On Surfaces in Three Dimensional Contact Manifolds

Lee, Paul W. Y.

Canadian journal of mathematics, Tome 65 (2013) no. 3, pp. 621-633

Résumé

In this paper, we introduce two quantities defined on a surface in a contact manifold. The first one is called degree of transversality $(\text{DOT})$ , which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality $(\text{COT})$ , is designed to give a comparison principle for $\text{DOT}$ along characteristic curves under bounds on $\text{COT}$ . In particular, this gives estimates on lengths of characteristic curves, assuming $\text{COT}$ is bounded below by a positive constant.We show that surfaces with constant $\text{COT}$ exist, and we classify all graphs in the Heisenberg group with vanishing $\text{COT}$ . This is accomplished by showing that the equation for graphs with zero $\text{COT}$ 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.

Détail

DOI : 10.4153/CJM-2012-027-5

Mots-clés : 35R03, contact manifolds, subriemannian manifolds, surfaces