On Analytic Interpolation Manifolds in Boundaries of Weakly Pseudoconvex Domains

Let Ω be a bounded, weakly pseudoconvex domain in C ⁿ , n ≤ 2, with real-analytic boundary. A real-analytic submanifold M ? ?Ω is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to (Ω¯). We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of ?Ω.     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

A Support Theorem for the Geodesic Ray Transform on Functions

Let (M,g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f∈L ²(M) vanishes on an appropriate open set of geodesics, then f=0 on the set of points lying on these geodesics. The approach is based on analytic microlocal analysis.     

Results on value distribution of L‐functions

We will establish a theorem concerning value distribution of L‐functions in the Selberg class, which shows how an L‐function and a meromorphic function are uniquely determined by their c‐values and which, as a consequence, proves a result on the uniqueness of the Riemann zeta function. The results in this paper also extend the corresponding ones in Li 6 .     

Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$² at a point p M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$³.We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$².     

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

In this paper, we study the Ruelle zeta function and the Selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. In particular, we show that they have meromorphic extensions to \mathbbC{\mathbb{C}} and satisfy functional equations. We also derive the order of the singularity of the Ruelle zeta function at the origin. To prove these results, we completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

On the smallest poles of Igusa's p-adic zeta functions

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kⁿ. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.     

Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.     

Multiple <Emphasis Type="Italic">q</Emphasis>-zeta functions and multiple <Emphasis Type="Italic">q</Emphasis>-polylogarithms

For every positive integer d we define the q-analog of multiple zeta function of depth d and study its properties, generalizing the work of Kaneko et al. who dealt with the case d=1. We first analytically continue it to a meromorphic function on ℂ ^d with explicit poles. In our Main Theorem we show that its limit when q ↑1 is the ordinary multiple zeta function. Then we consider some special values of these functions when d=2. At the end of the paper we also propose the q-analogs of multiple polylogarithms by using Jackson’s q-iterated integrals and then study some of their properties. Our definition is motivated by those of Koornwinder and Schlesinger although theirs are slightly different from ours. Partially supported by NSF grant DMS0139813 and DMS0348258.     

On Formal Maps Between Generic Submanifolds in Complex Space

Let H:(M,p)→(M ^′,p ^′) be a formal mapping between two germs of real-analytic generic submanifolds in ? ^N with nonvanishing Jacobian. Assuming M to be minimal at p and M ^′ holomorphically nondegenerate at p ^′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ^′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.     

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.     

Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

An elementary approach to the meromorphic continuation of some classical Dirichlet series

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.     

Spectral zeta functions of fractals and the complex dynamics of polynomials

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

One-Parameter Fixed-Point Theory and Gradient Flows of Closed 1-Forms

We use the one-parameter fixed-point theory of Geoghegan and Nicas to get information about the closed orbit structure of transverse gradient flows of closed 1-forms on a closed manifold M. We define a noncommutative zeta function in an object related to the first Hochschild homology group of the Novikov ring associated to the 1-form and relate it to the torsion of a natural chain homotopy equivalence between the Novikov complex and a completed simplicial chain complex of the universal cover of M.     

On the p-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is p-adic meromorphic.     

Analyticity of Higher-Order Moduli of Continuity of Real-Analytic Functions

The Perel'man's result according to which the first modulus of continuity of any real-analytic function f is a function analytic in a certain neighborhood of the origin is generalized to the case of arbitrary moduli of continuity of higher order.