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On Surfaces in Three Dimensional Contact Manifolds

  Published:2012-07-17
 Printed: Jun 2013
  • Paul W. Y. Lee,
    The Chinese University of Hong Kong, Shatin, Hong Kong
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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.

Keywords: contact manifolds, subriemannian manifolds, surfaces contact manifolds, subriemannian manifolds, surfaces
MSC Classifications: 35R03 show english descriptions Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc. 35R03 - Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.
 

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