arXiv research

Study classifies triharmonic surfaces in 3D homogeneous spaces.

Higher order higher spin operators are generalizations of $kth$-powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.

We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 …

Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial …

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf…

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part,…

Researchers create metrics on hyperbolic space's tangent bundle.

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete c…

This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the ca…

We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct…

The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowu…

We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when su…

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimens…

Minimal cylinders in Heisenberg group characterized using loop group method.

Holomorphic cylinders converge to disks joined by flow lines.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

The paper develops a method to map knots in a cylinder to virtual-flat knots.

In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manif…

Cylinders in warped product spaces have zero curvature.

We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$ Sobolev--Poincaré inequality developed by L.~Shartser.

The paper examines the stability of Killing cylinders in hyperbolic space.

Knots parametrized in cylinder coordinates by t -> (st, 3 + cos(nt), cos(mt + φ)) share properties of Lissajous and billiard knots in a cylinder. We use these 'billiard knots in a flat solid torus' to study two topics: when is Z(s,n,m) equal to Z(s,m,n)? And: why are the determinants of certain Lissajous and billiard k…

This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.

Study decomposes geometric surfaces, finding special curves.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

Stable cylinders found in hyperbolic groups and curve graphs.

Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.

Analyses cohomology relations for moving frames and coframes.

Study on quantum particle evolution on Grushin cylinder, embedding in R^3.

We consider cylinders in ${\cal H}^2\times R$ (see definitions in the introduction) and prove that a complete and connected surface in ${\cal H}^2\times R$ with the vanishing of the Gauss and extrinsic curvatures is a cylinder.

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. Th…

Study constructs closed curves with constant curvature on cylinders and tori.

Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.

In this article we prove that a connected and properly embedded translating soliton in $\mathbb{R}^3$ with uniformly bounded genus on compact sets which is $C^1$-asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.

Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…

Automorphism groups of Hopf manifolds are finite and have a bounded order.

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q < 3 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core cu…

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

Algorithm calculates Hopf invariant for simplicial mappings.

We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

New method proves inequalities for self-shrinkers using perturbation.