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Josip Globevnik

Publications and source records attributed to Josip Globevnik.

At least 19 recordsLinked to original sources
arXiv2020

On discs in bidiscs

Let U be the closed unit disc in C. We show that there is no continuous map F:U-->U^2, holomorphic on Int(U) and such that F(bU) = b(U^2).

math.CV↗
arXiv2016

A construction of complete complex hypersurfaces in the ball with control on the topology

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a complete proper holomorphic embedding from $D$ into the unit ball of $\mathbb{C}^{N+1}$. For $N=1$, we derive the existence of complete properly embedded complex curves in the unit ball of $\mathbb{C}^2$, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc $\mathbb{D}\subset \mathbb{C}$ into the unit ball of $\mathbb{C}^2$. These are the first known examples of complete bounded embedded complex hypersurfaces in $\mathbb{C}^{N+1}$ with any control on the topology.

math.CV↗
arXiv2016

Embedding complete holomorphic discs through discrete sets

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for any path p:[0,1)-->U such that |p(t)|-->1 as t-->1, the path t--> f(p(t)) from [0,1) to B has infinite length.

math.CV↗
arXiv2014

Holomorphic functions unbounded on curves of finite length

Given a pseudoconvex domain D in C^N, N>1, we prove that there is a holomorphic function f on D such that the lengths of paths p: [0,1]--> D along which Re f is bounded above, with p(0) fixed, grow arbitrarily fast as p(1)--> bD. A consequence is the existence of a complete closed complex hypersurface M in D such that the lengths of paths p:[0,1]--> M, with p(0) fixed, grow arbitrarily fast as p(1)-->bD.

math.CV↗
arXiv2014

A complete complex hypersurface in the ball of C^N

In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal result of the present paper is a construction of a holomorphic function on the open unit ball B_N of C^N whose real part is unbounded on every path in B_N of finite length that ends on the boundary of B_N. A consequence is the existence of a complete, closed, complex hypersurface in B_N. This gives a positive answer to Yang's question in all dimensions k, N, 1\leq k<N, by providing properly embedded complete complex manifolds.

math.CV↗
arXiv2012

The winding number of PF+1 for polynomials P and meromorphic extendibility of F

Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD then f extends holomorphically through D. We prove a related result on meromorphic extendibility for smooth functions with finitely many zeros of finite order, which, in particular, implies that the conjecture holds for real analytic functions.

math.CV↗
arXiv2011

Meromorphic extensions from small families of circles and holomorphic extensions from spheres

Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that is different from 1 if one of the points a, b is in B and the other in the complement of B and such that at least one of the numbers , is different from 1. We prove that if a continuous function f on the sphere bB extends holomorphically into B along each complex line which passes through one of the points a, b, c then f extends holomorphically through B. This generalizes recent work of L.Baracco who proved such a result in the case when the points a, b, c are contained in B. The proof is different from the one of Baracco and uses the following one variable result which we also prove in the paper and which in the real analytic case follows from the work of M.Agranovsky: Let D be the open unit disc in C. Given a in D let C(a) be the family of all circles in D obtained as the images of circles centered at the origin under an automorphism of D that maps the origin to a. Given distinct points a, b in D and a positive integer n, a continuous function f on the closed unit disc extends meromorphically from every circle T in either C(a) or C(b) through the disc bounded by T with the only pole at the center of T of degree not exceeding n if and only if f is of the form f(z) = g_0(z)+g_1(z)\bar z +...+ g_n(z)\bar z^n where the functions g_0, g_1, ..., g_n are holomorphic on D.

math.CV↗
arXiv2009

Small families of complex lines for testing holomorphic extendibility

Let B be the open unit ball in C^2 and let a, b be two points in B. It is known that for every positive integer k there is a function f in C^k(bB) which extends holomorphically into B along any complex line passing through either a or b yet f does not extend holomorphically through B. In the paper we show that there is no such function in C^\infty (bB). Moreover, we obtain a fairly complete description of pairs of points a, b in C^2 such that if a function f in C^\infty(bB) extends holomorphically into B along each complex line passing through either a or b that meets B, then f extends holomorphically through B.

math.CV↗
arXiv2009

Analyticity of functions analytic on circles

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U and passing through the point p. Then f is holomorphic in the interior of U.

math.CV↗
arXiv2008

On meromorphic extendibility

Let D be a bounded domain in the complex plane whose boundary bD consists of finitely many pairwise disjoint real analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on the closure of D which are holomorphic on D. For continuous functions f on bD we obtain a characterization of meromorphic extendibility in terms of the argument principle: f extends meromorphically through D if and only if there is a nonnegative integer N such that the winding number of Pf+Q along bD is bounded below by -N for all P, Q in A(D) such that Pf+Q has no zero on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.

math.CV↗
arXiv2006

Meromorphic Extendibility and the Argument Principle

Let U be the open unit disc in C. Given a continuous function g: bU --> C-{0} denote by W(g) the winding number of g around the origin. We prove that a continuous function f: bU --> C extends meromorphically through U if and only if there is a nonnegative integer N such that W(Pf+Q) is greater than or equal to -N for every pair P,Q of polynomials such that Pf+Q has no zero on bU. If this is the case then the meromorphic extension of f has at most N poles in U, counting multiplicity.

math.CV↗
arXiv2006

Degree and holomorphic extensions

Let D be a bounded convex domain in C^N, N\geq 2. We prove that a continous map F from bD to C^N extends holomorphically through D if and only if for every polynomial map P from C^N to C^N such that F+P has no zero on bD, the degree of F+P|bD is nonnegative. We also prove another such theorem for more general domains.

math.CV↗
arXiv2006

Holomorphic Extendibility and Mapping Degree

Let D be a bounded, finitely connected domain in the complex plane without isolated points in the boundary and let f be a continuous function on the boundary bD. Let F be a continuous extension of f to the closure of D. We prove that f extends holomorphically through D if and only if the degree of F+h is nonnegative for every holomorphic function h on D such that F+h is bounded away from zero near bD.

math.CV↗
arXiv2005

Single valued conjugates and holomorphic extendibility

Let D be a bounded domain in the complex plane whose boundary consists of m pairwise disjoint simple closed curves where m is greater than one. Let A(bD) be the algebra of all continuous functions on bD which extend holomorphically through D. We show that a continuous function f on bD belongs to A(bD) if for each g in A(bD) the harmonic extension of Re(fg) to D has a single valued conjugate.

math.CV↗