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MAPPED NULL HYPERSURFACES AND LEGENDRIAN MAPS
, 2007
"... Abstract. For an (m+1)dimensional spacetime (X m+1, g), define a mapped null hypersurface to be a smooth map ν: N m → X m+1 (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and gorthogonal to ν. We study relations between map ..."
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Abstract. For an (m+1)dimensional spacetime (X m+1, g), define a mapped null hypersurface to be a smooth map ν: N m → X m+1 (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and gorthogonal to ν. We study relations between
Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian Curves
, 1995
"... Etienne Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations [8]. The points in question are the ones in which the 3jet of ..."
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Cited by 14 (5 self)
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Etienne Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations [8]. The points in question are the ones in which the 3jet
Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 61 (10 self)
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are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the pmean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining
FLOER HOMOLOGY FOR NEGATIVE LINE BUNDLES AND REEB CHORDS IN PREQUANTIZATION SPACES
, 808
"... Abstract. In this article we prove existence of Reeb orbits for BohrSommerfeld Legendrians in certain prequantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibrewise quadratic Hamiltonian functions on negative line bundles. ..."
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Cited by 2 (0 self)
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Abstract. In this article we prove existence of Reeb orbits for BohrSommerfeld Legendrians in certain prequantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibrewise quadratic Hamiltonian functions on negative line bundles
Generalizations of the SzemerédiTrotter Theorem
 DISCRETE AND COMPUTATIONAL GEOMETRY
"... We generalize the SzemerédiTrotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each i = 0, 1,..., d − 1, we are given a finite set Si of iflats in Rd or in Cd, and a (complete) flag is a tuple (f0, f1,..., fd−1), where fi ∈ Si for each i and ..."
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(1) of them can be coplanar, (ii) incidences with Legendrian lines in R3, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in R3 (involving points, lines, and planes), where no given line can contain too many points or lie on too
Minimal surfaces in pseudohermitian geometry
, 2004
"... We consider surfaces immersed in threedimensional pseudohermitian manifolds. We define the notion of (p)mean curvature and of the associated (p)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the pmean curvature not only as the tangential ..."
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are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set
Cyclic and ruled Lagrangian surfaces in complex Euclidean space
, 2007
"... We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3sphere or a Legendrian curve of the ..."
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Cited by 1 (1 self)
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We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3sphere or a Legendrian curve
Cyclic and ruled Lagrangian surfaces in complex Euclidean space
, 2008
"... Anciaux and Pascal Romon We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3sphere or ..."
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Anciaux and Pascal Romon We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3sphere
Regularity of C 1 smooth surfaces with prescribed pmean curvature in the Heisenberg group
, 2007
"... Abstract. We consider a C 1 smooth surface with prescribed p(or H)mean curvature in the 3dimensional Heisenberg group. Assuming only the prescribed pmean curvature H ∈ C 0, we show that any characteristic curve is C 2 smooth and its (line) curvature equals −H in the nonsingular domain. By introdu ..."
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Cited by 5 (1 self)
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Abstract. We consider a C 1 smooth surface with prescribed p(or H)mean curvature in the 3dimensional Heisenberg group. Assuming only the prescribed pmean curvature H ∈ C 0, we show that any characteristic curve is C 2 smooth and its (line) curvature equals −H in the nonsingular domain
A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic
"... Abstract. The AleskerPoincaré pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cospherebundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally selfad ..."
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Cited by 11 (6 self)
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adjoint with respect to this pairing. As an application, the product structure of the space of SU(2) and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line H is stated and proved. 1. Smooth valuations on manifolds Let M be a smooth manifold
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