arXiv research

Study geometry of surfaces glued along a curve.

The paper classifies surfaces formed by quadrilateral gluings.

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

Characterizes metrics on triangulated surfaces using glued Euclidean triangles.

Extends 4D cornered skein theory to surfaces, proving gluing formulas.

Constructs minimal surfaces in a 3-ball using PDE gluing.

We construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

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

Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.

The paper extends gluing theorems for linearized gravitational fields in static spacetimes with cosmological constant.

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' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing tec…

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…

New signs and gradings enable detailed comparison in Heegaard Floer theory.

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two $4$-manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the $4$-manifold resulting as a connected sum along the surface. We…

New minimal surfaces found from vortex crystals.

Given a $4$-manifold $\hat{M}$ and two homeomorphic surfaces $Σ_1, Σ_2$ smoothly embedded in $\hat{M}$ with genus more than 1, we remove the neighborhoods of the surfaces and obtain a new $4$-manifold $M$ from gluing two boundaries $S^1 \times Σ_1$ and $S^1 \times Σ_1.$ In this artice, we prove a gluing formula which d…

Constructs minimal surfaces by gluing saddle towers with Scherk ends.

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

The study glues $CAT(0)$ subsets of the plane under certain curvature conditions.

New cosmological spacetimes without CMC Cauchy surfaces found.

The paper derives formulas for symplectic volume forms on surface representation varieties.

This paper solves minimal surface equations near Hardt-Simon foliations.

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scala…

Continuous process closes cusps in complex algebraic surfaces.

In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. In particular, we state conditions for the existence of outer tr…

Let $M_1$ and $M_2$ be two $n$-dimensional smooth manifolds with boundary. Suppose we glue $M_1$ and $M_2$ along some boundary components (which are, therefore, diffeomorphic). Call the result $N.$ If we have a group $G$ acting continuously on $M_1,$ and also acting continuously on $M_2,$ such that the actions are comp…

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

New vacuum spacetimes without CMC Cauchy surfaces found.

Existence of Ricci flat metric on Kummer K3 surface proven.

A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.

In this paper we define a $\mathbf{QP}^1$-valued class function on the mapping class group $\mathcal{M}_{g,2}$ of a surface $Σ_{g,2}$ of genus $g$ with two boundary components. Let $E$ be a $Σ_{g,2}$ bundle over a pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we …

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 …

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and Y, two connected small manifolds with incompressible boundary, along a closed sur…

Constructs minimal surfaces near the boundary of a ball.

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is t…

We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.

Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is t…

We show the existence of various families of properly embedded singly periodic minimal surfaces in R^3 with finite arbitrary genus and Scherk type ends in the quotient. The proof of our results is based on the gluing of small perturbations of pieces of already known minimal surfaces.

New minimal surfaces derived from helicoids.

New gravitational instantons and non-holomorphic minimal spheres discovered.

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…

New method constructs small symplectic 4-manifolds via contact gluing.

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

Random hyperbolic surfaces have low Cheeger constants.

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half…