arXiv subjects
Publications and source records attributed to Nikolai Nikolov.
Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.
Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors.
We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the "derivatives" of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds.
This note should clarify how the behavior of certain invariant objects reflects the geometric convexity of balanced domains.
Given $A\inΩ_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $Ω_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $Ω_3,$ the zero set of this metric is completely described.
We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the general case the given weights determine uniquely a point X and the solution to the Fermat problem is the point Y, which is isogonally conjugate of type I to the point X. We obtain a similar characterization of the solution to the Fermat problem in the case of mixed weights as well.
Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given.
We show that the symmetrized bidisc is a $\Bbb C$-convex domain. This provides an example of a bounded $\Bbb C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.
Some extensions of an inequality from IMO'2001 are proven by means of the Lagrange multiplier criterion.
We present the Carathéodory-Reiffen metric and the inner Carathéodory distance on generalized parabolas. It turns out that on such parabolas the Carathéodory distance is not inner.
We prove that the $(2n-1)$-th Kobayashi pseudometric of any domain $D\subset\Bbb C^n$ coincides with the Kobayashi--Buseman pseudometric of $D,$ and that $2n-1$ is the optimal number, in general.
We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.
A generalization of an inequality from IMO is proven.
We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros.
We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains.
We prove the the multipole Lempert function is monotone under inclusion of pole sets.
Let $F\subset\Bbb C^n$ be a proper closed subset of $\Bbb C^n$ and $A\subset\Bbb C^n\setminus F$ at most countable ($n\geq 2$). We give conditions of $F$ and $A$, under which there exists a holomorphic immersion (or a proper holomorphic embedding) $ϕ:\Bbb C\to\Bbb C^n$ with $A\subsetϕ(\Bbb C)\subset\Bbb C^n\setminus F$.
The boundary behavior of the Bergman metric near a convex boundary point $z_0$ of a pseudoconvex domain $D\subset\CC^n$ is studied; it turns out that the Bergman metric at points $z\in D$ in direction of a fixed vector $X_0\in\CC^n$ tends to infinite, when z is approaching $z_0$, if and only if the boundary of D does not contain any analytic disc through $z_0$ in direction of $X_0$.