2,515 research outputs found
An extension of heat hierarchy
We propose a formally completely integrable extension of heat hierarchy based
on the space of symmetries isomorphic to the Weyl algebra . The
extended heat hierarchy will be the basic model for the analysis of the
extension of KP hierarchy, and other integrable equations.Comment: This note is incorporated into arXiv:1409.7024, arXiv:1408.324
Legendrian Gronwall conjecture
The Gronwall conjecture states that a planar 3-web of foliations which admits
more than one distinct linearizations is locally equivalent to an algebraic
web. We propose an analogue of the Gronwall conjecture for the 3-web of
foliations by Legendrian curves in a contact three manifold. The Legendrian
Gronwall conjecture states that a Legendrian 3-web admits at most one distinct
local linearization, with the only exception when it is locally equivalent to
the dual linear Legendrian 3-web of the Legendrian twisted cubic in \,\PP^3.
We give a partial answer to the conjecture in the affirmative for the class of
Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web
which is sufficiently flat at a reference point is rigid under local linear
Legendrian deformation.Comment: 15 page
Projectively deformable Legendrian surfaces
Consider an immersed Legendrian surface in the five dimensional complex
projective space equipped with the standard homogeneous contact structure. We
introduce a class of fourth order projective Legendrian deformation called
\emph{-deformation}, and give a differential geometric characterization
of surfaces admitting maximum three parameter family of such deformations. Two
explicit examples of maximally -deformable surfaces are constructed;
the first one is given by a Legendrian map from \, \PP^2 blown up at three
distinct collinear points, which is an embedding away from the -2-curve and
degenerates to a point along the -2-curve. The second one is a Legendrian
embedding of the degree 6 del Pezzo surface, \, \PP^2 blown up at three
non-collinear points. In both cases, the Legendrian map is given by a system of
cubics through the three points, which is a subsystem of the anti-canonical
system.Comment: 33 page
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