This paper completes the classification of discrete conformal structures on surfaces.
problem Classifying discrete conformal structures on surfaces.
method Axiomatic approach and study of existing structures.
result Find new classes of discrete conformal structures, including generalized circle packing metrics.
Classifies Kähler surfaces with special properties.
problem Classifying Kähler surfaces with specific properties.
method Classification based on generalized Calabi type.
result Classification of Kähler surfaces with generalized Calabi type.
New classifications of totally real surfaces in nearly Kähler C⁴.
problem Classifying totally real surfaces in nearly Kähler C⁴.
method Investigation under various assumptions, including extrinsic homogeneity, minimality, total umbilicity, and Codazzi-like properties.
result Multiple new examples of totally real surfaces, including parallel and non-parallel Codazzi-like cases.
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…
Classifies non-Kahler surfaces and proves local conformal Kahler property.
problem Classifying non-Kahler surfaces and understanding their geometric properties.
method Proof based on Buchdahl-Lamari theorem.
result All non-Kahler surfaces that are not of class VII are locally conformally Kahler.
Simple proof for special surface classification.
problem Classifying surfaces with specific curvature properties.
method Elementary proof for parallel mean curvature surfaces.
result A lemma is proven for surfaces in complex space forms.
Simplified proof classifies surfaces using normal curves.
problem Classifying surfaces using algebraic topology invariants.
method Using normal curves and analogy to 3-manifolds classification.
result Simple proof without using algebraic topology invariants.
The paper classifies surfaces formed by quadrilateral gluings.
problem Classifying topological surfaces formed by quadrilateral gluings.
method Review of graphs embedded into surfaces, algorithms based on labeling schemes of fundamental polygons.
result Computing numbers of possible gluings for classification.
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.
problem Classifying surfaces with vanishing Gaussian curvature.
method Analyzes Willmore surfaces, studies Willmore cones, gives a Bernstein-type theorem.
result Classifies simply-connected, complete Willmore surfaces with vanishing Gaussian curvature.
Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation −Δu=eu on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth. result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
The paper proves bounded curvature for Kähler Ricci shrinker surfaces.
problem Understanding the curvature of Kähler Ricci shrinker surfaces.
method Proved bounded sectional curvature using earlier work.
result Complete classification of all 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.
problem Classifying fake surfaces for low-dimensional topology.
method Derived properties of fake surfaces, classified up to complexity 5.
result Proved conjectures about fake surfaces up to complexity 5.
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 $…
We give a complete topological classification of minimal surfaces in Euclidian three-space.
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.
problem Classifying homeomorphisms on surfaces of infinite type.
method Introduce tame homeomorphisms and prove a Nielsen-Thurston type classification theorem.
result For tame homeomorphisms, surfaces decompose into invariant subsurfaces with canonical decompositions.
Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.
problem Classifying periodic automorphisms on surfaces that commute with certain involutions.
method Analyzes irreducible periodic automorphisms on surfaces Σg that commute with hyperelliptic involutions. result A classification up to conjugacy for irreducible periodic automorphisms of a surface Σg commuting with involutions ι such that Σg/⟨ιangle is homeomorphic to T2. Paper uses Gromov-Hausdorff convergence to re-examine surface classification.
problem Classifying complete Riemannian surfaces with finite total curvature.
method Gromov-Hausdorff convergence theory applied to surfaces.
result New understanding of Huber's classification theorem for surfaces.
Local classification of surfaces and hypersurfaces with radial mean curvature.
problem Classifying surfaces and hypersurfaces with specific curvature properties.
method Local classification and recursive construction method.
result Local classification of hypersurfaces with vanishing second mean curvature.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in R3. method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in R3 is flat. Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.
problem Classifying semi-algebraic surfaces with isolated singularities.
method Bi-Lipschitz homeomorphisms with inner distance.
result Complete classifications for Nash surfaces and complex algebraic curves.
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.
problem Classifying stationary surfaces in Euclidean space based on their energy.
method Analyzing ruled and foliated surfaces, using critical point theory.
result Classification of stationary surfaces including vector planes, elongated helicoids, and specific types of surfaces.
The paper classifies 2D complete λ-surfaces in 3D space.
problem Classifying complete λ-surfaces in R3. method Complete classification of 2D complete λ-surfaces with constant squared norm of the second fundamental form. result A complete classification for 2-dimensional complete λ-surfaces in Euclidean space R3 with constant squared norm of the second fundamental form. Unique minimal translation surfaces found in Heisenberg group.
problem Classifying minimal translation surfaces in Heisenberg group.
method Complete classification through generating curves analysis.
result Uniqueness of minimal translation surfaces up to isometries.
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…
Classifies orientation-reversing homeomorphisms of even periods on surfaces.
problem Classifying orientation-reversing homeomorphisms of even periods on surfaces.
method Following the approach of [1] and correcting errors in [1] for the case of periods multiple of 4.
result Classification for orientation-reversing homeomorphisms of periods multiple of 4.
The paper classifies translation surfaces in a specific hyperelliptic component and finds the maximum number of disjoint geodesics.
problem Classifying and understanding translation surfaces in a specific hyperelliptic component.
method Analyzing geodesics and Euclidean structures on hyperelliptic surfaces.
result A classification theorem for translation surfaces in Hhyp(4) and the maximum number of disjoint geodesics. This paper classifies fundamental groups of perforated surfaces.
problem Classifying the fundamental groups of perforated surfaces.
method Using the classification theorem for surfaces and covering spaces.
result Fundamental groups of perforated surfaces are large and not Hopfian.
In this paper, helicoidal flat surfaces in the 3-dimensional sphere S3 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…
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.
problem Classifying timelike Bonnet surfaces in Lorentzian 3-manifolds.
method Provided a necessary and sufficient condition for timelike Bonnet surfaces in Lorentzian 3-spaces.
result Theorem classifying timelike Bonnet surfaces in a Lorentzian 3-space.
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.
problem Existence of rational curves on complex surfaces.
method Classification of complex surfaces and systematic study of rational curves in each class.
result Highlighting the different approaches to study rational curves on complex surfaces.
Classifies surfaces in Euclidean space with specific curvature relations.
problem Classifying surfaces with linear curvature relations.
method Variational characterization of surfaces through their generating curve.
result Closed, embedded, and periodic surfaces with Delaunay-like behavior.
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.
Deep learning improves defect classification in real-time surface inspection.
problem Real-time defect classification in manufacturing industry using limited datasets.
method Convolutional Neural Networks (CNNs) designed for speed and accuracy, neural data augmentation for class imbalance.
result 98.0% accuracy in binary defect classification with 22,000 labeled images.
Classifies pure mapping class groups based on surface properties.
problem Classifying pure mapping class groups based on surface properties.
method Complete classification without additional assumptions.
result PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface.
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…
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for K>KP. Classifies when homeomorphism groups of stable surfaces have automatic continuity.
problem Determining when homeomorphism groups of stable surfaces are continuous.
method Developed a general framework to prove automatic continuity for homeomorphism groups, applied to stable surfaces and Stone spaces.
result Classification of stable surfaces with respect to automatic continuity of their homeomorphism groups.
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 Sg of genus g≥2 determines the group up to a topological conjugation, provided that N≥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 surface types for autonomous indoor robots using inertial data.
problem Classifying surface types for wheeled robots in indoor environments.
method Prepared a time series dataset of inertial measurements, used deep learning and ensemble machine learning models.
result Baseline model achieved over 68% accuracy on a nine-category surface type dataset.
Proof of surface homeomorphism classification theorem.
problem Classify homeomorphisms of surfaces of finite type.
method Neilsen-Thurston classification theorem proof.
result Proof of Nielsen-Thurston theorem for surfaces of finite type.