Hyperbolic algebraic and analytic curves

Abstract

A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions about the existence of biholomorphic maps. Conversely, we show that the algebraic-geometric viewpoint leads to interesting questions in classical analysis. IntroductionBy a hyperbolic algebraic curve we mean a set V that is the intersection of a one dimensional algebraic set C with a bounded open set in C n . We call them hyperbolic to emphasize that we wish to study the geometry and function theory on V , not the global theory on C, nor just the local theory on small neighborhoods of points of V . Note that by a holomorphic function on V we mean a function that in a neighborhood of every point of V is the restriction of a holomorphic function on a neighborhood in C n .

The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary

“…A geometric argument then gives the original claim. For a more detailed understanding of such varieties, which has been rapidly developed in recent years, see [1,3,2,7,5].…”

On polynomial $n$-tuples of commuting isometries

“…. , h n of h As discussed in the beginning section 3 of [1], it follows from [3] that there is a finite subset Y of S such that any function in A(S) which vanishes to sufficiently high order on Y also belongs to A h (S). It follows from this that A h (S) has finite codimension in A(S), and if g ∈ A h (S) vanishes to sufficiently high order on Y , then g ·A(S) ⊆ A h (S).…”

On the Isomorphism Question for Complete Pick Multiplier Algebras

“…In [4], the following definition was introduced and studied: Then the map Proof: The main idea of the proof goes back to [6]. To show that α given by the formula f → f • h is a well defined bounded and invertible map from F W onto H 2 h (Σ), we will formally compute α * and R := αα * , and show that R is an injective Fredholm operator.…”

“…One can think of W as the space Σ with a finite number of pairs of points identified, or limiting cases of this where one gets cusps. See[4] for a detailed description.…”