The bi-Lipschitz constant of an isothermal coordinate chart

The purpose of the following discussion is to obtain the classical theorem in the title of this paper as an application of our previous considerations in [29: (i), (ii), (iv), (v)] (an early announcement, under the same title, has been given in [29: (iii)] as well).These, including of course the present study, concern in effect an abstract (axiomatic) approach to the standard differential geometry of -manifolds and/or of comple(analytic) ones without employing differential calculus at all. So here again one realizes, and essentially in a strengthened way, that «certain [fundamental] quantities which a priori depend on the local diffeerential geometry are actually global topological invariants» (see e.g. [8: Introduction]). Indeed, our treatment is quite topological-algebraic in nature, to the extent that this is accomplished via sheaf theory and, in particular, through sheaf cohomology. Thus, our study might also be viewed as algebraically (viz. operator-theoretically ) oriented. Yet, to make the exposition more comprehensible, we do develop, more or less, the necessary framework for the treatment of the theorem in question, material which, otherwise, is fully discussed in [31]. On the other hand, the connection of the classical Weil's theorem [45] with the theory of geometric quantization is standard (see e.g. [19]). So as a consequence of our study, we also exhibit, in brief (in the final section 9), the result of a similar application of our formulation of the latter theorem (see Theorem 7.1 in the sequel), in conjunction with an interpretation of elementery (free) particles through (sections of) vector sheaves; the latter point of view has been essentially advocated by S.A. Selesnick (cf., for instance, [38]). Finally, we also give in section 8 an outline of particular concrete cases, apart of course from that of the classical differential geometry (real and/or complex), where the present point of view can (in part, see e.g. (8.4) below) be applied. In this respect, it is probably worth noting too that these specific applications come from abstract (commutative)harmonic analysis (cf., for instance, [35], [36] as well as [41],[42]).

Pseudoanalytic functions F are constructed as complex combinations of real-valued analytic solutions of a generalized Stokes-Beltrami system. Function-theoretic methods identify the radius of the maximal open disk of analyticity of F, and the growth of F at its boundary from local pseudoanalytic polynomial approximates in the sense of S. N. Bernstein's classical theorem in analytic function theory.

We generalize Abel's classical theorem on linear\\break equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \\subset X^n$ in a compact oriented Riemannian $n$-manifold, or more generally for any $d$-cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$-gerbe $\\Lambda_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of ‘linear equivalence to zero’. In this setting, Abel's theorem remains valid. Moreover, we generalize the classical Inversion theorem for the Abel–Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.

Holonomic quantum field theory of bosons in the Poincaré disk and the zero curvature limit

Suppose/is meromorphic in a punctured neighborhood of the origin and has an essential singularity at the origin.Given any e > 0 we show that the Riemann surface of/contains an unramified disk of spherical radius w/3 -e.The number w/3 can be replaced by w/2 if /is locally schlicht and this value is best possible.If/ is actually holomorphic, then the Riemann surface of / contains arbitrarily large unramified euclidean disks.These results generalize theorems of Valiron and Ahlfors dealing with holomorphic and meromorphic functions, respectively, on the complex plane which have an essential singularity at infinity.1. Introduction.In considering the behavior of a meromorphic function in a neighborhood of an isolated singularity, we shall normalize to the case of a function meromorphic on the punctured unit disk D* = {z: 0 <| z |< 1}.We begin by briefly establishing some notation; for more details see [7 or 8].For / meromorphic on D*, let Rf denote the Riemann surface of/, viewed as spread over the Riemann sphere P.For z E D* let rP(z, f) denote the spherical radius of the largest unramified disk in Rf with center f(z).Of course, rP(z, f) = 0 if f(z) is a branch point of Rf.Let rP(f) = sup{/P(z, /): z E D*}.If/is actually holomorphic in D*, then we regard Rf as being spread over the complex plane C. In this situation, rc(z, f) designates the euclidean radius of the largest umramified disk in Rf with center f(z) and rc(/) = sup{/-c(z,/):zED*}.Now, assume / has an essential singularity at the origin.We shall derive Bloch constants, that is, positive lower bounds independent of/, for rc(f) and rP(f), depending on whether/is holomorphic or meromorphic in D*.This localizes to the case of an isolated essential singularity known results for transcendental entire and meromorphic functions on C. If/is holomorphic in D*, we show that rc(f) = oo.Valiron estabished the analogous result for transcendental esntire functions; Bloch's theorem for normalized holomorphic functions in the unit disk is a generalization of this result [3].In case / is meromorphic in D* we obtain rp(f)> 77/3.This is improved to the sharp inequality rP(f)> tr/2 when/is also locally schlicht.For a meromorphic function on C with an essential singularity at oo, Ahlfors [1] showed that its Riemann surface contains an unramified disk of spherical radius 77/4 -e for any e > 0. Minda [8] showed that 77/4 could be replaced by 77/3 in the general case and by 77/2 for locally schlicht meromorphic functions on C.

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In a companion paper the authors also described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform.

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand–Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo’s classic theorem that (–1)-lines on a smooth surface can be contracted. The resulting noncommutative blown-down algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth. In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.

The classical Theorem of Bloch (1924) asserts that if f is a holomorphic function on a region that contains the closed unit disk ¦z¦ ≤ 1 such that $f(0)=0 {\rm and}\mid f'(0)\mid=1$ , then the image domain contains discs of radius $${3\over 2}-{\sqrt 2}>{1\over 12}$$ The optimal value is known as Bloch’s constant and 1/12 is not the best possible. In this paper we give a direct generalization of Bloch’s Theorem to the three-dimensional Euclidean space in the framework of quaternion analysis. We compute explicitly a lower bound for the Bloch constant.

On the compression of a rigid disc by finitely deformed elastic halfspaces

The classical theorem that topological surfaces can be triangulated is proved using the torus trick of Kirby plus a few basic facts about smooth or PL surfaces.

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L 2 (ℝ 2 ).

The massive scalar field theory and the chiral Schwinger model are quantized on a Poincaré disk of radius ρ. The amplitudes are derived in terms of Legendre functions. The behavior at long distances and near the boundary of some of the relevant correlation functions is studied. The exact computation of the chiral determinant appearing in the Schwinger model is obtained exploiting perturbation theory. This calculation poses interesting mathematical problems, as the Poincaré disk is a noncompact manifold with a metric tensor which diverges when it approaches the boundary. The results presented in this paper are very useful in view of possible extensions to general Riemann surfaces. Moreover, they could also shed some light in the quantization of field theories on manifolds with constant curvature scalars in higher dimensions.

Complex analysis is a fundamental branch of mathematics with wide-ranging applications in various fields. This paper provides an overview of recent progress in complex analysis, focusing on two key areas: Riemann surfaces and holomorphic dynamics. We begin by discussing the historical development of complex analysis, highlighting the contributions of Cauchy, Riemann, and Weierstrass. We then delve into the theory of Riemann surfaces, including their definition, basic properties, and classification theorems. Next, we explore holomorphic dynamics, examining its definition, fundamental concepts, and recent advances. We also explore the interactions between Riemann surfaces and holomorphic dynamics, showcasing the unifying principles in complex analysis. Finally, we discuss the applications of complex analysis in quantum mechanics, number theory, and other areas of mathematics and physics. This paper aims to provide a comprehensive overview of recent developments in complex analysis and its implications for mathematics and beyond.

The classical Bloch constant is defined for the family of functions f which are holomorphic on the open unit disk IB and normalized by f ' ( 0 ) = 1. The Bloch constant involves the size of unramified disks which lie on RI, the Riemann surface of f viewed as spread over the complex plane C. The radius of the unramified disk is measured relative to the euclidean distance function. We shall be concerned with various Bloch constants associated with families of meromorphic functions which are defined on either C or a compact Riemann surface. These meromorphic functions are not required to be normalized at some fixed point. For a meromorphic function f we view the Riemann surface R I as being spread over the Riemann sphere IP and measure the radius of unramified disks relative to the spherical distance. Let us state our main results. We show that the Bloch constant for the family of locally schlicht meromorphic functions on C is ~/2. The Bloch constant for the family of all nonconstant meromorphic functions on II? lies between ~/3 and 2 arctan (1/l~). In addition, we consider Bloch constants for other families of meromorphic functions on ~2. Our second group of results concerns the Bloch constant for the family of nonconstant meromorphic functions on a compact Riemann surface of genus g. We show that there is a lower bound N(g) for this Bloch constant that depends only on the genus g and not on the conformal type of the compact surface. This is a consequence of a theorem about branched coverings of IP that might be of independent interest. We discuss upper and lower bounds for ~(g). Finally, we outline an extension of our results to compact bordered Riemann surfaces that involves the family of holomorphic functions which have modulus one on the boundary.

Given a smooth-plane Jordan curve with bounded absolute curvature κ&gt;0, we determine equivalence classes of distinctive disks of radius 1/κ included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely symmetric trajectory behaviour with respect to the curve turning. These lead to a decomposition of the plane into a finite number of maximal regions with respect to set inclusion leading to natural lower bounds for the length an area enclosed by the curve. We present a “half version” of the Pestov–Ionin theorem, and subsequently a generalisation of the classical Blaschke rolling disk theorem. An interesting consequence is that we describe geometric conditions relying exclusively on curvature and independent of any kind of convexity that allows us to give necessary and sufficient conditions for the existence of families of rolling disks for planar domains that are not necessarily convex. We expect this approach would lead to further generalisations as, for example, characterising volumetric objects in closed surfaces as first studied by Lagunov. Although this is a classical problem in differential geometry, recent developments in industrial manufacturing when cutting along some prescribed shapes on prescribed materials have revived the necessity of a deeper understanding on disks enclosed by sufficiently smooth Jordan curves.