CR Geometry in 3-D(Dedicated to Professor Haim Brezis with admiration)

CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

On the CR Poincaré–Lelong Equation,Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds

In this paper, we solve the so-called CR Poincaré–Lelong equation by solving the CR Poisson equation on a complete noncompact CR(2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of K?hler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.     

Spacelike submanifolds of codimension two in anti-de Sitter space

In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.     

Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings. J.-W. Oh was partially supported by BK21-Yonsei University.     

楼与群I（英文）

This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two definitions of buildings.We study in the second part the spherical and affine buildings of Chevalley groups.In the third part we deal with Bruhat-Tits theory of reductive groups over local fields.Finally we discuss the construction of the p-adic flag manifolds.     

Conformal quaternionic contact curvature and the local sphere theorem

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern–Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-Sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes.     

Weyl connections and the local sphere theorem for quaternionic contact structures

We apply the theory of Weyl structures for parabolic geometries developed by Čap and Slovák (Math Scand 93(1):53–90, 2003) to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot–Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold, cf. (Geom Appl 28(4):376–394, 2010). In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern–Moser tensor in CR geometry. This tensor was first discovered via different methods by Ivanov and Vasillev (J Math Pures Appl 93:277–307, 2010), and we also get an independent proof of their Local Sphere Theorem. However, as a result of our derivation of this tensor, its fundamental properties—conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure—follow as easy corollaries from the general parabolic theory.     

A note on Pythagorean hodograph quartic spiral

By using the geometric constraints on the control polygon of a Pythagorean hodograph (PH) quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G² contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.     

The derivative nonlinear Schrödinger equation on the half line

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\mathbb{R}$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.     

THE CAUCHY PROBLEMS FOR DISSIPATIVE HYPERBOLIC MEAN CURVATURE FLOW

In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Amp`ere equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.     

Statistical manifolds with almost contact structures and its statistical submersions

In this paper, we discuss statistical manifolds with almost contact sturctures. We define a Sasaki-like statistical manifold. Moreover, we consider Sasaki-like statistical submersions, and we study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12).     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

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.     

带有线性外力场的双曲平均曲率流Cauchy问题经典解的生命跨度

本文提出并研究带有线性外力场的双曲平均曲率流,通过凸曲线的支撑函数,导出一个双曲型Monge-Ampère 方程并将其转化成Riemann 不变量满足的拟线性双曲方程组。利用拟线性双曲方程组Cauchy 问题的局部解理论,讨论带有线性外力场的双曲平均曲率流Cauchy 问题经典解的生命跨度（即局部解存在的最大时间区间）。     

Contact Pseudo-Metric Manifolds of Constant Curvature and CR Geometry

In this paper, we show that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature \({\kappa}\) , then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}\) , where \({\xi}\) is the Reeb vector field. We note that the notion of contact pseudo-metric structure is equivalent to the notion of non-degenerate almost CR manifold, then an equivalent statement of this result holds in terms of CR geometry. Moreover, we study the pseudohermitian torsion \({\tau}\) of a non-degenerate almost CR manifold.     

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

On semidiscrete constant mean curvature surfaces and their associated families

The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete \(\sinh \)-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.     

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S ³. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.