The study classifies minimal translation surfaces in 3D and 3D_1.
problem Classifying minimal translation surfaces in specific geometric settings.
method Defined and classified minimal translation surfaces with semi-symmetric connections.
result New classification of minimal translation surfaces in R3 and R13. Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
Study of minimal surfaces in 3D space with special connections.
problem Classification of minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
method Analysis of singular minimal translation surfaces in a 3D Euclidean space with a semi-symmetric connection.
result Classification of singular minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
Study of sectorial decompositions in symmetric products of surfaces for symplectic geometry.
problem Symplectic topology of symmetric products of Riemann surfaces.
method Liouville sectorial techniques.
result New geometric proof of Homological Mirror Symmetry for a specific case.
Two related constructions are studied: (1) The diagonal complex D and its barycentric subdivision BD related to a \textit{punctured} oriented surface F equipped with a number of labeled marked points. (2) The symmetric diagonal complex Dinv and its barycentric subdivision $\math…
DPW method reconstructs minimal and symmetric CMC surfaces in 3-sphere.
problem Reconstructing minimal and symmetric CMC surfaces in S3. method DPW method for reconstructing minimal surfaces and extending to symmetric CMC surfaces.
result DPW potential for Lawson surfaces reconstructs minimal immersions in S3. Study geodesic flow on symmetric surfaces to determine parabolic type.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Analyze Fenchel-Nielsen coordinates and covering group properties.
result Conditions for a surface to be parabolic are equivalent to specific properties of its Fenchel-Nielsen coordinates.
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
Theory of symmetric rigidity in hyperbolic geometry.
problem Symmetric rigidity in hyperbolic frameworks.
method Gain graphs and orbit rigidity matrix, matroidal sparsity conditions.
result Characterization of infinitesimal rigidity for Gamma-symmetric frameworks.
New surfaces described that are symmetric and solve a specific equation.
problem Understanding symmetric shapes of membranes.
method Characterized and described axially symmetric Helfrich spheres using the reduced membrane equation.
result These surfaces are symmetric and belong to a specific family.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
New minimal surface theory disproves a conjecture in symmetric spaces.
problem Proving the existence of unstable minimal maps in symmetric spaces.
method Using Hitchin representations and equivariant maps, with a new index bound.
result Disproves the Labourie conjecture for PSL(n,R) with n≥4. Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.
We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded…
We show existence of centrally symmetric maps on surfaces all of whose faces are quadrangles and pentagons for each orientable genus g≥0. We also show existence of centrally symmetric maps on surfaces all of whose faces are hexagons for each orientable genus g=2k−1, k∈N. We enumerate centrally …
Study curvature and symplectic properties of symmetric products of surfaces.
problem Distinguishing between macroscopic dimensions in Riemannian manifolds.
method Detailed study of curvature and symplectic properties using symmetric products of surfaces.
result Symmetric products of surfaces sharply distinguish between two macroscopic dimensions.
The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.
problem Conditions for torsion elements to generate symmetric or alternating subgroups.
method Analyzes mapping class groups of surfaces, derives necessary and sufficient conditions for conjugates of torsion elements to generate symmetric or alternating subgroups.
result Symmetric or alternating subgroups cannot contain irreducible mapping classes and hyperelliptic involutions.
Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.
problem Maximally symmetric (2,3,5)-distribution of An-Nurowski surface rolling without slipping or twisting. method Calculated vector fields defining a split g2 Lie algebra and projected to an action of SL(3,R). result Obtained an action of SL(3,R) on the configuration space without a surface. The paper extends topological field theory to noncompact surfaces using symmetric powers.
problem Extending topological field theory to noncompact surfaces without closed boundaries.
method Constructing sectorial covers with combinatorics of the bar resolution.
result Recovering results of Rouquier and Manion on extending Heegaard-Floer theory.
Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong's canonical coordinate form of such connections is simplified.
The study finds points on surfaces where a tensor is conformal to a metric.
problem Existence of conformal points on surfaces.
method Analyzes symmetric bilinear two-tensor fields and Riemannian metrics.
result Provides conditions for the existence of conformal points.
Study on surfaces with constant curvature under a specific connection.
problem Classifying surfaces with constant sectional curvature under a semi-symmetric non-metric connection.
method Analyzing surfaces in R3 with a canonical semi-symmetric non-metric connection determined by a vector field. result Classification of surfaces under various conditions (cylindrical, rotational) with constant sectional curvature.
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
The paper explores harmonic maps and their properties in symmetric spaces.
problem Understanding harmonic maps in symmetric spaces.
method Discussion of associated family of harmonic maps from Riemann surfaces into symmetric spaces.
result Comparison and conjecture on harmonic maps and totally symmetric harmonic maps.
We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus g>1 by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.
The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.
problem Computing colored Jones and Alexander polynomials.
method Using two Lagrangians in a symmetric power of a surface to compute polynomials.
result Colored Jones and Alexander polynomials are special cases of a graded intersection between Lagrangians.
Polyhedral surfaces can be broken down into parallelograms.
problem Decomposing polyhedral surfaces into simpler shapes.
method Analyzing moduli spaces and using geometric properties.
result Polyhedral surfaces with 8 vertices can be decomposed into at most 20 parallelograms.
We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric plane…
An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surf…
We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is …
In this paper we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addit…
The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
Symmetric quandles provide new insights into link colorings.
problem Understanding link colorings using quandles.
method Construction of symmetric quandles and isomorphism of their cohomology groups.
result Homology groups of quandles are isomorphic to those of their symmetric doubles.
We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of t…
We study symmetric affine surfaces which have non-vanishing torsion tensor. We give a complete classification of the local geometries possible if the torsion is assumed parallel. This generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not…
Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a…
The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
problem Finding the shape of symmetric sets with minimal Gaussian surface area.
method Analyzing the boundary of symmetric sets and applying isoperimetric inequalities.
result Symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
Given a reductive representation ρ:π1(S)→G, there exists a ρ-equivariant harmonic map f from the universal cover of a fixed Riemann surface Σ to the symmetric space G/K associated to G. If the Hopf differential of f vanishes, the harmonic map is then minimal. In this paper, we investigate the…
In this paper, we develop a loop group description of harmonic maps F:M→G/K ``of finite uniton type", from a Riemann surface M into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…
The article explores symmetric maps on surfaces, focusing on semi-equivelar maps.
problem Identifying and classifying semi-equivelar maps on surfaces with specific Euler characteristics.
method Analyzing automorphisms and symmetry groups of maps on higher genus surfaces.
result There are at least 39 types of semi-equivelar maps on surfaces with Euler characteristic -2m, m ≥ 2, with symmetry groups isomorphic to dihedral or cyclic groups.