Geometric compactification of moduli spaces of half-translation structures on surfaces

In this paper, we give an equivariant compactification of the space \({\mathbb {P}}{\text {Flat}}(\Sigma )\) of homothety classes of half-translation structures on a compact, connected, orientable surface \(\Sigma \). We introduce the space \({\mathbb {P}}{\text {Mix}}(\Sigma )\) of homothety classes of mixed structures on \(\Sigma \), that are \({\text {CAT}}(0)\) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are \({\mathbb {R}}\)-trees and completions of surfaces endowed with half-translation structures. Endowing \({\text {Mix}}(\Sigma )\) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that \({\mathbb {P}}{\text {Mix}}(\Sigma )\) is an equivariant compactification of \({\mathbb {P}}{\text {Flat}}(\Sigma )\), thus allowing us to understand in a geometric way the degenerations of half-translation structures on \(\Sigma \). We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on \(\Sigma \), by the mean of the translation distances of the elements of the covering group of \(\Sigma \).     

Motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics two dimensional-plane single mass mechanical and mathematical vibration models in a higher order dimensional space with any virtual sectional curvature-positive or negative, constant or variable. A characterization of the Riemannian manifolds is performed by means of curvature operators. The computer codes Mathematica and MATLAB are used in the numerical simulation. The objects of the investigation are a sphere – with a positive constant sectional curvature, a cylinder-with a zero constant sectional curvature, helicoid-with a negative variable sectional curvature, a torus-with a variable (±) sectional curvature, any virtual surface of second order-with a variable (±) sectional curvature, pseudo-sphere – with a negative constant sectional curvature and a saddle-with a negative variable sectional curvature. The system motion is investigated in a qualitative aspect in time and frequency domain on the cited surfaces. The common algorithm derived in the paper can transform any motion from 3D space to curved manifold. We can derive the trajectory in an explicit form on the curved manifold. We can change the trajectory by a suitable variation of the curved manifold.     

Timelike surfaces with zero mean curvature in Minkowski 4-space

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.     

Some Global Theorems on Closed Surfaces With Qenus Zero in E~3

ＳｏｍｅＧｌｏｂａｌＴｈｅｏｒｅｍｓｏｎＣｌｏｓｅｄＳｕｒｆａｃｅｓＷｉｔｈＱｅｎｕｓＺｅｒｏｉｎＥ~３￥ＺｈｏｕＳｈｅｎｇｗｕ；ＬｉｕＪｉｎｌｕ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ，ＣｈｉｎａＵｎｉｖｅｒｓｉｔｙ...     

Lengths of geodesics on non-orientable hyperbolic surfaces

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces with boundary.      

Quasi-isometries of groups, graphs and surfaces

We give a characterization of virtual surface groups as groups quasi-isometric to complete simply-connected Riemannian surfaces. Results on the equivalence up to quasi-isometry of various bounded geometry conditions for Riemannian surfaces are also obtained. Received: January 18, 2000     

On Ordinarity and Lifting Fibrations on Surfaces

We investigate the relation between the ordinarity of a surface and of its Picard scheme in connection with the problem of lifting fibrations of genus g ≥ 2 on surfaces to characteristic zero.     

On a question of Zariski on Zariski surfaces

We give a new method to construct unirational surfaces which may be applied to the following question posed by Zariski in his studies on unirational surfaces. Is any Zariski surface with geometric genus zero rational? Our main result is a negative answer to this question in any characteristic case.     

Uniqueness of the Riemann minimal examples

We prove that a properly embedded minimal surface in R ³ of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the language of Hurwitz schemes to understand the underlying moduli space of surfaces in our setting. Oblatum 30-V-1997 & 5-VIII-1997     

Lattice surfaces and smallest triangles

We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than \(1\over 20\) is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects.     

Circle decompositions of surfaces

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

Classification of Two-Dimensional Surfaces with Zero Normal Torsion in Four-Dimensional Spaces of Constant Curvature

We introduce the notion of normal torsion of a surface at a point along a given direction in a Riemannian space. We give a complete classification of surfaces in four-spaces of constant curvature having zero normal torsion.     

Surfaces in Sol3 Space Foliated by Circles

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol₃. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.     

Incompressible pairwise incompressible surfaces in almost alternating knot complements

In this paper, we deal with incompressible pairwise incompressible surfaces in almost alternating knot complements. We show that the genus of a surface in an almost alternating knot exterior equals zero if there are two, four or six boundary components in the surface.     

Conformal cochains

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author's paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.

     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

Surface Reconstruction of 3D Scattered Data with Radial Basis Functions

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.     

Smooth surface reconstruction via natural neighbour interpolation of distance functions

We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and accommodates non-uniform samples. The reconstructed surface interpolates the data points and is implicitly represented as the zero set of some pseudo-distance function. It can be meshed so as to satisfy a user-defined error bound, which makes the method especially relevant for small point sets. Experimental results are presented for surfaces in .