Expand all Collapse all | Results 1 - 22 of 22 |
1. CMB Online first
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 |
2. CMB Online first
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 SeppÃ¤lÃ¤. 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 |
3. CMB Online first
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 |
4. CMB 2012 (vol 56 pp. 466)
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 |
5. CMB 2011 (vol 56 pp. 306)
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel |
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 |
6. CMB 2011 (vol 56 pp. 500)
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 |
7. CMB 2011 (vol 55 pp. 26)
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, LivnÃ© criterion, Rankin-Cohen brackets Categories:11, 14D, 14J |
8. CMB 2011 (vol 55 pp. 176)
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 |
9. CMB 2011 (vol 55 pp. 114)
On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$-Parallel Shape Operator |
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 |
10. CMB 2011 (vol 54 pp. 422)
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 |
11. CMB 2011 (vol 54 pp. 311)
Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows We give some extension to theorems of JimÃ©nez LÃ³pez and Soler LÃ³pez 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 |
12. CMB 2009 (vol 53 pp. 218)
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 |
13. CMB 2009 (vol 52 pp. 493)
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 |
14. CMB 2009 (vol 52 pp. 257)
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 |
15. CMB 2009 (vol 52 pp. 66)
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 |
16. CMB 2008 (vol 51 pp. 359)
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 |
17. CMB 2006 (vol 49 pp. 560)
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, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 |
18. CMB 2006 (vol 49 pp. 624)
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 |
19. CMB 2004 (vol 47 pp. 398)
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 |
20. CMB 2004 (vol 47 pp. 22)
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 |
21. CMB 2002 (vol 45 pp. 154)
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 |
22. CMB 2000 (vol 43 pp. 427)
Helices, Hasimoto Surfaces and BÃ¤cklund 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, BÃ¤cklund transformations Categories:53A05, 58F37, 52C42, 58A15 |