arXiv subjects
Publications and source records attributed to Nikolai Nikolov.
We prove two separate lower bounds -- one for nondegenerate convex domains and the other for nondegenerate $\mathbb{C}$-convex (but not necessarily convex) domains -- for the squeezing function that hold true for all domains in $\mathbb{C}^n$, for a fixed $n\geq 2$, of the stated class. We provide explicit expressions in terms of $n$ for these estimates.
Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.
In this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of the underlying random variable. In particular we study the minima and the maxima of these sequences and prove that they are attained for indices of the sequences smaller or equal than the support of the underlying random variable. Noting that the minimum itself is a minimum of multi-variate polynomials we conjecture that one universal polynomial envelopes the minimum from below and that it is maximal in some sense and largest in another. We prove this conjecture in a special case.
In the last decade, G. Bharali and A. Zimmer defined a class of domains called Goldilocks domains and they showed that such a domain satisfies a visibility condition with respect to the Kobayashi extremal curves. Inspired by their construction, we define a subclass of Goldilocks domains called strongly Goldilocks domains and we prove a quantitative visibility result on strongly Goldilocks domains. Using our quantitative visibility result, we extend the Gehring-Hayman theorem on simply connected planar domains to strongly Goldilocks domains. As an application of our construction, we also give lower estimates to the Kobayashi distance.
Recently, the visibility property of Kobayashi (almost) geodesics has been used to provide localizations of the Kobayashi distance. In this note, we provide sufficient growth conditions for the Kobayashi distance to obtain new strong multiplicative and additive localization results. Curiously, the conditions we provide are deeply related to the behaviour of the Kobayashi geodesics.
We prove that in a strongly pseudoconvex domain with smooth boundary, then the length of a geodesic for the Kobayashi-Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the points.
It is proved for a strongly pseudoconvex domain $D$ in $\Bbb C^d$ with $\mathcal C^{2,α}$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\partial D$ and whose difference is non-tangential to $\partial D$ intersect a compact subset of $D$ that depends only on the rate of non-tangentiality. As an application, a lower bound for the Kobayashi distance is obtained.
We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain a geometric description of all possible areas obtained in this way given the ratios of the lengths of consecutive segments the lines divide these two opposite sides.
We introduce the notion of locally visible and locally Gromov hyperbolic domains in $\mathbb C^d$. We prove that a bounded domain in $\mathbb C^d$ is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.
The main result of this paper is that for any norm on a complex or real $n$-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor $2^n-1$. Furthermore, the constant $2^n-1$ is tight. We also prove that the norms of any two extremal bases are comparable with a factor of $2^n-1$, which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.
Comparison and localization results for the Lempert function, the Carathéodory distance and their infinitesimal forms on strongly pseudoconvex domains are obtained. Related results for visible and strongly complete domains are proved.
In this note, we introduce the notion of visible boundary points with respect to Kobayashi distance for domains in complex euclidean space. Following the work of Sarkar, we obtain additive and multiplicative localization results about Kobayashi distance near visible boundary points. Then using the additive localization result, we show that visibility property with respect to Kobayashi distance is a local property of the boundary points and it does not depend on the domain.
We obtain explicit bounds on the difference and ratio between "local" and "global" Kobayashi distances in a domain of $\mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.
Let $D\subset \mathbb C^n$ be a bounded domain. A pair of distinct boundary points $\{p,q\}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}\subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.
A quantitative version of strong localization of the Kobayashi, Azukawa and Sibony metrics, as well as of the squeezing function, near a plurisubharmonic peak boundary point of a domain in $\Bbb C^n$ is given. As an application, the behavior of these metrics near a strictly pseudoconvex boundary point is studied. A weak localization of the three metrics and the squeezing function is also given near a plurisubharmonic antipeak boundary point.
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $\mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $\mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the $\mathcal C^2$ case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.
In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem.
An extension of the estimates for the squeezing function of strictly pseudoconvex domains obtained recently by J. E. Fornæss and E. Wold in \cite{FW1} is applied to derive a sharp boundary behaviour of invariant metrics and Bergman curvatures.