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.
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.
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.
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…
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map from a moduli space of gluing data to the moduli space of torsion-free G_2-structu…
New method for constructing gluing parameterizations in geometric analysis.
problem Constructing gluing parameterizations for anti-self-dual connections.
method Using C1 maps of smooth Banach manifolds with corners. result Provides a method applicable to other nonlinear PDE problems.
Study shows simplicial volumes add for certain manifold gluings.
problem Understanding simplicial volumes in manifold gluings.
method Proves additivity of locally finite simplicial volume and Lipschitz simplicial volume for connected sums and specific boundary components.
result Additivity of simplicial volumes in dimension 3 and above for certain gluings.
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…
Two tropical gluing formulas help calculate Gromov-Witten invariants.
problem Calculating Gromov-Witten invariants of symplectic manifolds.
method Tropical geometry applied to exploded manifolds.
result Generalizes existing formulas for Gromov-Witten invariants.
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…
Extends 4D cornered skein theory to surfaces, proving gluing formulas.
problem Formulating gluing formulas for 4-manifolds with corners and boundaries.
method Develops a categorical framework and introduces bicategories for closed surfaces.
result Proves gluing formulas for categories associated with 3-manifolds with boundary.
Formula connects invariants of 4-manifolds after gluing.
problem Relating invariants of 4-manifolds after gluing.
method Proved a gluing formula for Seiberg-Witten invariants.
result Formula connects invariants of 4-manifolds after gluing.
In the previous article "Refined Analytic Torsion on Manifolds with Boundary" we have presented a construction of refined analytic torsion in the spirit of Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We now derive a gluing formula for our construction, which can be viewed as…
Extends ASD connection existence to 4-manifolds with cylindrical ends.
problem Existence of ASD connections on 4-manifolds with specific ends.
method Extends gluing theorems to cylindrical ends, using mASD connections.
result Establishes existence of ASD connections on 4-manifolds with cylindrical ends.
Gluing instantons on 4-manifolds with group action.
problem Gluing invariant ASD connections on 4-manifolds with group action.
method Equivariant gluing theory for instanton moduli spaces.
result Construction of Γ-invariant ASD connections on connected sums. In a previous article we studied PGL(n,C)-representations of a 3-manifold via a generalization of Thurston's gluing equations. Neumann has proved some symplectic properties of Thurston's gluing equations that play an important role in recent developments of exact and perturbative Chern-Simons theory. In this paper, we …
The paper constructs solutions to Kapustin-Witten equations with Nahm poles over manifolds with boundaries.
problem Constructing solutions to Kapustin-Witten equations with Nahm poles over manifolds with boundaries.
method Establishing a gluing construction for Nahm pole solutions over manifolds with boundaries and cylindrical ends.
result There exists an obstruction class for gluing Nahm pole solutions along cylindrical ends.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
Formula calculates invariant for 3-manifolds with torus boundaries.
problem Calculating an invariant for 3-manifolds with torus boundaries.
method Generalized Chern-Simons invariant and provided a gluing formula.
result A gluing formula for the invariant d(M,ρ). Extends invariants to 4-manifolds with boundary.
problem Defining invariants for 4-manifolds with boundary.
method Using unfolded Seiberg-Witten Floer spectra.
result Detailed proof of gluing theorem for relative invariants.
Study of hyperbolic 3-manifolds using gluing equations and torsion.
problem Understanding the geometry and topology of hyperbolic 3-manifolds.
method Linking gluing equations to cohomology of infinitesimal isometries and geometrically reformulating torsion.
result Verification of generalized 1-loop Conjecture for sister manifold of figure-eight knot complement.
Study G_2-manifolds from glued circles and Calabi-Yau manifolds.
problem Analyse G_2-manifolds from glued circles and Calabi-Yau manifolds.
method Express nu-invariant in terms of fixpoints and gluing contributions, involving Dedekind sums and eta-function.
result Existence of compact G_2-manifolds that are not G_2-nullbordant.
Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic me…
The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.
problem Preserving the Turaev-Viro invariant volume conjecture under gluings of toroidal boundary components.
method Using a construction of hyperbolic cusped 3-manifolds by Agol, the authors show that the asymptotics of Turaev-Viro invariants are additive under certain gluings of elementary pieces.
result The Turaev-Viro invariant volume conjecture is preserved under specific gluings of 3-manifolds.
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
Study series invariants for plumbed 3-manifolds and their properties.
problem Understanding series invariants for plumbed 3-manifolds and their applications.
method Twisted root lattice, gluing and splitting properties, explicit description of lens spaces and Brieskorn spheres.
result Series verify gluing and splitting properties of 3-manifolds.
We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely general gluing formula for Reidemeister torsion of 3-manifolds. Lastly, we descri…
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.
problem Define and prove gluing theorem for monopole Floer homology.
method Define monopole Floer homology, establish gluing theorem for disconnected boundary.
result Floer homology detects Thurston norm and fiberness of 3-manifolds.
Formula for combining Seiberg-Witten invariants of 4-manifolds.
problem Combining invariants of 4-manifolds obtained by connected sum.
method Proved a gluing formula for families Seiberg-Witten invariants.
result Formula expresses invariants of connected sum in terms of one summand.
Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten invariants in normal-crossing degenerations. I gave the original lecture series…
Local gluing connects flow lines in finite time intervals.
problem Connecting flow lines in finite time intervals.
method Functional analytic approach to define local gluing map.
result Explicit construction of local gluing map in Euclidean case; intricate construction in non-Euclidean case.
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…
Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of the hyperbolic gluing equations, which enables the construction of hy…
This paper looks at a class of closed orientable 3-manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is π1-injective. This construction is the generalisation to handlebodies of the condition for gluing three solid tori to produce non-Haken Seifert fibered 3-manifol…
The paper constructs Ricci-positive metrics on complex manifolds.
problem Creating Ricci-positive metrics on connected sums of products of spheres.
method Two new technical theorems are introduced: a gluing construction for Ricci-positive manifolds with corners and a boundary deformation theorem.
result Ricci-positive metrics are constructed on the connected sum of products of arbitrarily many spheres.
Researchers glue and deform Calabi-Yau 3-folds, proving local diffeomorphisms.
problem Deforming and gluing asymptotically cylindrical Calabi-Yau 3-folds.
method Developed new proofs for gluing and deformation, extending results to the asymptotically cylindrical case.
result Proved the gluing map is a local diffeomorphism, connecting moduli spaces of Calabi-Yau structures.
Smoothly attaches manifolds with controlled curvature.
problem Attaching manifolds with corners and controlling curvature.
method Desingularization theorem to smoothly attach manifolds with controlled scalar and mean curvature.
result Smoothly attaches manifolds with controlled scalar curvature and mean curvature of boundary.
We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP2 and the product metric on S2×S2. Using these met…
We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a t…
Built the smallest non-commensurable hyperbolic 4-manifold.
problem Finding the smallest non-commensurable hyperbolic 4-manifold.
method Gluing copies of a polytope to build the manifold.
result The constructed manifold has twice the minimal volume.
The paper defines equations for constructing projective structures on 3-manifolds.
problem Constructing projective structures on 3-manifolds.
method Defining a system of real valued equations and inequalities.
result These equations can be used to detect properly convex structures on 3-manifolds.
Existence of non-trivial monopoles on 3-spheres proven.
problem Existence of non-trivial SU(2)-monopoles on 3-manifolds.
method Gluing construction to prove existence on rational homology 3-spheres.
result Existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities.
This paper constructs Seiberg-Witten monopoles from harmonic spinors on 3-manifolds.
problem Constructing Seiberg-Witten monopoles from Z2-harmonic spinors on 3-manifolds. method Gluing construction with alternating iteration to cancel infinite-dimensional obstruction bundle.
result A 1-parameter family of Seiberg-Witten monopoles converging to a Z2-harmonic spinor. Paper discusses gluing formula for zeta-determinants with Robin boundary condition.
problem Computing zeta-determinants with Robin boundary condition.
method Uses BFK type gluing formula and computes differences with Dirichlet boundary condition.
result Computes zeta-determinant on a cylinder with Robin boundary condition.