New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.
problem Modeling STSs with specific horizontal restrictions.
method Modified model with conjugacy classes of permutations to restrict horizontal gluings.
result Asymptotic analysis of components, genus distribution, and saddle connections.
Researchers create new triply periodic minimal surfaces by gluing saddle towers.
problem Creating triply periodic minimal surfaces without symmetry constraints.
method Gluing Karcher-Scherk saddle towers with phase differences and balancing under vertical interaction.
result Expands known triply periodic minimal surfaces into new 5-parameter families.
This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.
problem Creating cmc 1/2 surfaces with positive genus in H2imesR. method Analytic gluing construction, solving mean curvature equation via perturbative methods and linear analysis.
result Construction of cmc 1/2 annuli asymptotic to horizontal catenoids, proving convergence to horocylinders.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Constructs covariant derivatives for Ehresmann connections.
problem Developing a method for covariant derivatives in fibre bundles.
method Introducing a vertical endomorphism to construct covariant derivatives on vertical and horizontal distributions.
result Covariant derivatives can be constructed separately on vertical and horizontal distributions and then glued together.
We construct the first examples of complete, properly embedded minimal surfaces in H2×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.
problem Embedding minimal surfaces with Scherk ends.
method Gluing saddle towers with prescribed phase differences, analyzing slight bendings.
result Correctly identifies scenarios where constructed surfaces are embedded.
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.
problem Conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
method Proposes a Gluing Conjecture and proves it under certain conditions.
result The Gluing Conjecture is true under specific conditions, generalizing Petrunin's Gluing Theorem.
Study on knots formed by gluing ellipses, defining gluing degree.
problem Properties of glued knots.
method Defined gluing degree to relate to knot properties.
result Classified all knots up to gluing degree 6.
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…
Study geometry of surfaces glued along a curve.
problem Geometry of surfaces glued along a curve.
method Moving frame along the curve, developable surfaces defined.
result Developed geometric properties of glued surfaces.
New method to prove nonarithmeticity of hyperbolic gluings.
problem Proving nonarithmeticity of hyperbolic gluings.
method Determining adjoint trace fields of gluings.
result Many new examples of nonarithmetic gluings.
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 P over a Hopf algebra H with a mapping ΔR:P→P⊗H. In our case P…
Contact gluing maps are shown to be equivalent in sutured Floer homology.
problem Equivalence of contact gluing maps in sutured Floer homology.
method Established a conjecture by Zarev, showing the contact gluing map is equivalent to the sutured Floer homology gluing map.
result Contact gluing maps are equivalent in sutured Floer homology.
A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε>0, there is a C1 horizontal curve Γ such that Γ=γ and Γ′=γ′ outside a set of measure at most ε. We verify this property for free Carno…
Defines conformal submersion with horizontal distribution and provides necessary conditions for its existence.
problem Existence and conditions for conformal submersion with horizontal distribution.
method Definition and analysis of conformal submersion with horizontal distribution, dual connections, and necessary conditions.
result Necessary and sufficient conditions for conformal submersion with horizontal distribution and geodesics.
The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.
problem Studying spectral properties of the horizontal Laplacian
method Interpreting the horizontal Laplacian as a twisted Laplacian acting on a flat vector bundle
result The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold
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.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Detailed characteristic gluing of strongly asymptotically flat data to Kerr spacetime.
result Alternative proof of spacelike gluing construction for strongly asymptotically flat spacelike initial data.
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…
The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
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.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
Researchers prove a nonlinear gluing theorem for gravitational fields near static backgrounds.
problem Proving a nonlinear gluing theorem for gravitational fields near static backgrounds.
method Proved a nonlinear characteristic Ck-gluing theorem for vacuum gravitational fields in Bondi gauge. result Generalized the C2-gluing theorem near light cones to a wider class of hypersurfaces. Formulae connect heat kernels on glued manifolds.
problem Connecting heat kernels on joined manifolds.
method Proved gluing formulae for Laplacian heat kernels.
result Formulae linking heat kernels on joined manifolds.
Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
The study develops inequalities for Riemannian foliations without bundle-like assumptions.
problem Developing inequalities for Riemannian foliations without restrictive conditions.
method Bochner theory and Bakry-Emery calculus for horizontal Laplacians, derived explicit Bochner formulas, generalized curvature dimension inequalities.
result Established generalized curvature dimension inequalities for Riemannian foliations.
Solves C^3 null gluing problem for Einstein vacuum equations.
problem Null gluing of up to third-order derivatives of the metric in Einstein vacuum equations.
method Linear and nonlinear analysis of characteristic data close to Minkowski data.
result Solvable up to a 20-dimensional space of obstructions, 10 of which are novel.
Affine and conformal submersions with horizontal distribution are studied in statistical manifolds.
problem Characterizing submersions and geodesics in statistical manifolds.
method Introducing conformal submersions with horizontal distribution and proving conditions for statistical manifold properties.
result Necessary and sufficient conditions for submersions and geodesics in statistical manifolds.
Paper solves Einstein vacuum equations gluing problem with applications.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Codimension-10 gluing construction, relating charges to ADM energy, and localized version.
result Asymptotically flat data can be glued to Kerr spacetime with 10 charges.
The complexity of horizontality in twistor spaces on tori is infinite.
problem Complexity of horizontality in twistor spaces on tori.
method Analyzing the complexity of horizontality in the twistor space associated with an oriented vector bundle over a torus.
result The complexity of horizontality in the twistor space is expressed by a dense subset of S2 when it 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.
problem Proving the gluing formula for analytic torsion forms.
method Extending prior work to the family case, presenting a new proof.
result Simplified proof of the gluing formula.
Constructs a solution operator for hyperbolic gluing in higher dimensions.
problem Solving the linearized constant scalar curvature equation at hyperbolic space.
method Constructs a solution operator with good support propagation properties.
result Extends the Mao-Oh-Tao gluing method to the hyperbolic setting, gaining two derivatives.
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.
problem Proving quasi-arithmeticity of building blocks in hyperbolic gluings.
method Generalized gluings of hyperbolic orbifolds, proving quasi-arithmeticity.
result Building blocks of quasi-arithmetic gluings must also be quasi-arithmetic.
Smooth 4-manifolds have simple horizontal decompositions.
problem Classifying smooth, closed, orientable 4-manifolds.
method Horizontal handlebody decomposition.
result Simplest horizontal decompositions classify closed 4-manifolds.
The paper classifies surfaces formed by quadrilateral gluings.
problem Classifying topological surfaces formed by quadrilateral gluings.
method Review of graphs embedded into surfaces, algorithms based on labeling schemes of fundamental polygons.
result Computing numbers of possible gluings for classification.
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.
problem Gluing charged black holes into de Sitter space.
method Extends Hintz's result to Einstein-Maxwell system with positive cosmological constant, using Reissner-Nordström or Kerr--Newman--de Sitter black holes.
result Determines conditions for gluing black holes into 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…
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
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…