arXiv subjects

Buma L. Fridman

Publications and source records attributed to Buma L. Fridman.

8 recordsLinked to original sources

arXiv2011

Nonlinear Convergence Sets of Divergent Power Series

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh, is introduced. Given a family y=ϕ_{s}(t,x)=sb_{1}(x)t+b_{2}(x)t^{2}+... of analytic curves in C\timesC^{n} passing through the origin, Conv_ϕ(f) of a formal power series f(y,t,x)\inC[[y,t,x]] is defined to be the set of all s\inC for which the power series f(ϕ_{s}(t,x),t,x) converges as a series in (t,x). We prove that for a subset E\subsetC there exists a divergent formal power series f(y,t,x)\inC[[y,t,x]] such that E=Conv_ϕ(f) if and only if E is a F_{σ} set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case ϕ_{s}(t,x)=st.

math.CV↗

arXiv2011

Holomorphic functions on subsets of C

Let $Γ$ be a $C^\infty $ curve in $\Bbb{C}$ containing 0; it becomes $Γ_θ$ after rotation by angle $θ$ about 0. Suppose a $C^\infty $ function $f$ can be extended holomorphically to a neighborhood of each element of the family $\{Γ_θ\}$. We prove that under some conditions on $Γ$ the function $f$ is necessarily holomorphic in a neighborhood of the origin. In case $Γ$ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for \textit{real analyticity}. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\Bbb{C}$.

math.CV↗

arXiv2011

Osgood-Hartogs type properties of power series and smooth functions

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given $C^\infty $ function $f$ is proved when the restriction of $f$ on analytic curves belonging to some family is analytic. Our results generalize two known statements: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogs-type" problems.

math.CV↗

arXiv2011

Testing holomorphy on curves

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.

math.CV↗

arXiv2006

Fixed points and Determining Sets for Holomorphic Self-Maps of a Hyperbolic Manifold

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different.

math.CV↗

arXiv2006

Isolated fixed point sets for holomorphic maps

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed.

math.CV↗

arXiv2002

Perturbation of domains and automorphism groups

The paper is devoted to the description of changes of the structure of the holomorphic automorphism group of a bounded domain in ${\Bbb C}^n$ under small perturbation of this domain in the Hausdorff metric. We consider a number of examples when an arbitrary small perturbation can lead to a domain with a larger group, present theorems concerning upper semicontinuity property of some invariants of automorphism groups. We also prove that the dimension of an abelian subgroup of the automorphism group of a bounded domain in ${\Bbb C}^n$ does not exceed $n$.

math.CV↗

arXiv2002

Upper semicontinuity of the dimensions of automorphism groups of domains in $C^n$

Let $H^n$ be the metric space of all bounded domains in $C^n$ with the metric equal to the Hausdorff distance between boundaries of domains. We prove that the dimension of the group of automorphisms of domains is an upper semicontinuous function on $\H^n$. We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in $H^n$.

math.CV↗