arXiv research

We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…

New flat surfaces found in 3D sphere space.

Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.

Curved flats linked to pairs of Lie applicable surfaces.

We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…

Study links' flat-virtual diagrams to create link invariants.

Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

The study generalizes origamis to flat surfaces, exploring their combinatorial and geometric properties.

The paper proves ideal triangulations and disk unfolding for singular flat surfaces.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the s…

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k…

Flat semigroups can represent normal weighted homogeneous surface singularities.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal…

Flat surfaces in Lie groups with constant curvature are flat.

Paper connects hypersurfaces in 4D to surfaces in 3-sphere.

Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of bands required to present the link as boundaries of an unknotted (flat) banded surf…

Proof that stable minimal surfaces in 3D are flat.

The paper characterizes surfaces in 4D space forms with flat normal connection.

Study K3 surfaces and their metrics, focusing on dynamics.

Ray marching method visualizes flat surfaces efficiently.

Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.

The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hy…

Simply connected moduli space of Ricci flat metrics on K3 surfaces.

We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat (pseudo-) Riemannian geometries.

SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…

In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be 1 (mod 3). Next, we construct an explicit example in the higher codimensional ca…

We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.

Given a smooth curve $γ$ in some $m$-dimensional surface $M$ in $\mathbb{R}^{m+1}$, we study existence and uniqueness of a flat surface $H$ having the same field of normal vectors as $M$ along $γ$, which we call a flat approximation of $M$ along $γ$. In particular, the well-known characterisation of flat surfaces as to…

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…

In this paper, helicoidal flat surfaces in the $3$-dimensional sphere $\mathbb{S}^3$ are considered. A complete classification of such surfaces is given in terms of their first and second fundamental forms and by linear solutions of the corresponding angle function. The classification is obtained by using the Bianchi-S…

Minimal surfaces in harmonic conformally flat space are studied.

Exposes two methods for constructing flat surfaces in 4D spaces.

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain…

The paper studies special surfaces in 4D space forms with specific geometric properties.

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.

In this expository article, we illustrate how two independent flat structures on minimal surfaces induce a harmonic function, which captures the uniqueness of Enneper's surface.

New complex manifolds found with flat structure.

In this paper we study general rotational surfaces in the 4- dimensional Euclidean space E4 and give a characterization of flat general rotation surface with pointwise 1-type Gauss map. Also, we show that a non-planar flat general rotation surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Cl…

Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.