1. 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
2. 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
3. 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