New complete embedded minimal surfaces in $${{\mathbb {H} ^2\times \mathbb {R}}}$$

We construct three kinds of complete embedded minimal surfaces in \({\mathbb {H}^2\times \mathbb {R}}\) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in \({\mathbb {R}^3}\) . The third one is a finite total curvature surface which is conformal to \({\mathbb {S}^2\setminus\{p_1,\ldots,p_k\}, k\geq3.}\)     

Explicit classes of permutation polynomials of $$ \mathbb{F}_{3^{3m} } $$

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions. This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China (Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405)     

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.     

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 Willmore Legendrian surfaces in $$\mathbb {S}^5$$ and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in \(\mathbb {S}^5\), and then we use this relation to prove a classification result for Willmore Legendrian spheres in \(\mathbb {S}^5\). We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in \(\mathbb {S}^5\) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let \(\Sigma \) be a closed surface and \((M,\alpha ,g_\alpha ,J)\) a 5-dimensional Sasakian manifold with a contact form \(\alpha \), an associated metric \(g_\alpha \) and an almost complex structure J. Assume that \(f:\Sigma \mapsto M\) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if \((M,\alpha ,g_\alpha ,J)\) is a Sasakian Einstein manifold, in particular \(\mathbb {S}^5\).     

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.     

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).     

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.     

Complete negatively curved immersed ends in $$\mathbb {R}^3$$

This paper extends, in a sharp way, the famous Efimov’s Theorem to immersed ends in \(\mathbb {R}^3\). More precisely, let M be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of M into \(\mathbb {R}^3\) satisfying that \(\int _M |K|=+\infty \) and \(K\le -\kappa <0\), where \(\kappa \) is a positive constant and K is the Gaussian curvature of M. In particular Efimov’s Theorem holds for complete Hadamard immersed surfaces, whose Gaussian curvature K is bounded away from zero outside a compact set.     

Global Bifurcations in Generic One-parameter Families on $$\mathbb{S}^2$$

In this paper we prove that generic one-parameter families of vector fields on \(\mathbb{S}^2\) in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.     

Generalized constant ratio surfaces in \mathbb{E}^3

It iswell-known that the positionvector function is themost basic geometric object for a surface immersed in the three dimensional Euclidean space $\mathbb{E}^3 $ . In 2001, B.-Y. Chen defined constant ratio hypersurfaces in Euclidean n-spaces. Independently, in 2010, by using another approach in dimension 3, the second author classified constant slope surfaces. In this paper, we extend this concept in order to study surfaces with the property that the tangential component of the position vector is a principal direction on the surface.     

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 \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

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...     

Clifford Cauchy type integrals on Ahlfors-David regular surfaces in $$\mathbb{R}^{m + 1} $$

The main goal of this paper is centred around the study of the behavior of the Cauchy type integral and its corresponding singular version, both over nonsmooth domains in Euclidean space. This approach is based on a recently developed quaternionic Cauchy integrals theory [1, 5, 7] within the three-dimensional setting. The present work involves the extension of fundamental results of the already cited references showing that the Clifford singular integral operator has a proper invariant subspace of generalized H?lder continuous functions defined in a surface of the (m+1)-dimensional Euclidean space.     

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.     

The Hilbert functions ofk-configurations in\mathbb{P}^2 and\mathbb{P}^3

In this paper, we proved the set of points which are the vertices of then-gon in $\mathbb{P}^2 $ (n ≥ 3) has the Uniform Position Property and what the graded free resolutions of the ideals ofk-configurations in $\mathbb{P}^3 $ are.     

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...     

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 ${\mathcal{N}.}$