Related papers: Gluing Karcher-Scherk saddle towers II: Singly per…

We construct minimal surfaces by gluing simply periodic Karcher--Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase…

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes…

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the…

Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T}\times\mathbb{R}$.…

Starting from works by Scherk (1835) and by Enneper-Weierstra\ss \ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive…

We construct minimal surfaces by stacking doubly periodic Scherk surfaces one above another and gluing them along their ends. It is previously known that the Karcher--Meeks--Rosenberg (KMR) doubly periodic minimal surfaces and Meeks' family…

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…

We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend…

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The…

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's…

We prove the existence of nonperiodic, properly embedded minimal surfaces in $\mathbb{R}^2\times\mathbb{S}^1$ with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical…

We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to…

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The…

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…

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry…

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus…

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz…

In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean…

We present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before.