%% Papers Published
@article{fds360690,
Author = {Porter, C and Zelenko, I},
Title = {Absolute parallelism for 2-nondegenerate CR structures via
bigraded Tanaka prolongation},
Journal = {Journal Fur Die Reine Und Angewandte Mathematik},
Volume = {2021},
Number = {777},
Pages = {195-250},
Year = {2021},
Month = {August},
url = {http://dx.doi.org/10.1515/crelle-2021-0012},
Abstract = {This article is devoted to the local geometry of everywhere
2-nondegenerate CR manifolds M of hypersurface type. An
absolute parallelism for such structures was recently
constructed independently by Isaev and Zaitsev, Medori and
Spiro, and Pocchiola in the minimal possible dimension
(dimM=5{\dim M=5}), and for dimM=7{\dim M=7} in certain
cases by the first author. In the present paper, we develop
a bigraded (i.e., ×{\mathbb{Z}\times\mathbb{Z}}-graded)
analog of Tanaka's prolongation procedure to construct an
absolute parallelism for these CR structures in arbitrary
(odd) dimension with Levi kernel of arbitrary admissible
dimension. We introduce the notion of a bigraded Tanaka
symbol-a complex bigraded vector space-containing all
essential information about the CR structure. Under the
additional regularity assumption that the symbol is a Lie
algebra, we define a bigraded analog of the Tanaka universal
algebraic prolongation, endowed with an anti-linear
involution, and prove that for any CR structure with a given
regular symbol there exists a canonical absolute parallelism
on a bundle whose dimension is that of the bigraded
universal algebraic prolongation. Moreover, we show that for
each regular symbol there is a unique (up to local
equivalence) such CR structure whose algebra of
infinitesimal symmetries has maximal possible dimension, and
the latter algebra is isomorphic to the real part of the
bigraded universal algebraic prolongation of the symbol. In
the case of 1-dimensional Levi kernel we classify all
regular symbols and calculate their bigraded universal
algebraic prolongations. In this case, the regular symbols
can be subdivided into nilpotent, strongly non-nilpotent,
and weakly non-nilpotent. The bigraded universal algebraic
prolongation of strongly non-nilpotent regular symbols is
isomorphic to the complex orthogonal algebra (m,â
){\mathfrak{so}(m,\mathbb{C})}, where m=12(dimM+5){m=\tfrac{1}{2}(\dim
M+5)}. Any real form of this algebra-except
(m){\mathfrak{so}(m)} and (m-1,1){\mathfrak{so}(m-1,1)}-corresponds
to the real part of the bigraded universal algebraic
prolongation of exactly one strongly non-nilpotent regular
CR symbol. However, for a fixed dimM≥7{\dim M\geq 7} the
dimension of the bigraded universal algebraic prolongations
of all possible regular CR symbols achieves its maximum on
one of the nilpotent regular symbols, and this maximal
dimension is 14(dimM-1)2+7{\frac{1}{4}(\dim
M-1)^{2}+7}.},
Doi = {10.1515/crelle-2021-0012},
Key = {fds360690}
}
@article{fds355197,
Author = {Porter, C},
Title = {3-folds CR-embedded in 5-dimensional real
hyperquadrics},
Journal = {Journal of Geometry and Physics},
Volume = {163},
Year = {2021},
Month = {May},
url = {http://dx.doi.org/10.1016/j.geomphys.2021.104107},
Abstract = {E. Cartan's method of moving frames is applied to
3-dimensional manifolds M which are CR-embedded in
5-dimensional real hyperquadrics Q in order to classify M up
to CR symmetries of Q given by the action of one of the Lie
groups SU(3,1) or SU(2,2). In the latter case, the CR
structure of M derives from a shear-free null geodesic
congruence on Minkowski spacetime, and the relationship to
relativity is discussed. In both cases, we compute which
homogeneous CR 3-folds appear in Q.},
Doi = {10.1016/j.geomphys.2021.104107},
Key = {fds355197}
}
@article{fds361358,
Author = {Porter, C},
Title = {Unit Tangent Bundles, CR Leaf Spaces, and Hypercomplex
Structures},
Year = {2021},
Month = {February},
Abstract = {Unit tangent bundles $UM$ of semi-Riemannian manifolds $M$
are shown to be examples of dynamical Legendrian contact
structures, which were defined in recent work [25] of
Sykes-Zelenko to generalize leaf spaces of 2-nondegenerate
CR manifolds. In doing so, Sykes-Zelenko extended the
classification in Porter-Zelenko [20] of regular,
2-nondegenerate CR structures to those that can be recovered
from their leaf space. The present paper treats dynamical
Legendrian contact structures associated with
2-nondegenerate CR structures which were called "strongly
regular" in Porter-Zelenko, named "L-contact structures."
Closely related to Lie-contact structures, L-contact
manifolds have homogeneous models given by isotropic
Grassmannians of complex 2-planes whose algebra of
infinitesimal symmetries is one of $\mathfrak{so}(p+2,q+2)$
or $\mathfrak{so}^*(2p+4)$ for $p\geq1$, $q\geq0$. Each
2-plane in the homogeneous model is a split-quaternionic or
quaternionic line, respectively, and more general L-contact
structures arise on contact manifolds with hypercomplex
structures, unit tangent bundles being a prime example. The
Ricci curvature tensor of $M$ is used to define the
"Ricci-shifted" L-contact structure on $UM$, whose Nijenhuis
tensor vanishes when $M$ is conformally flat. In the
language of Sykes-Zelenko (for $M$ analytic), such $UM$ is
the leaf space of a 2-nondegenerate CR manifold which is
"recoverable" from $UM$, providing a new source of examples
of 2-nondegenerate CR structures.},
Key = {fds361358}
}
@article{fds353837,
Author = {Porter, C},
Title = {The local equivalence problem for $7$-dimensional,
$2$-nondegenerate $\operatorname{CR}$ manifolds},
Journal = {Communications in Analysis and Geometry},
Volume = {27},
Number = {7},
Pages = {1583-1638},
Publisher = {International Press of Boston},
Year = {2019},
url = {http://dx.doi.org/10.4310/cag.2019.v27.n7.a5},
Doi = {10.4310/cag.2019.v27.n7.a5},
Key = {fds353837}
}
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Mathematics Department