arXiv research

New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.

Researchers create new triply periodic minimal surfaces by gluing saddle towers.

This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.

New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.

Constructs covariant derivatives for Ehresmann connections.

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegen…

Constructs minimal surfaces by gluing saddle towers with Scherk ends.

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …

We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus. We illustrate multiple new examples for each genus g>2. In th…

The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.

Study on knots formed by gluing ellipses, defining gluing degree.

Study geometry of surfaces glued along a curve.

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…

Contact gluing maps are shown to be equivalent in sutured Floer homology.

In this short article we review how the classical theory of principal fibre bundles (PFB) transcribes in an algebraic formalism. In this dual formulation, a PFB is given by a right co-module algebra ${\cal P}$ over a Hopf algebra ${\cal H}$ with a mapping $Δ_R:{\cal P}\to{\cal P}\otimes{\cal H}$. In our case ${\cal P}$…

A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $γ$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of measure at most $\varepsilon$. We verify this property for free Carno…

Defines conformal submersion with horizontal distribution and provides necessary conditions for its existence.

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.

We describe explicit horizontal open books on some Seifert fibered 3--manifolds. We show that the contact structures compatible with these horizontal open books are Stein fillable and horizontal as well. Moreover we draw surgery diagrams for some of these contact structures.

Paper glues characteristic data to Kerr spacetime, proving spacelike gluing.

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…

We determine the adjoint trace field of gluings of general hyperbolic manifolds. This provides a new method to prove the nonarithmeticity of gluings, which can be applied to the classical construction of Gromov and Piatetski-Shapiro (and generalizations) as well as certain gluings of pieces of commensurable arithmetic …

The paper studies critical points of horizontal energy functional in Riemannian foliations.

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth st…

The paper constructs gluing maps for harmonic maps between Riemannian manifolds.

Researchers prove a nonlinear gluing theorem for gravitational fields near static backgrounds.

Formulae connect heat kernels on glued manifolds.

Study horizontal discs in fat distributions, proving their existence.

The study develops inequalities for Riemannian foliations without bundle-like assumptions.

Solves C^3 null gluing problem for Einstein vacuum equations.

Affine and conformal submersions with horizontal distribution are studied in statistical manifolds.

Paper solves Einstein vacuum equations gluing problem with applications.

The complexity of horizontality in twistor spaces on tori is infinite.

We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface times I, and (3) gluing a handlebody to th…

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…

Simplified proof of gluing formula for analytic torsion forms.

Constructs a solution operator for hyperbolic gluing in higher dimensions.

In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($π$)1-form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. This result generalizes the existence and u…

Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.

Smooth 4-manifolds have simple horizontal decompositions.

The paper classifies surfaces formed by quadrilateral gluings.

We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.

We present a gluing formula for Gromov-Witten invariants in the case of a triple product. This gluing formula is a simple case of a much more general gluing formula proved and stated using exploded manifolds. We present this simple case because it is relatively easy to explain without any knowledge of exploded manifold…

Adds charged black holes to de Sitter space.

This paper studies the associativity of gluing of trajectories in Morse theory. We show that the associativity of gluing follows from of the existence of compatible manifold with face structures on the compactified moduli spaces. Using our previous work, we obtain the associativity of gluing in certain cases. In partic…

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a conseque…

Study on curvatures of surfaces in specific Lie groups.