Surfaces with constant curvature in S<Superscript>2</Superscript>×R and H<Superscript>2</Superscript>×R. Height estimates and representation

We obtain optimal height estimates for surfaces in ℍ² × ℝ and × ℝ with constant Gaussian curvature K(I) and positive extrinsic curvature, characterizing the extreme cases as the revolution ones. Moreover, we get a representation for surfaces with constant Gaussian curvature in such ambient spaces, paying special attention to the cases of K(I) = 1 in × ℝ and K(I) = −1 in ℍ² × ℝ. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant No. PAI-05-034. The authors are partially supported by MEC-FEDER, Grant No. MTM2007-65249.     

A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ℍ<Superscript>2</Superscript> × ℝ<Subscript>1</Subscript>

There are examples of complete spacelike surfaces in the Lorentzian product ℍ² × ℝ₁ with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ² × ℝ₁ with constant Gaussian curvature K > −1.     

Constant angle surfaces in ℍ<Superscript>2</Superscript> × ℝ

We classify all surfaces in ℍ² × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ² is the hyperbolic plane. The author was supported by grants CEEX ET 5883/2006-2008 and PNII ID_ 398/2007-2010 ANCS (Romania).     

Tight Lagrangian surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

We determine all tight Lagrangian surfaces in S ² × S ². In particular, globally tight Lagrangian surfaces in S ² × S ² are nothing but real forms of this symmetric space.     

Complete hypersurfaces with constant laguerre scalar curvature in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let x: M ^n?1 → R ⁿ be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of R ⁿ are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R ⁿ , denote the trace-free Laguerre tensor by ?\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ?x: H ¹ × S ^n?2 → R ⁿ .     

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S<Superscript>3</Superscript> × S<Superscript>3</Superscript>

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S³ × S³ is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((?h)(v, v, v), Jv) = λ holds for all unit tangent vector v.     

Quantization of curvature for compact surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n⋆</Emphasis></Superscript>

For minimal surfaces in spheres, there is a well known conjecture about the quantization of intrinsic curvature which has been solved only in special cases so far. We recall an intrinsic and an extrinsic version for the known results and extend them to compact non-minimal surfaces in spheres. In particular we discuss special classes like Willmore surfaces and surfaces with parallel mean curvature vector. Mathematics Subject Classification (2000):53C42, 53A10.H.Li is partially supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC. U. Simon is partially supported by DFG 163/Si-7-2 and a Chinese–German research cooperation of NSFC and DFG.     

Weierstrass representation for discrete isotropic surfaces in ℝ<Superscript>2,1</Superscript>, ℝ<Superscript>3,1</Superscript>, and ℝ<Superscript>2,2</Superscript>

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation for hyperbolic surfaces parameterized along isotropic directions in ℝ^2,1, ℝ^3,1, and ℝ^2,2. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.     

Lorentzian Sasaki-Einstein metrics on connected sums of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not #kS ² × S ³ are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki η-Einstein and Lorentzian Sasaki-Einstein metrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors \({\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_i}\) such that the D _i are rational curves.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

Regularity results for boundaries in ℝ<Superscript>2</Superscript> with prescribed anisotropic curvature

In this paper we consider the anisotropic perimeter

Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ<Superscript>3</Superscript>

Generic singularities of the boundary of the local transitivity set of a control system on two- and three-dimensional manifolds are classified. The indicatrices of the system are assumed to be given by generic equations and inequalities.     

“Minimal Surfaces in ℍ<Superscript>2</Superscript> × ℝ”

In ² × ℝ one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ² with arbitrary continuous asymptotic values. Finally, a graph on a domain of ² cannot have an isolated singularity. Received: 20 June 2002     

Moving Object in ℝ<Superscript>2</Superscript> and a Group of Observers

We formulate an extremal problem of constructing a trajectory of a moving object that is farthest from a group of observers with fixed visibility cones. Under some constraints on the arrangement of the observers, we give a characterization and a method of construction of an optimal trajectory.     

Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ<Superscript>2</Superscript>

Given a tiling T, one may form a related tiling, called the derived Voronoi tiling of T, based on a patch of tiles in T. Similarly, for a tiling space X, one can identify a patch which appears regularly in all tilings in X, and form a derived Voronoi space of tilings, based on that patch.     

Analytic properties for holomorphic matrix-valued maps in ℂ<Superscript>2×2</Superscript>

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on C^2×2 and Julia set of one dimensional complex dynamic system.     

Elementary modifications and line configurations in ℙ<Superscript>2</Superscript>

Associated to a projective arrangement of hyperplanes ${\mathcal A}$ ⁿ is the module D$({\mathcal A})$, which consists of derivations tangent to ${\mathcal A}$. We study D$({\mathcal A})$ when ${\mathcal A}$ is a configuration of lines in ². In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D$({\mathcal A})$. We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose constraints on Teraos conjecture.     

Semiclassical short strings in AdS<Subscript>5</Subscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

In the short-string limit, we present results for the one-loop correction to the energy of string solutions in AdS ₅ × S ⁵ that belong to a certain class. The computations are based on the observation that the fluctuation operators, just as for rigid spinning-string elliptic solutions, can be written in the single-gap Lamé form. Based on these computations, we reveal a remarkable universality of the expression for the energy of short semiclassical strings, which may help in better understanding the structure of the strongcoupling expansion of the anomalous dimensions of dual gauge theory operators.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).