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

Stefano Montaldo; Irene I. Onnis

Displaying similar documents to “A Note on Surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ ”

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.

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.

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

Thomas E. Cecil (1975)

Colloquium Mathematicae

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On the projective genus of surfaces

Pietro Sabatino (2006)

Bollettino dell'Unione Matematica Italiana

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Let $X\subset\mathbb{P}^{N}$ be a smooth irreducible non degenerate surface over the complex numbers, $N\geq 4$ . We define the projective genus of $X$ , denoted by $PG(X)$ , as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$ . In this paper we conjecture that the only surfaces for which $PG(X)=g(X)-1$ are the del Pezzo surface in $\mathbb{P}^{4}$ , in $\mathbb{P}^{5}$ and a conic bundle of degree 5 in $\mathbb{P}^{4}$ . We prove that for $N\geq 5$ if $PG(X)=g(X)-1+\lambda$ , $\lambda$ a non negative integer, then $g(X)\leq\lambda+1+\alpha$ where...

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.

Cylindrical real hyper surfaces in $\mathbf{C}^{n}$

Claudio Rea (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si stabiliscono due condizioni sufficienti per un germe di ipersuperficie reale di classe $c^{\infty}$ in $\mathbf{C}^{n}$ affinchè esistano coordinate olomorfe rispetto alle quali l'ipersuperficie risulti essere il luogo di zeri di una funzione di $k<n$ variabili e $k$ sia minimale rispetto a questa proprietà. In altre parole si vuole che l'ipersuperficie, a meno di una trasformazione bi-olomorfa, sia l’unione di sottovarietà lineari complesse, parallele di dimensione $n — k$ .

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...

Conductor and separating degrees for sets of points in $\mathbb{P}^{r}$ and in $\mathbb{P}^{1}\times\mathbb{P}^{1}$

Lucia Marino (2006)

Bollettino dell'Unione Matematica Italiana

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We attempt to generalize conductor degree's results, known in $\mathbb{P}^{2}$ , to the case of 0-dimensional schemes of $\mathbb{P}^{r}$ . In the first part of this paper, we consider the problem of characterizing the sequences generators's degrees of the conductor which are compatible with a fixed postulation (or Hilbert function) for a set of points in $\mathbb{P}^{r}$ and we determine the conductor degree of every point in a $r$ -partial intersection. In addition, we define the separating degree of a point for a 0-dimensional subscheme...

Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

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Let us consider two closed surfaces $ℳ$ , $𝒩$ of class $C^{1}$ and two functions $ϕ : ℳ \to ℝ$ , $ψ : 𝒩 \to ℝ$ of class $C^{1}$ , called measuring functions. The natural pseudodistance $d$ between the pairs $(ℳ,)$ , $(𝒩, ψ)$ is defined as the infimum of $Θ (f) : = {max}_{P \in ℳ} | ϕ (P) - ψ (f (P)) |$ as $f$ varies in the set of all homeomorphisms from $ℳ$ onto $𝒩$ . In this paper we prove that the natural pseudodistance equals either $| c_{1} - c_{2} |$ , $\frac{1}{2} | c_{1} - c_{2} |$ , or $\frac{1}{3} | c_{1} - c_{2} |$ , where $c_{1}$ and $c_{2}$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and...

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...

Isomorphisms of Royden Type Algebras Over $\mathbb{S}^{1}$

Teresa Radice, Eero Saksman, Gabriella Zecca (2009)

Bollettino dell'Unione Matematica Italiana

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Let $\mathbb{S}^{1}$ and $\mathbb{D}$ be the unit circle and the unit disc in the plane and let us denote by $\mathcal{A}(\mathbb{S}^{1})$ the algebra of the complex-valued continuous functions on $\mathbb{S}^{1}$ which are traces of functions in the Sobolev class $W^{1,2}(\mathbb{D})$ . On $\mathcal{A}(\mathbb{S}^{1})$ we define the following norm $\|u\|=\|u\|_{L^{\infty}(\mathbb{S}^{1})}+\left(\iint_{\mathbb{D}}|\nabla\tilde% {u}|^{2}\right)^{1/2}$ where is the harmonic extension of $u$ to $\mathbb{D}$ . We prove that every isomorphism of the functional algebra $\mathcal{A}(\mathbb{S}^{1})$ is a quasitsymmetric change of variables on $\mathbb{S}^{1}$ .

Congruences between modular forms and related modules

Miriam Ciavarella (2006)

Bollettino dell'Unione Matematica Italiana

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We fix $\ell$ a prime and let $M$ be an integer such that $\ell\operatorname{\not|}M$ ; let $f\in S_{2}(\Gamma_{1}(M\ell^{2}))$ be a newform supercuspidal of fixed type at $\ell$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$ , of discriminant dividing the level of $f$ , there is a local quaternionic Hecke algebra $T$ associated to $f$ . The algebra $T$ acts on a module $M_{f}$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global...

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 Relative $\gamma_{k}$ -Sets

Maddalena Bonanzinga, Filippo Cammaroto, Bruno A. Pansera (2007)

Bollettino dell'Unione Matematica Italiana

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In this note we show a relative version of $\gamma_{k}$ -set introduced and studied in [12]. We give several characterizations of this property; in particular one of the characterizations is Ramsey theoretical. Also we give a result involving a property of the corresponding mapping between function spaces.

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

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For any collection of graphs $G ₁, . . ., G_{N}$ we find the minimal dimension d such that the product $G ₁ \times . . . \times G_{N}$ is embeddable into $ℝ^{d}$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3, 3}) ⁿ$ are not embeddable into $ℝ^{2 n}$ , where K₅ and $K_{3, 3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $L k O \to S^{2 n - 1}$ , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Cylindrical real hyper surfaces in $\mathbf{C}^{n}$

Claudio Rea (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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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 Regular Threefold of General Type with $p_{g}=0$ and $P_{2}=6$

M. Cristina Ronconi (2009)

Bollettino dell'Unione Matematica Italiana

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The range of the bigenus $P_{2}$ is one of the unsolved problems concerning smooth complex projective regular threefolds of general type with $p_{g}=0$ : The examples in the literature have $P_{2}\leq 5$ . In the present paper we present a non-singular threefold with $p_{g}=q_{1}=q_{2}=0$ ; $P_{2}=6$ ; the bicanonical map is stably birational.

A New $L^{1}$ -Lower Semicontinuity Result

Daniele Graziani (2007)

Bollettino dell'Unione Matematica Italiana

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The aim of this work is to prove a chain rule and an $L^{1}$ -lower semicontinuity theorems for integral functional defined on $BV(\Omega)$ . Moreover we apply this result in order to obtain new relaxation and $\Gamma$ -convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$ .

The Schreier Property and Gauss' Lemma

Daniel D. Anderson, Muhammad Zafrullah (2007)

Bollettino dell'Unione Matematica Italiana

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Let $D$ be an integral domain with quotient field $D$ . Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus\{0\}$ , $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$ . A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus\{0\}$ , there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_{v}$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$ . We also show that $D$ is Schreier if and only if every...

The Probabilistic Zeta Function of the Alternating Group $\operatorname{Alt}(p+1)$

Marilena Massa (2007)

Bollettino dell'Unione Matematica Italiana

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We study the irreducibility of the Dirichlet polynomial $P_{G}(s)$ when $G$ is the alternating group on $p+1$ elements with $p$ prime and we prove that $P_{G}(s)$ is irreducible for infinitely many choiches of $p$ .

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...

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .

A Montel Type Result for Subharmonic Functions

R. Supper (2009)

Bollettino dell'Unione Matematica Italiana

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This article is devoted to sequences $(u_{n})_{n}$ of subharmonic functions in $\mathbb{R}^{N}$ , with finite order, whose means $J_{u_{n}}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r>0$ , $\exists A_{r}>0$ such that $J_{u_{n}}(r)\leq A_{r}$ , $\forall n\in\mathbf{N}$ . The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.

A Note on Sectorial and R-Sectorial Operators

Alberto Venni (2008)

Bollettino dell'Unione Matematica Italiana

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The following results are proved: (i) if $\alpha$ , $\beta\in\mathbb{R}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha}A^{\beta}(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.

Eisenstein ideal and reducible $\lambda$ -adic Representations Unramified Outside a Finite Number of Primes.

Miriam Ciavarella (2006)

Bollettino dell'Unione Matematica Italiana

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The object of this note is to study certain 2-dimensional $\lambda$ -adic representations of $\operatorname{Gal}(\bar{Q}/Q)$ ; fixed $p_{1},\ldots,p_{n}$ distinct primes, we will consider representations $\rho\colon G\to GL_{2}(A)$ , given by the matrix $\rho=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$ which are unramified outside $p_{1},\ldots,p_{n},\infty$ and the residue characteristic of $\lambda$ , which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$ . In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke...

Threefolds with Kodaira Dimension 0 or 3

Ezio Stagnaro (2007)

Bollettino dell'Unione Matematica Italiana

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Using the theory of adjoints and pluricanonical adjoints, we construct three nonsingular threefolds, as desingularizations of degree six hypersurfaces in $\mathbb{P}^{4}$ , having the irregularities $q_{1}=q_{2}=0$ and the following periodical sequences of plurigenera respectively $(p_{g},P_{2},P_{3},\ldots,P_{m},\ldots)=(0,0,1,0,0,1,\ldots),(0,0,0,1,0,0,0,1,% \ldots),(0,0,0,0,1,0,0,0,0,1,\ldots).$ In the Appendix, starting from the second above-mentioned example, we construct a threefold of general type with $qq_{1}=q_{2}=0,p_{g}=1$ , $P_{2}=2$ whose m-canonical transformation is birational if and only if $m\geq 11$ .

Correctors for Parabolic Equations in a Heterogeneous Fibered Medium

Mourad Sfaxi, Ali Sili (2007)

Bollettino dell'Unione Matematica Italiana

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We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\Omega$ made of two materials. The first one is located in a set $F_{\epsilon}$ of cylindrical parallel fibers periodically distributed with a period of size $\epsilon$ , and the second one is located in the "matrix" $M_{\epsilon}=\Omega\setminus F_{\epsilon}$ . The ratio between the conductivity coefficients of the two materials is of order $1/\epsilon^{2}$ . After writing the homogenized problem, we give a corrector result and prove...

$L^{p}$ Maximal Regularity for Second Order Cauchy Problems is Independent of $p$

Ralph Chill, Sachi Srivastava (2008)

Bollettino dell'Unione Matematica Italiana

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If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^{p}$ maximal regularity for some $p\in(1,\infty)$ , then it has $L^{p}$ maximal regularity for every $p\in(1,\infty)$ .

Some Separation Axioms Via Ideals

D. Sivaraj, V. Renuka Devi (2007)

Bollettino dell'Unione Matematica Italiana

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We introduce a new class of spaces, called Hausdorff modulo $\mathcal{I}$ or $T_{2}$ mod $\mathcal{I}$ spaces with respect to an ideal $\mathcal{I}$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $\mathcal{I}$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $\mathcal{I}$ -compact subsets in Hausdorff modulo $\mathcal{I}$ spaces and generalize some results of Hamlett and Jankovic....

Cohomology of Tango bundle on $\mathbb{P}^{5}$

Daniele Faenzi (2006)

Bollettino dell'Unione Matematica Italiana

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The Tango bundle $T$ is defined as the pull-back of the Cayley bundle over a smooth quadric $Q_{5}$ in $\mathbb{P}_{6}$ via a map $f$ existing only in characteristic 2 and factorizing the Frobenius $\varphi$ . The cohomology of $T$ is computed in terms of $S\otimes C$ , $\varphi^{*}(C)$ , $\text{Sym}^{2}(C)$ and $C$ , which we handle with Borel-Bott-Weil theorem.

Geometry of Syzygies via Poncelet Varieties

Giovanna Ilardi, Paola Supino, Jean Vallès (2009)

Bollettino dell'Unione Matematica Italiana

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We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$ -dimensional linear subspaces of $V_{n}=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}_{1}}(n))$ . We define $\mathfrak{X}_{k,r,d}$ as the classifying space of the $k$ -dimensional linear systems of degree $n$ on $\mathbb{P}^{1}$ , whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$ , say $r$ syzygies of degree $d$ . Firstly, we compute the dimension of $\mathfrak{X}_{k,r,d}$ . In the second part we make a link between $\mathfrak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

Displaying similar documents to “A Note on Surfaces in ℍ 2 × ℝ ”

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