Oscillatory integral operators in scattering theory

We consider a particular Fourier integral operator with folding canonical relations, which arises in scattering theory: the Radon Transform of Melrose and Taylor. We obtain the regularity properties of this operator when the obstacle admits tangent planes with contact of precise order k (Theorem 1.1 and its Corollary).

For these purposes, we derive asymptotic estimates for oscillatory integral operators in Wn with folding canonical relations (Theorem 2.2). Asymptotics correspond to vanishing principal curvature of a fold of one of the projections from the canonical relation, and to small support of the localization of oscillatory integral operator.     

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

Integrability of canonic affinor structures of homogeneous periodic Φ-spaces

We study connections between the Lie bracket on the tangent space of a homogeneous periodic Φ-space and the operators of canonical affinor structures of this space. The relations obtained allowed us to single out several cases of integrability of the structures under consideration.     

On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.     

Curves in Affine and Semi-Euclidean Spaces

In this paper we define centroaffine invariant arc length and curvature functions of a curve in affine n-space. Then we consider the properties and relations of the curves in affine space and Semi-Euclidean space. Using these notions and conclusions, by solving certain differential equations, we give some examples and classifications of the curves in affine 2-space and 3-space.     

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

Connections on contact manifolds and contact twistor space

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.     

Remarks on η-Einstein unit tangent bundles

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2. Authors’ addresses: Y. D. Chai, S. H. Chun, J. H. Park, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea; K. Sekigawa, Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan     

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.     

三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面

本文主要给出了三维Minkowski空间中非类光曲线的双曲达布像和从切高斯曲面的奇点分类,并且建立了奇点和曲线几何不变量之间的联系,其中曲线几何不变量与曲线同螺线切触的阶数密切相关.     

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

Normal almost contact metric structures of hyperbolic type of the first kind on hypersurfaces of a space of constant holomorphic curvature of hyperbolic type

We consider normal integrable Sasakian almost contact metric structures of hyperbolic type of the first kind on hypersurfaces of a space of constant holomorphic curvature of hyperbolic type, in particular, on hypersurfaces of a flat A-space of hyperbolic type.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 22–32, January–March, 2005.Translated by V. Mackeviius     

Symplectic enumeration

Every complex projective space of odd dimension carries a natural contact structure. We give first steps towards the enumeration of curves in ℙ³ tangent to the contact structure. Such a curve is involutive in the sense that its homogeneous ideal is closed under Poisson bracket. Involutive curves in ℙ³ contained in a plane split as a union of concurrent lines. We give a formula for the number of plane involutive curves of a given degree in ℙ³ meeting the appropriate number of lines. We also discuss strategies to deal with the enumeration of involutive rational curves.     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

Proper Holomorphic Maps from Domains in C2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space,where the domain is of finite type and admits a transverse circle action.The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

Proper Holomorphic Maps from Domains in ?2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

Jørgensen's Inequality for Complex Hyperbolic Space

Jørgensen's inequality gives a necessary condition for a nonelementary two generator group of isometries of hyperbolic space to be discrete. We give analogues of Jørgensen's inequality for nonelementary groups of isometries of complex hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. These results give an improvement over earlier results of Basmajian and Miner.     

Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.