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Peter Pflug

Publications and source records attributed to Peter Pflug.

At least 37 records · Page 2Linked to original sources
arXiv2009

Cross theorems with singularities

We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.

math.CV↗
arXiv2008

An elementary proof of the cross theorem in the Reinhardt case

We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image.

math.CV↗
arXiv2008

On the derivatives of the Lempert functions

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.

math.CV↗
arXiv2007

Boundary cross theorem in dimension 1 with singularities

Let $D$ and $G$ be copies of the open unit disc in $\C,$ let $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp. $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy" $\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\setminus M,$ measurable on $A\times B,$ and separately holomorphic on $\big((A\times G) \cup (D\times B)\big)\setminus M$ "extends" to a function holomorphic on $\hat{W\setminus M}.$

math.CV↗
arXiv2007

Envelope of holomorphy for boundary cross sets

Let $D\subset \C^n,$ $G\subset \C^m$ be open sets, let $A$ (resp. $B$) be a subset of the boundary $\partial D$ (resp. $\partial G$) and let $W$ be the 2-fold boundary cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ An open subset $X\subset\C^{n+m} $ is said to be the ``envelope of holomorphy" of $W$ if it is, in some sense, the maximal open set with the following property: Any function locally bounded on $W$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a holomorphic function defined on $X$ which admits the boundary values $f$ a.e. on $W.$ In this work we will determine the envelope of holomorphy of some boundary crosses.

math.CV↗
arXiv2006

Boundary cross theorem in dimension 1

Let $X, Y$ be two complex manifolds of dimension 1 which are countable at infinity, let $D\subset X,$ $ G\subset Y$ be two open sets, let $A$ (resp. $B$) be a subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that $D$ (resp. $G$) is {\it Jordan-curve-like on $A$} (resp. $B$) and that $A$ and $B$ are {\it of positive length}. We determine the "envelope of holomorphy" $\hat{W}$ of $W$ in the sense that any function locally bounded on $W,$ measurable on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function holomorphic on the interior of $\hat{W}.$

math.CV↗
arXiv2006

Generalization of a theorem of Gonchar

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $ G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Under a geometric condition on the boundary sets $A$ and $B,$ we show that every function locally bounded, separately continuous on $W,$ continuous on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and holomorphic on the interior of $\hat{W}.$

math.CV↗
arXiv2005

A remark on separate holomorphy

Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice $X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the family $\{f|_{X_a}: f\in\mathcal F\}$.

math.CV↗