arXiv research

Study geometry of surfaces glued along a curve.

We describe a new approach to the problem of constructing gluing parameterizations for open neighborhoods of boundary points of moduli spaces of anti-self-dual connections over closed four-dimensional manifolds. Our approach employs general results from differential topology for $C^1$ maps of smooth Banach manifolds wi…

Geometric model of unbounded sl3 laminations with tropical coordinates.

In this paper we announce a gluing theorem for conformal structures with anti-self-dual (ASD) Weyl tensor that applies in geometrical situations that are more general than those considered by previous authors. By adapting a method proposed by Floer, sufficient conditions are given for the existence of ASD conformal str…

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely …

The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in geometric analysis to construct new solutions to a given nonlinear partial diffe…

The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by glu…

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express…

For a compact Riemannian manifold $(M, g_2)$ with constant $Q$-curvature of dimension $n\geq 6$ satisfying nondegeneracy condition, we show that one can construct many examples of constant $Q$-curvature manifolds by gluing construction. We provide a general procedure of gluing together $(M,g_2)$ with any compact manifo…

We construct a new class of complete constant mean curvature surfaces in R^3. These are geometrically different than the surfaces constructed by Kapouleas' gluing technique. These are obtained by piecing together half-Delaunay surfaces to the truncations of minimal k-noids. The gluing techniques are new: the surfaces a…

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth soluti…

Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter (AdS) geometry. Our approach involves solving Thurston's gluing …

We describe a construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued t…

For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.

Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.

Method constructs rigid associative submanifolds in twisted G2-manifolds.

This is an extended write-up of a talk given in April, 1993 in honor of Raoul Bott's 70th birthday. We first illustrate how some traditional topological and geometric invariants obey ``gluing laws'' inspired by those in classical and quantum field theory. Here we discuss characteristic numbers, particularly the Euler n…

With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…

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

We establish a link between the holomorphic derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. Moreover, we provide a geometric reformulation of the non-abelian Reidemeister torsion correspondi…

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we produce the generalized connected sum along $K$. On this third manifold we produ…

Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a manifold with a triangulated boundary. These invariants can be naturally united in a …

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

Proposes integrating global and local entropy for more reliable LLMs.

Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries" (see Introduction and Figure 1). These geometric componen…

New method for computing hyperbolic structures on 3-manifolds with torus boundaries.

Study finds eigenvalue bounds for non-convex domains using cohomology.

This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)

Proves modular functors for SO(3) have integral Hodge structures.

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional…

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, an…

New Morse theory for shapes at distances.

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

Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. In the present paper, a similar interpolation theorem for pseudo-holomorphic maps from the cylinder S to an almost-complex manifold (M,J) is proved. Properties of the space of pseudo-holomorphic ma…

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

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 connects orbifold singularities to higher symmetries in SQFTs.

Study examines heart and football-shaped metrics, verifying geometric structure.

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.

In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total curvature. II. Complete embedde…

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

Paper solves Einstein vacuum equations gluing problem with applications.

This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen 7-dimensional orbifolds, with the help of asymptotically locally Euclidean spaces. Anoth…

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…