On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Classifying ACM sets of points in $${\mathbb{P}^{1} \times \mathbb{P}^{1}}$$ via separators

The purpose of this note is to give a new, short proof of a classification of ACM sets of points in in terms of separators.     

On Hamiltonian Minimality of Isotropic Nonhomogeneous Tori in $$\mathbb{H}^n$$ and $$\mathbb{C} \mathrm{P}^{2n+1}$$

Mathematical Notes - We construct a family of flat isotropic nonhomogeneous tori in $$\mathbb{H}^n$$ and $$\mathbb{C}\mathrm{P}^{2n+1}$$ and find necessary and sufficient conditions for their...     

Surfaces in $${\mathbb{S}^2\times\mathbb{R}}$$ with a canonical principal direction

We show a way to choose nice coordinates on a surface in and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in are both minimal and flat. We also show that the condition that the projection of the direction tangent to onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in . We present classification theorems under the extra assumption of minimality or flatness. J. Fastenakels is a research assistant of the Research Foundation—Flanders (FWO). J. Van der Veken is a postdoctoral researcher supported by the Research Foundation—Flanders (FWO). This work was partially supported by project G.0432.07 of the Research Foundation—Flanders (FWO).     

Constructing infinitely many smooth structures on $3\mathbb{CP}^2 \# n\overline{\mathbb{CP}}^2$

Using Seiberg-Witten theory and rational blow-down procedures of R. Fintushel and R.J. Stern, we construct infinitely many irreducible smooth structures, both symplectic and non-symplectic, on the four-manifold for each integer n lying in the interval . Received: 17 January 2000 / Published online: 18 January 2002     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

Hypersurfaces in nearly Kaehler manifold $$\pmb {\mathbb {S}}^{\varvec{3}}\varvec{\times } \pmb {\mathbb {S}}^{\varvec{3}}$$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold \({\mathbb {S}}^3\times {\mathbb {S}}^3\) with a conformal vector field. There are three almost contact metric structures on a hypersurface of \({\mathbb {S}}^3\times {\mathbb {S}}^3\), and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Dirac Operators with Singular Potentials Supported on Unbounded Surfaces in $$\mathbb{R}^{3}$$

Functional Analysis and Its Applications - We consider the self-adjointness and essential spectrum of 3D Dirac operators with bounded variable magnetic and electrostatic potentials and with...     

Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system

Proper trajectories of type {\mathbb{C}^{\ast}} of a polynomial vector field on {\mathbb{C}^2}

We prove that if a polynomial vector field on ${\mathbb{C}^2}$ has a proper and non-algebraic trajectory analytically isomorphic to ${\mathbb{C}^{\ast}}$ all its trajectories are proper, and except at most one which is contained in an algebraic curve of type ${\mathbb{C}}$ all of them are of type ${\mathbb{C}^{\ast}}$ . As corollary we obtain an analytic version of Lin?CZa?denberg Theorem for polynomial foliations.     

The Nonexistence of Pseudoquaternions in {\mathbb{C}^{2\times 2}}

The field of quaternions, denoted by \mathbbH{\mathbb{H}} can be represented as an isomorphic four dimensional subspace of \mathbbR^4×4{\mathbb{R}^{4\times 4}}, the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in \mathbbR^4×4{\mathbb{R}^{4\times 4}} which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called field of pseudoquaternions. It exists in \mathbbR^4×4{\mathbb{R}^{4\times 4}} but not in \mathbbH{\mathbb{H}}. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in \mathbbR⁴{\mathbb{R}^4}. And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b.     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ to $$\mathbb {Z}_{m}$$Zm

In this paper, several nonexistence results on generalized bent functions \(f:\mathbb {Z}_{2}^{n} \rightarrow \mathbb {Z}_{m}\) are presented by using the knowledge on cyclotomic number fields and their imaginary quadratic subfields.     

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Rigid Embeddings of Sasakian Hyperquadrics in $$\mathbb {C}^{n+1}$$

We give a classification of Sasakian manifolds that are CR-equivalent to hyperquadrics by describing their exact parameter space. For “ half” of the parameter space, we find an explicit representation by defining equations. This problem is related to the problem of finding pseudo-Kähler potentials with prescribed Ricci curvature.