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1. CJM Online first
Abelian Surfaces with an Automorphism and Quaternionic Multiplication We construct one dimensional families of Abelian surfaces with
quaternionic multiplication
which also have an automorphism of order three or four. Using Barth's
description of the moduli space of $(2,4)$-polarized Abelian surfaces,
we find the Shimura curve parametrizing these Abelian surfaces in a
specific case.
We explicitly relate these surfaces to the Jacobians of genus two
curves studied by Hashimoto and Murabayashi.
We also describe a (Humbert) surface in Barth's moduli space which
parametrizes Abelian surfaces with real multiplication by
$\mathbf{Z}[\sqrt{2}]$.
Keywords:abelian surfaces, moduli, shimura curves Categories:14K10, 11G10, 14K20 |
2. CJM 2012 (vol 65 pp. 621)
On Surfaces in Three Dimensional Contact Manifolds In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all graphs in the Heisenberg group with vanishing COT. This is accomplished by showing that the equation for graphs with zero COT can be decomposed into two first order PDEs, one of which is the backward invisicid Burgers' equation. Finally we show that the p-minimal graph equation in the Heisenberg group also has such a decomposition. Moreover, we can use this decomposition to write down an explicit formula of a solution near a regular point. Keywords:contact manifolds, subriemannian manifolds, surfaces Category:35R03 |
3. CJM 2010 (vol 62 pp. 1201)
Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces
Very ampleness criteria for rank $2$ vector bundles over smooth, ruled
surfaces over rational and elliptic curves are given. The criteria are then
used to settle open existence questions for some special threefolds of low
degree.
Keywords:vector bundles, very ampleness, ruled surfaces Categories:14E05, 14J30 |
4. CJM 1999 (vol 51 pp. 470)
Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper half-sphere covered once or twice.
Keywords:harmonic mappings, minimal surfaces Categories:30C55, 30C62, 49Q05 |