发表机构
Indian Institute of Technology Bhubaneswar; School of Basic Sciences（布巴内斯瓦尔印度理工学院; 基础科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有界光滑$\mathbb{C}$-凸域的Kobayashi距离Gromov双曲性、D'Angelo型有限性与$\bar{\partial}$-Neumann问题次椭圆估计三者等价，并给出相关指数与边界等价性结果。
AI 中文摘要
本文研究了具有光滑边界的有限$\nmathbb{C}$-凸域$\Omega \subset \mathbb{C}^n$（$n \geq 2$）的三个性质，即其Kobayashi距离的Gromov双曲性、$\partial \Omega$的D'Angelo型的有限性，以及$(0,1)$-形式上$\bar{\partial}$-Neumann问题次椭圆估计的存在性。我们证明了这三个性质是等价的。次椭圆估计的获得无需Catlin的多次调和权函数构造。我们的权函数是Bergman核的对数的显式有界变换。一个单一的量控制着这三个性质，即通过$\Omega$中一点在给定复方向上的最大圆盘的半径。Gromov双曲性迫使这些半径满足幂次界。有限型给出幂次$1/M$，其中$M$是最大型，且该幂次不可改进。次椭圆估计通过作用于归一化Bergman核的典范解算子给出幂次界。解析估计不需要边界正则性。在有界伪凸域上，一个有界权函数，其诱导的Hessian在$(0,q)$-形式上被边界距离的负幂次从下方界定，可给出$(0,q)$-形式零延拓的Sobolev估计，以及$\bar{\partial}$-Neumann算子的特征值界。在有限型光滑$\mathbb{C}$-凸域上，我们证明了方向展开指数、法向展开指数和次椭圆增益的上确界均等于$1/M$。接下来，我们证明了在Kobayashi距离为Gromov双曲的Lipschitz $\mathbb{C}$-凸域上，欧几里得边界与Gromov边界是双Hölder等价的。最后，我们给出了环带度量Gromov双曲性的Hardy型判据。
英文摘要
In this article, we study three properties of a bounded $\mathbb{C}$-convex domain $Ω\subset \mathbb{C}^n$, $n \geq 2$, with smooth boundary, namely Gromov hyperbolicity of its Kobayashi distance, finiteness of the D'Angelo type of $\partial Ω$, and the existence of a subelliptic estimate for the $\bar{\partial}$-Neumann problem on $(0,1)$-forms. We prove that these three properties are equivalent. The subelliptic estimates are obtained without Catlin's construction of plurisubharmonic weights. Our weights are explicit bounded transforms of the logarithm of the Bergman kernel. A single quantity controls all three properties, namely the radius of the largest disc through a point of $Ω$ in a given complex direction. Gromov hyperbolicity forces a power bound for these radii. Finite type gives the power $1/M$, where $M$ is the maximal type, and this power cannot be improved. A subelliptic estimate gives a power bound through the canonical solution operator applied to normalized Bergman kernels. The analytic estimates need no boundary regularity. On a bounded pseudoconvex domain, a bounded weight whose induced Hessian on $(0,q)$-forms is bounded below by a negative power of the boundary distance gives a Sobolev estimate for the extension by zero of $(0,q)$-forms, and also eigenvalue bounds for the $\bar{\partial}$-Neumann operator. On smooth $\mathbb{C}$-convex domains of finite type, we show that the directional expansion exponent, the normal expansion exponent, and the supremum of the subelliptic gains are all equal to $1/M$. Next, we prove that on a Lipschitz $\mathbb{C}$-convex domain whose Kobayashi distance is Gromov hyperbolic, the Euclidean boundary and the Gromov boundary are bi-Hölder equivalent. Finally, we give a Hardy-type criterion for Gromov hyperbolicity of collar metrics.
CommentsPreliminary draft; remaining information will be added in the subsequent version
