Study of Demoulin surfaces using Gauss maps and conformal coordinates.
problem Characterizing Demoulin surfaces in real projective 3-space.
method Generalized Weierstrass type representation via primitive maps.
result Established a new representation for Demoulin surfaces.
For a surface in the 3-dimensional real projective space, we define a Gauss map, which is a quadric in R4 and called the first-order Gauss map. It will be shown that the surface is a Demoulin surface if and only if the first-order Gauss map is conformal, and the surface is a projective minimal coincidence …
Gauge theory explains Lie applicable surfaces.
problem Understanding Lie applicable surfaces.
method Gauge-theoretic approach.
result Coincides with classical notions of Ω- and Ω0-surfaces. Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
Discretizes surface theory preserving integrable structure.
problem Discretizing surface theory in projective differential geometry.
method Introduces a canonical frame and derives a Backlund transformation for discrete Demoulin surfaces.
result Derives a two-component generalization of the integrable discrete Tzitzeica equation.
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
Discretizes special surfaces using Koenigs nets.
problem Integrable structure of special surfaces.
method Discretisation via Koenigs nets.
result Preserves integrable structure in discretization.
Discrete linear Weingarten surfaces in space forms are characterized as special discrete Ω-nets, a discrete analogue of Demoulin's Ω-surfaces. It is shown that the Lie-geometric deformation of Ω-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…
Paper formulates governing equations for membrane O surfaces.
problem Formulating equations for membrane O surfaces.
method Formulated governing equations for membrane O surfaces of the 1st and 2nd kind.
result Membrane O surfaces are a subclass of Demoulin's Ω surfaces.
Discretizes projective minimal surfaces using geometric characterizations.
problem Classifying discrete projective minimal surfaces.
method Introduced canonical discrete models and line congruences.
result Discrete analogues of classical Lie quadrics and surfaces.