucci's result for the bidisc toward higher dimensional product domains. In particular, it answers an open question of Kerzman for continuous datum.
We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical Taylor expansion for Abelian differentials and period matrices.
For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_β$, and the analytic capacity $c_{B}$ satisfy the inequality chain $πK \geq c^2_β \geq c^2_B$; moreover, equality holds at a single point between any two of the three quantities if and only if $X$ is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that $c_{B}^2 \geq πv^{-1}(X)$ on planar domains, where $v(\cdot)$ is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green's function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.
Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly less a relatively closed polar set. In comparison with Guan and Zhou's proof, our proof essentially depends on Maitani and Yamaguchi's variation formula for the Bergman kernel, and we explore the harmonicity in such variations. As applications, we characterize the above surface by the constant Gaussian curvature property of the Bergman kernel or metric, and also find the relations with disc quotients. Additionally, we obtain results on planar domains without the Bergman-completeness assumption.
In light of the Suita conjecture, we explore various rigidity phenomena concerning the Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance and volume.
We use weighted $L^2$-methods to obtain sharp pointwise estimates for the canonical solution to the equation $\bar\partial u=f$ on smoothly bounded strictly convex domains and the Cartan classical domain domains when $f$ is bounded in the Bergman metric $g$. We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains $Ω$ of rank $2$ the maximum blow up order is greater than $-\log δ_Ω(z)$, which was obtained for the unit ball case by Berndtsson. For example, for IV$(n)$ with $n \geq 3$, the maximum blow up order is $δ(z)^{1 -{n \over 2}}$ because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on $f$.
We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set.
We use the Suita conjecture (now a theorem) to prove that for any domain $Ω\subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(Ω)^{-1}$ for some $z_0 \in Ω$ if and only if $Ω$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) closed polar set. When $Ω$ is bounded with $C^{\infty}$-boundary, we provide a simple proof of this using the zero set of the Szegö kernel. Finally, we show that this theorem fails to hold in $\mathbb{C}^n$ for $n > 1$ by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$.
For a once-punctured complex torus, we compare the Bergman kernel and the fundamental metric, by constructing explicitly the Evans-Selberg potential and discussing its asymptotic behaviors. This work aims to generalize the Suita type results to potential-theoretically parabolic Riemann surfaces.
We provide explicit formulas of Evans kernels, Evans-Selberg potentials and fundamental metrics on potential-theoretically parabolic planar domains.
The author proves that the generalized Suita conjecture holds for any complex torus, which means that $ απK \geq c^2(α\in\mathbb R)$, $c$ being the modified logarithmic capacity and $K$ being the Bergman kernel on the diagonal. The open problems for general compact Riemann surfaces with genus $\geq2$ is also elaborated. The proof relies in part on elliptic function theories.