Study dihedral spherical surfaces and their foliations.
problem Characterize dihedral spherical surfaces and their foliations.
method Define and analyze dihedral surfaces and their foliations, introduce geometric decompositions and deformations.
result Determine the dimension of the moduli space for dihedral surfaces.
Minimal surfaces with dihedral symmetry are studied as angles converge to zero.
problem Understanding minimal surfaces with dihedral symmetry as angles approach zero.
method Analyzing the limit of minimal surfaces in wedges with varying angles and using the implicit function theorem.
result New minimal surfaces are discovered and existence proofs are simplified.
We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume …
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
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 CMC surfaces with dihedral symmetry constructed from Darboux transforms.
problem Constructing closed CMC surfaces with specific symmetries.
method Using Darboux transforms on multiple covers to create new CMC surfaces.
result Explicit parametrisations of new closed CMC surfaces with dihedral symmetry.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of X branched along B. For X simply-connected, we deduce a necessary condition on the intersection form of a simply-connected i…
We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most 2n+1 ends, and with symmetry group the natural $\bfZ_2$ extension of the dihedral group Dn. The surfaces are …
Sharp obstruction found for extending knot group quotients over surfaces in B4.
problem Determining when a knot group quotient extends over a surface in B4. method Sharp obstruction using Seifert form and explicit construction when possible.
result Sharp obstruction and explicit construction of surfaces.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, lea…
Let K⊂S3 be a Fox p-colored knot and assume K bounds a locally flat surface S⊂B4 over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X→S4. Its branching set is a closed surface embedded in S4 locally flatly away from one singularity whose li…
The paper constructs surfaces of high genus with three ends.
problem Creating minimal surfaces of high genus with specific properties.
method One-parameter family of minimal surfaces constructed in Euclidean 3-space.
result The family includes the Costa-Hoffman-Meeks surfaces with two catenoidal ends and a flat middle end.
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.
Unique genus 2 minimal surface in S^3 with specific symmetry group.
problem Finding minimal surfaces in S^3 with specific symmetries.
method Analyzing isometry groups of known minimal surfaces and proving uniqueness.
result ξ_{2,1} is the unique minimal surface of genus 2 with bidihedral symmetry.
We construct new constant mean curvature surfaces in H2xR. They arise as sister surfaces of Plateau solutions. It is a family of MC 1/2 surfaces with k ends, genus 1 and k-fold dihedral symmetry, k greater 2. The surfaces are Alexandrov- embedded.
The study extends group actions from surfaces to 3-manifolds using G-cobordisms.
problem When a finite group action on a surface extends to a 3-manifold.
method Using Schur multiplier and G-cobordisms, the study examines principal actions. result Affirmative extension for abelian, dihedral, symmetric, and alternating groups.
We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.
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.
Consider a dihedral cover f:Y→X with X and Y four-manifolds and f branched along an oriented surface embedded in X with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover f:Y→S4 if Y is homotopy equivalent to $\mathbb{CP…
The study constructs minimal surfaces in a product space with specific properties.
problem Constructing minimal surfaces with specific topological and geometric properties in a product space.
method 1-parameter families of complete properly Alexandrov-embedded minimal surfaces with dihedral symmetry and finite total curvature.
result Examples of minimal surfaces with genus 1 and 2k ends in quotient spaces.
The study finds infinitely many twist knot complements with totally geodesic surfaces.
problem Finding infinitely many twist knot complements with a specific number of totally geodesic surfaces.
method Using a family of twist knot complements and their dihedral covers, the authors construct examples of hyperbolic 3-manifolds with totally geodesic surfaces.
result The construction of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces for any positive integer k.
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.
problem The challenge is to find conditions for links to admit surjective dihedral representations.
method The method involves introducing two-tone colorings and providing conditions for the link groups to admit such representations.
result Any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.
With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space H3, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an ess…
For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.
In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.
Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
problem Rigidity of 3D weakly convex domains with nonnegative scalar curvature.
method Capillary minimal surfaces and foliations with nonnegative mean curvature.
result Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
Symmetric elastic knots are found for certain classes with dihedral symmetry.
problem Finding elastic knots with specific symmetries.
method Minimizing bending energy under dihedral symmetry constraints.
result Existence of dihedral symmetric elastic knots, including a figure-eight union for the trefoil.
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits corr…
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
Dihedral linking invariant uses knot colorings to distinguish knots.
problem Distinguishing knots using knot colorings and linking numbers.
method Algorithm for computing linking numbers in dihedral branched covers.
result The dihedral linking invariant distinguishes more than 98% of prime knot pairs.
Classifies orbits of Hurwitz actions on dihedral quandles.
problem Classifying orbits of Hurwitz actions on dihedral quandles.
method Introduced three computable invariants to classify orbits.
result Complete classification of orbits under Hurwitz action.
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the…
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
Given a symmetry τ of a closed Riemann surface S, there exists an extended Kleinian group K, whose orientation-preserving half is a Schottky group Γ uniformizing S, such that K/Γ induces ⟨τ⟩; the group K is called an extended Schottky group. A geometrical structural description, in terms of…
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.
problem Computing the relationship between dihedral angles and edge lengths in tetrahedra.
method Computed the Wigner derivative and its inverse for spherical tetrahedra.
result The Wigner derivative and its inverse are equal for spherical tetrahedra.
The paper characterizes finite metacyclic subgroups in mapping class groups of surfaces.
problem Characterizing finite metacyclic subgroups in mapping class groups of surfaces.
method Derives necessary and sufficient conditions for conjugates of torsion elements to generate finite split non-abelian metacyclic subgroups.
result Complete classification of certain finite metacyclic subgroups in Mod(Sg) for g≥3.