arXiv research

This paper completes the classification of discrete conformal structures on surfaces.

New classifications of totally real surfaces in nearly Kähler C⁴.

We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be Kähler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we const…

Simple proof for special surface classification.

Simplified proof classifies surfaces using normal curves.

The paper classifies surfaces formed by quadrilateral gluings.

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.

Classifies surfaces with no Gaussian curvature.

Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.

The paper proves bounded curvature for Kähler Ricci shrinker surfaces.

We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.

Classifies fake surfaces up to complexity 5.

We give a complete topological classification of minimal surfaces in Euclidian three-space.

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $…

A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of trapped, umbilical and extremal surfaces, which split into several different cla…

Study of homeomorphisms on infinite type surfaces with a classification theorem.

Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.

Paper uses Gromov-Hausdorff convergence to re-examine surface classification.

Proof that stable minimal surfaces in 3D are flat.

Local classification of surfaces and hypersurfaces with radial mean curvature.

Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …

Two classification results for stationary surfaces of least moment of inertia.

Unique minimal translation surfaces found in Heisenberg group.

We give a mostly self-contained proof of the classification of non-Kahler surfaces based on Buchdahl-Lamari theorem. We also prove that all non-Kahler surfaces which are not of class VII are locally conformally Kahler.

We investigate bi-Hermitian metrics on compact complex surfaces with odd first Betti number producing new examples with connected anti-canonical divisor using the general construction of \cite{abd15}. The result is a complete classification for all \it unbranched \rm Kato surfaces and a classification up to logarithmic…

The paper classifies translation surfaces in a specific hyperelliptic component and finds the maximum number of disjoint geodesics.

Classifies orientation-reversing homeomorphisms of even periods on surfaces.

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…

This paper classifies fundamental groups of perforated surfaces.

It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable …

Paper classifies timelike Bonnet surfaces in Lorentzian 3-manifolds.

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces…

Survey on rational curves on complex surfaces, highlighting different approaches.

This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.

We present a local classification of smooth projective surfaces in 3-space via projective transformations in accordance with singularity types of central projections up to codimension 4. We also discuss relations between our classification of Monge forms and bifurcations of parabolic curves and flecnodal curves.

We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find clos…

Classifies pure mapping class groups based on surface properties.

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

Classifies when homeomorphism groups of stable surfaces have automatic continuity.

In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_g$ of genus $g \geq 2$ determines the group up to a topological conjugation, provided that $N\geq 3g$. The first author et al…

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.

Study mapping class groups of infinite type surfaces with noncompact boundaries.

The paper classifies helicoidal surfaces with specific curvature functions.

The classification of Willmore 2-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n=3,4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n=5$. There are three types of such surfaces up to Möbius transformations: (1) super-conformal…

Researchers classify cmc surfaces using Jacobi elliptic functions.

Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to re…