Robert Xin Dong, Yuan Zhang
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we prove that any degree $d$ deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so called Brieskorn lattice/Petrov module of the polynomial. Some applications on the cyclicity of c…
In [7], Dong and I proved that the domains $D \subset \mathbb{C}$ of finite volume whose on-diagonal Bergman kernels $K(\cdot, \cdot)$ satisfy $K(z_0, z_0) = Volume(D)^{-1}$ are disks minus closed polar sets. We utilized the solution of the Suita conjecture, a deep theorem of several complex variables. In this note, I present a significantly more elementary proof of this theorem that does not use several complex variables. As a corollary, a new lower bound for the on-diagonal Bergman kernel is given. Finally, I show that the only real ellipsoid in Webster normal form which satisfies $K(0, 0) =…
The logarithmic capacity (also called Chebyshev constant or transfinite diameter) of two real intervals $[-1,α]\cup[β,1]$ has been given explicitly with the help of Jacobi's elliptic and theta functions already by Achieser in 1930. By proving several inequalities for these elliptic and theta functions, an upper bound for the logarithmic capacity in terms of elementary functions of $α$ and $β$ is derived.
We continue the study in \cite{As18, AAZ18} by giving a multitude of applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel that was introduced and studied in \cite{As18, AAZ18}. We compare quantitatively the projective logarithmic capacity with the complex Monge-Ampère capacity on $\mathbb P^n$ and we deduce that the set of zero logarithmic capacity is of Monge-Ampère capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capa…
We verify an old conjecture of G. Polya and G. Szego saying that the regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area.
We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.
Potential theory on the complement of a subset of the real axis attracts a lot of attention both in function theory and applied sciences. The paper discusses one aspect of the theory - the logarithmic capacity of closed subsets of the real line. We give simple but precise upper and lower bounds for the logarithmic capacity of multiple intervals and a lower bound valid also for closed sets comprising an infinite number of intervals. Using some known methods to compute the exact values of capacity we demonstrate graphically how our estimates compare with them. The main machinery behind our resul…
Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non simply connected domains, generalizing a recent example of Fornæss.
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 conc…
Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set
该研究针对非Brjuno实数集，推导得出其对数Hausdorff测度与对数容量的精确临界指数均为2，并明确了不同参数范围下的测度与容量性质，给出其容量维数结果。
Fast and accurate computation of the logarithmic capacity of compact sets
本文提出了一种数值方法，用于计算由Jordan曲线边界和有限连通补集界定的复平面上紧集的对数容量，该方法基于共形映射到lemniscatic域和Neumann核的边界积分方程求解，展示了方法的高效性和准确性，并应用于Cantor中间三分之一集及其推广的对数容量估计。
A logarithmic Bogomolov--Sommese vanishing theorem on compact Kähler manifolds
本文在紧Kähler流形上建立对数Bogomolov–Sommese消失定理，并结合Iwai和Matsumura的结构定理得到对数余切丛第二陈类为零的刚性结果。
A new Poincaré type rigidity phenomenon with applications
本文在有界对称域上的圆丛间建立局部CR映射的最优刚性定理，证明非常值映射可延拓为有理双全纯映射，并由此给出反典范丛延拓结果及高维紧致CR超曲面新例子。
We study the behavior of various set-functions under holomorphic motions. We show that, under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.
Computing the logarithmic capacity of compact sets having (infinitely) many components with the Charge Simulation Method
本文提出用电荷模拟法计算由多个小分量组成的紧集的对数容量，每个分量仅用一个电荷点，结合预条件迭代和快速多极方法，高效且高精度，并在广义康托集和康托尘上验证。
Exceptional Sets of Positive Capacity in Gross's Star Theorem
本文证明Gross定理中的例外集不必有对数容度零，构造了超越整函数及其正则局部逆，使其例外集含正容度紧集，且对任意s<1，该紧集有正s维豪斯多夫测度。
Fermat's Last Theorem in star-invariant subspaces: capacity-zero boundary spectrum
该研究针对内函数θ边界谱具对数容量零的情况，证明指数n≥4的费马方程在星不变子空间K_θ^p中无非平凡解，肯定回答了Dyakonov提出的相关问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。