On surfaces in three dimensional contact manifolds

TL;DR

This paper introduces new geometric notions, DOT and COT, to analyze surfaces in contact manifolds, providing classification results, estimates on characteristic curves, and explicit solutions in the Heisenberg group.

Contribution

It defines the concepts of degree and curvature of transversality, classifies surfaces with zero COT in the Heisenberg group, and derives explicit solutions for related PDEs.

Findings

Surfaces with constant COT exist.

Classification of graphs with zero COT in the Heisenberg group.

Explicit formulas for solutions near regular points.

Abstract

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.