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Search: All articles in the CMB digital archive with keyword surface

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1. CMB Online first

Yang, Qingjie; Zhong, Weiting
Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$
In this paper we prove that there is only one conjugacy class of dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral symplectic group can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus $p-1$. A pair of representative generators of the realizable class is also given.

Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domain
Categories:20H25, 57M60

2. CMB Online first

da Silva, Rosângela Maria; Tenenblat, Keti
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids.

Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space
Categories:53A10, 53B40

3. CMB Online first

Parlier, Hugo
A short note on short pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22

4. CMB Online first

Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin
Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator
In this paper we give a characterization of a real hypersurface of Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb C}^{m+2})} } }$, which means a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in ${G_2({\mathbb C}^{m+2})}$, by the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.

Keywords:real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operator
Categories:53C40, 53C15

5. CMB 2012 (vol 56 pp. 466)

Aulaskari, Rauno; Rättyä, Jouni
Inclusion Relations for New Function Spaces on Riemann Surfaces
We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, BMOA or the recently defined $Q_p$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

Keywords:Bloch space, BMOA, $Q_p$, Green's function, hyperbolic Riemann surface
Categories:30F35, 30H25, 30H30

6. CMB 2011 (vol 56 pp. 306)

Pérez, Juan de Dios; Suh, Young Jin
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel
We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

Keywords:complex projective space, real hypersurface, structure Jacobi operator
Categories:53C15, 53C40

7. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
The Lang--Weil Estimate for Cubic Hypersurfaces
An improved estimate is provided for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

Keywords:cubic hypersurface, rational points, finite fields
Categories:11G25, 14G15

8. CMB 2011 (vol 55 pp. 26)

Bertin, Marie José
A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series
We present another example of a $3$-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series.

Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, Livné criterion, Rankin-Cohen brackets
Categories:11, 14D, 14J

9. CMB 2011 (vol 55 pp. 176)

Spirn, Daniel; Wright, J. Douglas
Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of $t^{-5/6}$. This rate is due to competition between surface tension and gravitation at $O(1)$ wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called "slowest wave". Additionally, we combine our dispersive estimates with $L^2$ type energy bounds to prove a family of Strichartz estimates.

Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimates
Categories:76B07, 76B15, 76B45

10. CMB 2011 (vol 55 pp. 114)

Kon, S. H.; Loo, Tee-How
On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$-Parallel Shape Operator
In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

Keywords:complex space form, Hopf hypersurfaces, ruled real hypersurfaces, $\eta$-parallel shape operator
Categories:53C40, 53C15

11. CMB 2011 (vol 54 pp. 422)

Pérez, Juan de Dios; Suh, Young Jin
Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space
We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Keywords:complex projective space, real hypersurface, structure Jacobi operator, two conditions
Categories:53C15, 53B25

12. CMB 2011 (vol 54 pp. 311)

Marzougui, Habib
Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows
We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder
Categories:37B20, 37E35

13. CMB 2009 (vol 53 pp. 218)

Biswas, Indranil
Restriction of the Tangent Bundle of $G/P$ to a Hypersurface
Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.

Keywords:tangent bundle, homogeneous space, semistability, hypersurface
Categories:14F05, 14J60, 14M15

14. CMB 2009 (vol 52 pp. 493)

Artebani, Michela
A One-Dimensional Family of $K3$ Surfaces with a $\Z_4$ Action
The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of $\Z_4$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Keywords:genus three curves, $K3$ surfaces
Categories:14J28, 14J50, 14J10

15. CMB 2009 (vol 52 pp. 257)

Ikeda, Toru
Essential Surfaces in Graph Link Exteriors
An irreducible graph manifold $M$ contains an essential torus if it is not a special Seifert manifold. Whether $M$ contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits $M$ into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors.

Keywords:Graph link, Graph manifold, Seifert manifold, Essential surface
Category:57M25

16. CMB 2009 (vol 52 pp. 66)

Dryden, Emily B.; Strohmaier, Alexander
Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces
Categories:58J53, 11F72

17. CMB 2008 (vol 51 pp. 359)

Cho, Jong Taek; Ki, U-Hang
Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator
Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $(A)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Keywords:complex space form, real hypersurface, structure Jacobi operator
Categories:53B20, 53C15, 53C25

18. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it.

Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory
Categories:14G05, 14J28, 11D41

19. CMB 2006 (vol 49 pp. 624)

Teragaito, Masakazu
On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces
For a non-trivial knot in the $3$-sphere, only integral Dehn surgery can create a closed $3$-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

Keywords:knot, Dehn surgery, non-orientable surface
Category:57M25

20. CMB 2004 (vol 47 pp. 398)

McKinnon, David
A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces
Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating rational curves such that the density of rational points on $U-Z_U$ is strictly less than the density of rational points on $Z_U$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets $Z_U$ for successively smaller sets $U$. In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of

Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height
Categories:11G35, 14G05

21. CMB 2004 (vol 47 pp. 22)

Goto, Yasuhiro
A Note on the Height of the Formal Brauer Group of a $K3$ Surface
Using weighted Delsarte surfaces, we give examples of $K3$ surfaces in positive characteristic whose formal Brauer groups have height equal to $5$, $8$ or $9$. These are among the four values of the height left open in the article of Yui \cite{Y}.

Keywords:formal Brauer groups, $K3$ surfaces in positive, characteristic, weighted Delsarte surfaces
Categories:14L05, 14J28

22. CMB 2002 (vol 45 pp. 154)

Weitsman, Allen
On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Keywords:harmonic mappings, dilatation, minimal surfaces
Categories:30C62, 31A05, 31A20, 49Q05

23. CMB 2000 (vol 43 pp. 427)

Ivey, Thomas A.
Helices, Hasimoto Surfaces and Bäcklund Transformations
Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in $\R^3$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in $\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in $H^3$ or $S^3$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

Keywords:surfaces, filament flow, Bäcklund transformations
Categories:53A05, 58F37, 52C42, 58A15

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