Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
Study of conformal Gauss map for Willmore surfaces in model spaces.
problem Characterizing Willmore surfaces and their geometric properties.
method Detailed study of conformal Gauss map and its application to Willmore surfaces.
result New characterizations of minimal and CMC surfaces using conformal Gauss map.
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Study singularities of constant curvature surfaces and harmonic maps.
problem Understanding singularities and bifurcations of constant curvature surfaces.
method Loop group methods to construct bifurcations and analyze harmonic maps.
result Determine which map germs can be represented by harmonic maps.
Wave maps connect to constant curvature surfaces, studying singularities and bifurcations.
problem Understanding singularities and bifurcations in wave maps and pseudospherical surfaces.
method Constructing germs of wave maps from their jets and using loop groups to construct pseudospherical surfaces.
result Obtained bifurcations in generic 1-parameter families of pseudospherical surfaces.
Criterion found for Teichmüller extremal maps on infinite Riemann surfaces.
problem Finding necessary and sufficient conditions for Teichmüller extremal maps.
method Established a criterion for Teichmüller-type extremal maps.
result Criterion for Teichmüller extremal maps on infinite Riemann surfaces.
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…
Study of Demoulin surfaces using Gauss maps and conformal coordinates.
problem Characterizing Demoulin surfaces in real projective 3-space.
method Generalized Weierstrass type representation via primitive maps.
result Established a new representation for Demoulin surfaces.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
This paper classifies semi-equivelar maps on a special surface.
problem Classifying semi-equivelar maps on a surface of Euler genus 3.
method Analyzing cyclic face sequences and presenting complete map types.
result Complete list of semi-equivelar maps on a surface of Euler characteristic -1.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. The Gauss map of a special surface is studied, leading to symmetry conclusions.
problem Understanding the Gauss map of free boundary minimal surfaces.
method Analyzing eigenfunctions of the Jacobi-Steklov operator.
result Rotationally symmetric surfaces have components of their Gauss map as eigenfunctions of the Jacobi-Steklov operator.
The paper studies branched surfaces and their properties.
problem Global topologies of branched surfaces and their maps.
method Explicit construction of maps and study of topological properties.
result New insights into the geography of branched surfaces and their maps.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.
problem Investigating the mapping classes of real rational surface automorphisms and their restrictions.
method Analysis of reducible maps, determination of pseudo-Anosov mapping classes, and comparison with Penner's construction.
result Realized Lehmer's number as the stretch factor of a pseudo-Anosov map on a specific surface.
Study shows isolated singularities in maps to K3 surfaces.
problem Characterizing singularities of triholomorphic maps.
method Analyzes weakly triholomorphic maps from hyperkähler to K3 surfaces.
result Maps have only isolated singularities.
We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H2. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…
We prove that the distortion function of the Gauss map of a harmonic surface coincides with the distortion function of the surface. Consequently, Gauss map of a harmonic surface is K quasiregular if and only if the surface is K quasiregular, provided that the Gauss map is regular or what is …
Uniformizes surfaces using discrete harmonic maps and hyperbolic metrics.
problem Uniformizing surfaces with complex geometries.
method Least Dirichlet energy harmonic embedding of graphs on surfaces.
result Existence of hyperbolic metrics realizing least energy embeddings.
Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.
problem Large geometric distortion when using spherical domain for genus-0 closed surfaces.
method Developed a novel method for ellipsoidal density-equalizing maps and combined with quasi-conformal maps.
result Significantly improved surface remeshing performance for genus-0 closed surfaces.
Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.
problem Preserving distances on hyperbolic surfaces with boundaries.
method Proving bijections between geodesics are isometries.
result A bijection between geodesics is an isometry.
Presented an infinite group presentation for non-orientable surfaces.
problem Presenting a group presentation for non-orientable surfaces.
method Used Dehn twists and crosscap pushing maps.
result An infinite presentation for the mapping class group of a non-orientable surface.
Paper finds a normal generating set for Torelli group of non-orientable surfaces.
problem Finding a normal generating set for the Torelli group of non-orientable surfaces.
method Explicit construction using BSCC and BP maps.
result An explicit normal generating set for the Torelli group of genus-g compact non-orientable surfaces with b boundary components. Gauss map of complete minimal surfaces avoids certain hypersurfaces.
problem Characterizing the range of Gauss maps of minimal surfaces.
method Analyzing the degree of hypersurfaces omitted by the Gauss map.
result Gauss map can omit hypersurfaces of degree at most nn+2(n+1)n+2. The study examines surfaces in isotropic space with specific Gauss map properties.
problem Understanding surfaces in simply isotropic space with degenerate metric.
method Investigates surfaces with Gauss map coordinates as eigenfunctions of the Laplace-Beltrami operator for minimal and parabolic normals.
result Identifies surfaces characterized by eigenfunction properties of the Gauss map.
Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
Unified study of harmonic maps between pseudo-Riemannian surfaces.
problem Classifying harmonic maps between pseudo-Riemannian surfaces.
method Unified formalism and Bäcklund transformation.
result Unified solutions to harmonic map equations and corresponding maps.
Study on Gauss maps of minimal surfaces in 4D space.
problem Understanding the Gauss map of minimal surfaces in Euclidean 4-space.
method Systematic study of Gauss map properties for complete minimal surfaces.
result Optimal results for the maximal number of exceptional values of the Gauss map.
Study non-orientable surfaces, find finitely generated abelian subgroups.
problem Characterize abelian subgroups in non-orientable surfaces.
method Apply Birman-Lubotzky-McCarthy's arguments to non-orientable surfaces.
result Find a finitely generated group isomorphic to a given subgroup.
Study on special surfaces in 4D pseudo-Euclidean space with specific Gauss maps.
problem Characterizing rotational surfaces with pointwise 1-type Gauss maps in E4-2.
method Analyzing surfaces with spacelike profile curves in E4-2.
result Characterizations for surfaces to have pointwise 1-type Gauss maps.
New stability theorem for nonorientable surfaces mapping class groups.
problem Stability of homology groups of mapping class groups of nonorientable surfaces.
method Galatius--Kupers--Randal-Williams framework of cellular E2-algebras. result New best known stability range for homology of nonorientable surfaces.
This paper characterizes Willmore surfaces in spheres using harmonic maps.
problem Characterizing Willmore surfaces in spheres using harmonic maps.
method Analyzes the conformal Gauss map of Willmore surfaces and its properties.
result Generically, strongly conformally harmonic maps from Riemann surfaces to the target space are associated with Willmore surfaces in spheres.
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.
problem Characterizing quasi-isometric embeddings of maps from cusped surfaces into moduli space.
method Investigates shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces, considering quasi-isometric embeddings with respect to Teichmüller distance and intrinsic distance.
result Characterizations of quasi-isometric embeddings are solely determined by the map's monodromy under mild conditions.
Generators found for non-orientable surface mapping class group.
problem Finding minimal generating set for non-orientable surfaces.
method Szepietowski's system of generators, Dehn twists, and Y-homemorphisms.
result Szepietowski's system is a minimal generating set of mapping class group.
Isomorphic mapping class groups on infinite-type surfaces imply homeomorphic surfaces.
problem Characterizing isomorphisms between mapping class groups of infinite-type surfaces.
method Proving that all simplicial automorphisms are induced by homeomorphisms and applying to mapping class groups.
result Isomorphic big mapping class groups imply homeomorphic underlying surfaces.
The study characterizes surfaces in 7D space from harmonic maps into 6D sphere.
problem Characterizing surfaces in 7D space from harmonic maps into 6D sphere.
method Using harmonic sequences and properties of immersions.
result Characterizes specific types of surfaces like minimal, parallel mean curvature, pseudo-umbilical, and isotropic surfaces.
For a surface in the 3-dimensional real projective space, we define a Gauss map, which is a quadric in R4 and called the first-order Gauss map. It will be shown that the surface is a Demoulin surface if and only if the first-order Gauss map is conformal, and the surface is a projective minimal coincidence …
In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces.
Proves homology of mapping class groups for infinite-type surfaces.
problem Homology of mapping class groups for infinite-type surfaces.
method Modification of Mather's argument and homological stability result.
result Homology of mapping class groups determined for binary tree surfaces.
Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.
problem Relating topological entropy of pseudo-Anosov maps to homology of mapping tori.
method Analyzing the topological entropy of pseudo-Anosov maps on surfaces with punctures and relating it to the rank of the first homology of their mapping tori.
result Entropy of a pseudo-Anosov map is bounded by a formula involving the genus, number of punctures, and homology rank.
The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.
problem Presentations of mapping class groups of surfaces stabilizing boundaries.
method Gave presentations of mapping class groups of marked surfaces stabilizing boundaries.
result Presented cluster automorphism groups of cluster algebras from surfaces.