Uniformity of holomorphic families of non-homeomorphic planar Riemann surfaces

Sachiko Hamano

Displaying similar documents to “Uniformity of holomorphic families of non-homeomorphic planar Riemann surfaces”

On the completeness of flat surfaces in $S^{3}$

Thomas E. Cecil (1975)

Colloquium Mathematicae

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A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli, Giovanni Felder (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $D$ be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an $n$ -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of $D$ as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology $H H^{2 n} (𝒟_{n}, 𝒟_{n}^{*})$ of the algebra of differential operators on a formal neighbourhood...

Counting lines on surfaces

Samuel Boissière, Alessandra Sarti (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper deals with surfaces with many lines. It is well-known that a cubic contains $27$ of them and that the maximal number for a quartic is $64$ . In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with $352$ lines, and give examples of surfaces of degree $d$ containing a sequence of $d (d - 2) + 4$ skew lines.

< $> 𝒜 <$ >-Schemes and Zariski-Riemann Spaces

Satoshi Takagi (2012)

Rendiconti del Seminario Matematico della Università di Padova

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Dehn twists on nonorientable surfaces

Michał Stukow (2006)

Fundamenta Mathematicae

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Let $t_{a}$ be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, $I (t ⁿ_{a} (b), b) = | n | I (a, b) ²$ , where I(·,·) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if ℳ(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup...

Numerical Campedelli surfaces with fundamental group of order 9

Margarida Mendes Lopes, Rita Pardini (2008)

Journal of the European Mathematical Society

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We give explicit constructions of all the numerical Campedelli surfaces, i.e. the minimal surfaces of general type with $K^{2} = 2$ and $p_{g} = 0$ , whose fundamental group has order 9. There are three families, one with $π_{1}^{alg} = ℤ_{9}$ and two with $π_{1}^{alg} = ℤ_{3}^{2}$ . We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with $π_{1}^{alg} = ℤ_{9}$ and for one of the families of surfaces with $π_{1}^{alg} = ℤ_{3}^{2}$ the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces...

Translation surfaces of finite type in ${Sol}_{3}$

Bendehiba Senoussi, Hassan Al-Zoubi (2020)

Commentationes Mathematicae Universitatis Carolinae

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In the homogeneous space Sol $_{3}$ , a translation surface is parametrized by $r (s, t) = γ_{1} (s) * γ_{2} (t)$ , where $γ_{1}$ and $γ_{2}$ are curves contained in coordinate planes. In this article, we study translation invariant surfaces in ${Sol}_{3}$ , which has finite type immersion.

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends

Gerd Dethloff, Pham Hoang Ha, Pham Duc Thoan (2016)

Colloquium Mathematicae

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We study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

Systole growth for finite area hyperbolic surfaces

Florent Balacheff, Eran Makover, Hugo Parlier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this note, we observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g, n)$ is greater than a function that grows logarithmically in terms of the ratio $g / n$ .

On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Matthias Schütt, Andreas Schweizer (2013)

Annales de l’institut Fourier

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We explicitly determine the elliptic $K 3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$ , for each of the two possible maximal fibre types, $I_{19}$ and $I_{14}^{*}$ , the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I_{13}^{*}$ , and there exist two (resp. one) one-parameter families of such surfaces.

Some surfaces with maximal Picard number

Arnaud Beauville (2014)

Journal de l’École polytechnique — Mathématiques

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For a smooth complex projective variety, the rank $ρ$ of the Néron-Severi group is bounded by the Hodge number $h^{1, 1}$ . Varieties with $ρ = h^{1, 1}$ have interesting properties, but are rather sparse, particularly in dimension $2$ . We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If $(M, \nabla)$ is a manifold with a symmetric linear connection, then $T^{*} M$ can be endowed with the natural Riemann extension $\bar{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*} M, \bar{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar{g}$ reduces...

A Note on Surfaces in $\mathbb{H}^{2}\times\mathbb{R}$

Stefano Montaldo, Irene I. Onnis (2007)

Bollettino dell'Unione Matematica Italiana

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In this article we consider surfaces in the product space $\mathbb{H}^{2}\times\mathbb{R}$ of the hyperbolic plane $\mathbb{H}^{2}$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the local classification of totally umbilical surfaces.

Even sets of nodes on sextic surfaces

Fabrizio Catanese, Fabio Tonoli (2007)

Journal of the European Mathematical Society

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We determine the possible even sets of nodes on sextic surfaces in $ℙ^{3}$ , showing in particular that their cardinalities are exactly the numbers in the set ${24, 32, 40, 56}$ . We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification...

The KSBA compactification for the moduli space of degree two $K 3$ pairs

Radu Laza (2016)

Journal of the European Mathematical Society

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X, H)$ consisting of a degree two $K 3$ surface $X$ and an ample divisor $H$ . Specifically, we construct and describe explicitly a geometric compactification ${\bar{P}}_{2}$ for the moduli of degree two $K 3$ pairs. This compactification...

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this article, we study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give...

Dimension vs. genus: A surface realization of the little k-cubes and an $E_{\infty}$ operad

Ralph M. Kaufmann (2009)

Banach Center Publications

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We define a new $E_{\infty}$ operad based on surfaces with foliations which contains $E_{k}$ suboperads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes (thus making contact with string topology), by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new Ω spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension...

Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka (2008)

Studia Mathematica

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The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on $^{ω}$ , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(a_{p}, s) = \prod_{p} {(1 - a_{p} p^{- s})}^{- 1}$ for $a_{p}$ in $^{ω}$ . Among other things, using the Haar measure on $^{ω}$ for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived...