Search

We adapt the ''royal road'' method used to simplify automatic analyticity theorems in noncommutative function theory to several complex variables. We show that certain families of functions must be real analytic if they have certain nice properties on one-dimensional slices. Let E ⊂ Rd be open. A function ƒ : E → R is matrix monotone lite if ƒ(φ1(t), . . . , φd(t)) is a matrix monotone function of t whenever t ∈ (0,1), the φi are automorphisms of the upper half plane, and the tuple (φ1(t), . . . , φd(t)) maps (0,1) into E. We use the ''royal road" to show that a function is matrix monotone lite if and only if it analytically continues to the multivariate upper half plane as a map into the upper half plane. Moreover, matrix monotone lite functions in two variables are locally matrix monotone in the sense of Agler-McCarthy-Young. [ABSTRACT FROM AUTHOR]

Robert L. Bryant (see [Astérisque 154-155 (1987), pp. 321-347, Proposition 4]) proved that an isolated singularity of a conformal metric of positive constant curvature with finite area on a Riemann surface is a conical one. Using complex analysis, we find all local models for an isolated singularity of a flat metric whose area satisfies some polynomial growth condition near the singularity. In particular, we show that an isolated singularity of a flat metric with finite area must be conic. [ABSTRACT FROM AUTHOR]

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry. [ABSTRACT FROM AUTHOR]

We show a residues formula for maps generically transversal to regular holomorphic distributions. [ABSTRACT FROM AUTHOR]

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in Cn. For a bounded strongly pseudoconvex domain in Cn and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to the identity map with respect to the C1-norm and maps the boundary point to a strongly convex boundary point. [ABSTRACT FROM AUTHOR]

Let X be a 1-convex manifold with the exceptional set S, which is also a manifold, Z a complex manifold, Z → X a holomorphic submersion, a : X → Z a holomorphic section and S ⊂ U ... X an open relatively compact 1-convex set. We construct a metric on a vector bundle E → Z restricted to a neighbourhood V of a(U), conic along a(S) with at most polynomial poles at a(S) and positive Nakano curvature tensor in bidegree (p, q). [ABSTRACT FROM AUTHOR]

When the moment problem is indeterminate, the Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization. [ABSTRACT FROM AUTHOR]

Let g be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of g. Here, we present a variation of this idea which leads to a new, and quite elementary, construction of g itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of g. [ABSTRACT FROM AUTHOR]

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least 2. [ABSTRACT FROM AUTHOR]

In this paper we present a variational approach to a class of elliptic problems with superlinear semipositone nonlinearities. We consider the parametrized family of problems ... with l > 0, a continuous, and f subcritical and superlinear at infinity. We obtain positive solutions of such problems for 0 < λ < λ0 by combining a suitable rescaling with a continuity argument. In doing so, we require f to be of regular variation at infinity, so that f does not need to be asymptotic to a power. Furthermore, a may vanish in open parts of O or change sign. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a new type of the classical Schwarz lemma at the boundary for holomorphic self-mappings of the unit ball in ℂn, and then give the boundary version of the rigidity theorem. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a new type of the classical Schwarz lemma at the boundary for holomorphic self-mappings of the unit ball in ℂn, and then give the boundary version of the rigidity theorem. [ABSTRACT FROM AUTHOR]

For random graphs, the containment problem considers the probability that a binomial random graph G(n, p) contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well known that the (sharp) threshold is at p = 1/n. We consider a natural analogue of this question for higher-dimensional random complexes Xk(n, p), first studied by Cohen, Costa, Farber and Kappeler for k = 2. Improving previous results, we show that p = Θ(1/ √ n) is the (coarse) threshold for containing a subdivision of any fixed complete 2-complex. For higher dimensions k > 2, we get that p = O(n-1/k) is an upper bound for the threshold probability of containing a subdivision of a fixed k-dimensional complex. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for a variational inequality involving the complex Monge-Ampère determinant and give applications to proper holomorphic mappings. We also clarify the proof of a sharp inequality about volumes of holomorphic images by placing it in this more general context. [ABSTRACT FROM AUTHOR]

Given a set E in ℂm and a point p ∈ E, there is a unique smallest complex-analytic germ Xp containing Ep, called the holomorphic closure of Ep. We study the holomorphic closure of semialgebraic arc-symmetric sets. Our main application concerns CR-continuation of semialgebraic arc-analytic mappings: A mapping f : M → ℂn on a connected real-analytic CR manifold which is semialgebraic arc-analytic and CR on a non-empty open subset of M is CR on the whole M. [ABSTRACT FROM AUTHOR]

In this paper, we extend the holomorphic L² torsion introduced by Carey, Farber and Mathai to the case without the determinant class condition. We compute the metric variation formula for the holomorphic L² torsion in our case. We also study the asymptotics of the holomorphic L² torsion associated with a power of a positive line bundle. [ABSTRACT FROM AUTHOR]

It is shown that if a proper holomorphic map f : ℂn → ℂN, 1 < n ≤ N, sends a pseudoconvex real analytic hypersurface M of finite type into another such hypersurface, then any (n - 1)-dimensional component of the critical locus of f intersects both sides of M. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in ℂn. [ABSTRACT FROM AUTHOR]

We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K). [ABSTRACT FROM AUTHOR]

Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials r(z,***) on Ćn X Ćn for which r(z,***) ||z||2d = ||h(z)||² for some natural number d and a holomorphic polynomial mapping h = (h1,...,hK) from Ćn to ĆK. When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in Ć[z1,...,zn] and apply a well-known result of Macaulay to estimate some natural quantities. [ABSTRACT FROM AUTHOR]

In this paper, we explore a method for associating L-series to weakly holomorphic modular forms and then proceed to study their L-values. As our main application, we prove a very curious limiting theorem which relates three "periods" of a mock modular form and its shadow to the ratio of their noncritical L-values. Critical L-values are then shown to fit nicely within the framework of period polynomials and an extended Eichler-Shimura theory recently studied by Guerzhoy, Ono and the first and third authors. [ABSTRACT FROM AUTHOR]

We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz's exponents in the real and complex setting. [ABSTRACT FROM AUTHOR]

We give a new application of the theory of holomorphic motions to the study of the distortion of holomorphic maps of disks and annuli establishing sharp distortion theorems. [ABSTRACT FROM AUTHOR]

Let A be a (complex, unital) semisimple Banach algebra and denote by ΩA its open spectral unit ball, that is, the set ΩA = {a ∈, A : σ(a) ⊆ D}, where σ (a) denotes the spectrum of a in A and D is the open unit disc in the complex plane. We prove that if F : ΩA → ΩA is a holomorphic map satisfying F (0) = 0 and F′ (0) = I (the identity of A), then for a in ΩA the intersection of all closed discs lying inside D and containing σ (a) equals the intersection of all closed discs lying inside D and containing σ (F (a)). When all the elements of A have an at most countable spectrum and F is biholomorphic, this implies that F preserves the convex hull of the spectrum. As an application of the same equality, we prove that if B is a semisimple Banach algebra and T : A → B is a unital surjective spectral isometry, then σ (T (a)) = σ (a) in the case when σ (a) has exactly two elements. [ABSTRACT FROM AUTHOR]

If Mn is a compact Riemannian manifold for which H1(Mn, Z) = 0, V is a continuous complex valued function whose imaginary part is of constant sign, and -Δψ+Vψ = 0 for some C2 complex valued function ψ on Mn, then either ψ vanishes somewhere or there is a constant c and an everywhere positive function F such that ψ = cF . [ABSTRACT FROM AUTHOR]