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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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 …
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{…
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.
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.
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.
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 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.
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.
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.
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.
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 …
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.
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.
We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space…
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
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.
Survey of Willmore surfaces in spheres using DPW method.
problem Global and local properties of Willmore surfaces in spheres.
method DPW method for conformal Gauss map.
result Characterizations of minimal surfaces and Willmore deformations.
Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.
problem Understanding the geometry of surfaces in Heisenberg group.
method Using Gans model of hyperbolic plane, relate tension field of Gauss map to surface's mean curvature.
result Established a relationship between tension field and mean curvature.
New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
This paper finds minimal sets of generators for mapping class groups of specific surfaces.
problem Finding minimal sets of generators for mapping class groups of infinite-type surfaces.
method Analyzing specific surfaces S(n) to determine minimal sets of generators. result Minimal sets of generators for Map(S(n)) are identified for n≥8 (3 elements), n≥3 (4 elements), and S(1) (2 elements). Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.
problem Characterizing one-harmonic maps from curved surfaces to hyperbolic spaces.
method Using Minkowski geometry and interpreting maps as Gauss maps of convex surfaces.
result One-harmonic maps have images confined to the interior of convex hulls.
Extends harmonic map theory to arbitrary surfaces.
problem Classifying harmonic maps from arbitrary surfaces.
method New theory of nilpotent cycles from diagrams, criterion for uniton number finiteness.
result New classification of minimal surfaces and constancy of finite type harmonic maps.
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.
Characterizes multitwists in surface mapping class groups.
problem Understanding multitwists in surface mapping class groups.
method Characterization through braid relations.
result Characterized multitwists satisfying braid relations.
Translation surfaces in Heisenberg group classified by Gauss map determinant.
problem Classifying translation surfaces with zero intrinsic curvature in Heisenberg group.
method Constructed surfaces as product of planar curves, classified by Gauss map determinant.
result Surfaces with vanishing intrinsic curvature identified.
Simple properties of the Gauss map characterise important classes of surfaces in $\Rq$: R-surfaces, the real version of plane complex curves; Lagrangean surfaces; isoclinic surfaces.
The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.
problem Understanding the factorization of harmonic maps between Riemann surfaces and manifolds.
method The proof relies on geometric properties of the Hopf differential and properties of holomorphic and anti-holomorphic diffeomorphisms.
result The theorem provides a factorization of harmonic maps under certain conditions involving holomorphic or anti-holomorphic diffeomorphisms.
New examples of surface bundles found over surfaces.
problem Finding compact atoroidal surface bundles.
method Type-preserving homomorphism from knot complement to mapping class group.
result Infinitely many commensurability classes of surface subgroups.