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Register a new userSurfaces in three-dimensional Lie groups in terms of spinors
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Authors
I.A. Taimanov
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This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference Geometry related to the theory of integrable systems (RIMS, Kyoto, September 2007).
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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.
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