Related papers: Adding one handle to half-plane layers

The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.

We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal $n$-gon -- so-called minimal reflection surfaces. The minimal $n$-gon solves a free boundary problem in a fundamental piece of…

We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy \cite{ET0}, we obtain a description of exceptional surfaces in terms of a set of…

This paper mathematically studies membranes and filaments adhering to periodic patterned substrates in a one-dimensional model. The problem is formulated by the minimizing problem of an elastic energy with a contact potential on graph…

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a…

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the…

We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the four dimensional…

Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the…

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4\pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant…

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new…

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian.…

Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and…

The paper provides an elementary proof of Kenyon's necessary condition for the existence of a periodic tiling of the plane by squares with given periods. A similar new result on covering both sides of a rectangle by nonoverlaping squares is…

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.

We introduce a square tiling/tetragonal strip representation to the P, D, and G triply periodic minimal surfaces. This approach is useful in identifying mixtures and grain boundaries of these surfaces that might be useful for material…

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of…

We propose a new exact approach to the generalized graph layering problem that is based on a particular quadratic assignment formulation. It expresses, in a natural way, the associated layout restrictions and several possible objectives,…

A problem of an interface crack between two semi-planes made out of different materials under an action of an in-plane loading of general tensile-shear type is treated in a semi-analytical manner with the help of Dirichlet-to-Neumann…