Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

We study translation surfaces with rich groups of affine diffeomorphisms—“prelattice” surfaces. These include the lattice translation surfaces studied by W. Veech. We show that there exist prelattice but nonlattice translation surfaces. We characterize arithmetic surfaces among prelattice surfaces by the infinite cardinality of their set of points periodic under affine diffeomorphisms. We give examples of translation surfaces whose periodic points and Weierstrass points coincide.     

Some special surfaces in the pseudo-Galilean space

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.      

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

Energy functions on moduli spaces of flat surfaces with erasing forest

This paper follows on from Nguyen (Geom Funct Anal 20(1):192–228, 2010), in which we study flat surfaces with erasing forest, these surfaces are obtained by deforming the metric structure of translation surfaces, and their moduli space can be viewed as a deformation of the moduli space of translation surfaces. We showed that the moduli spaces of such surfaces are complex orbifolds, and admit a natural volume form μ _Tr. The aim of this paper is to show that the volume of those moduli spaces with respect to μ _Tr, normalized by some energy function involving the area and the total length of the erasing forest, is finite. Note that translation surfaces and flat surfaces of genus zero can be viewed as special cases of flat surfaces with erasing forest, and on their moduli space, the volume form μ _Tr equals the usual ones up to a multiplicative constant. Using this result we obtain new proofs for some classical results due to Masur-Veech, and Thurston concerning the finiteness of the volume of the moduli space of translation sufaces, and of the moduli space of polyhedral flat surfaces.     

Möbius Geometry of Surfaces of Constant Mean Curvature 1 in Hyperbolic Space

An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Möbius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.     

General blowups of ruled surfaces

In this note we study blowups of algebraic surfaces of Kodaira dimension κ = - ∞ at general points, their embeddings and secant varieties of the embedded surfaces.     

Lokale Untersuchungen an Normalenkongruenzen im dreidimensionalen elliptischen Raum

In this paper we study rectlinear congruence surfaces of rectlinear normal congruences in the three dimensional elliptic space. First we characterize normal congruences by their principal parameters of distribution d₁ and d₂ and establish relations between d₁, d₂ and the principal curvatures k₁ and k₂ of the middle surfaces along the asymptotic lines and the lines of curvature of the middle surfaces. Finally we show that the congruence surfaces with a constant parameter of distribution cross the middle surfaces along curves whose strips are certain CESARO-strips.     

Notes on braidzel surfaces for links

As a generalization of pretzel surfaces, L. Rudolph has introduced a notion of braidzel surfaces in his study of the quasipositivity for pretzel surfaces. In this paper, we show that any oriented link has a braidzel surface. We also introduce a new geometric numerical invariant of links with respect to their braidzel surface and study relationships among them and other ``genus' for links.

     

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?

     

Topological triviality of smoothly knotted surfaces in -manifolds

Some generalizations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are standard, then these surfaces are topologically unknotted.

     

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.     

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

Some remarks on the symmetry of self-conjugate minimal surfaces inR 3

Symmetry properties of self-conjugate minimal surfaces, i.e. minimal surfaces which are congruent with their conjugate ones inR ³ are studied.Supported by ICTP and NNSF of China.Supported by ICTP and SAREC.     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integral Geometry of Real Surfaces in the Complex Projective Plane

We give a Poincaré formula for any real surfaces in the complex projective plane which states that the mean value of the intersection numbers of two real surfaces is equal to the integral of some terms of their Kähler angles.     

On Principally Polarized Adler-van Moerbeke Surfaces

In the present paper we study Kummer surfaces in IP⁶ arising as projections of a certain family of principally polarized abelian surfaces in IP⁶. This family was introduced by Adler and van Moerbeke in connection with some Hamiltonian dynamical systems. In this paper we give explicit equations of the associated Kummer surfaces. This enables us to deacribe their moduli space and to give the full list of degenerations.     

Conformal Wasserstein distances: Comparing surfaces in polynomial time

We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the “similarity” of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Möbius transformations. We present in detail the case where the surfaces to compare are disk-like; we also sketch how the approach can be generalized to other types of surfaces.     

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron–Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus. Received: 29 January 1999     

Stability of n-dimensional extremal surfaces of revolution

In this paper we consider extremal surfaces of revolution of area-type functionals. For the latter we calculate the first and second variations. We prove stability and instability criteria for n-dimensional surfaces of revolution based on their definition and in terms of special integrals.