Laurent Mazet, M. Magdalena Rodriguez, Martin Traizet
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).
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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 Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain Jenkins-Serrin graphs over ideal polygonal domains (with total intrinsic curvature 2 pi(1-k)); we also get properly embedded minimal surfaces which are symmetric with respect to a horizontal slice and have total intrinsic curvature 4 pi(1-k), genus zero and k vertical planar ends.
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 ensure the convergence of a family of bounded minimal disks. Such a barrier is actually a periodic deformation of a minimal plane with prescribed asymptotic behavior. A consequence of the barrier construction is that the number of disjoint minimal graphs suppoerted on domains is not bounded in Heisenberg space.
Starting from works by Scherk (1835) and by Enneper-Weierstraß\ (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 genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}). However, Traizet's construction is implicit and excludes {\it towers}, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martin and Ramos Batista (see \cite{Martin}), all of them with genus one. For genus two, t…
Gluing Karcher-Scherk saddle towers I: Triply periodic minimal surfaces
本文通过沿翼粘合单周期 Karcher-Scherk 鞍塔并引入相位差打破对称性，发现了一种微妙的垂直平衡条件，成功构造了扩展为 5 参数族的三重周期极小曲面。
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 intersect at the same line. In our paper, by a {\it saddle-tower} we call the desingularization of such ``forbidden'' planes into an embedded singly periodic minimal surface. We give explicit examples of genus two and discuss some advances regarding this problem. Moreover, our examples are the first ones containing {\it Gaussian geodesics}, and for the first tim…
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 quotient, called {\it saddle towers}. They have been recently classified by Pérez and Traizet \cite{PeTra1} as the only properly embedded singly periodic minimal surfaces in $\R^3$ with genus zero and finitely many Scherk-type ends in the quotient. In this paper we obtain as a limit of saddle towers: the catenoid; the doubly periodic Scherk minimal surface …
Gluing Karcher-Scherk saddle towers II: Singly periodic minimal surfaces
本文通过沿翼面粘合单周期 Karcher-Scherk 鞍塔构造具有 Scherk 端的单周期极小曲面，规定相位差获得无水平反射平面的新实例，并仔细分析轻微弯曲以纠正近期关于平行 Scherk 端嵌入性的错误，确定曲面确实嵌入的场景。
For each $k\geq 3$, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times\mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb{H}^2\times\mathbb{R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to $k$ vertical planes, as well as finite total curvature $-4kπ$. Finally, we also provide examples of complete properly Alexandrov-embedded minimal surfa…
Morse index of Karcher saddle towers in $\mathbb{R}^2 \times \mathbb{S}^1(m)$
本文研究R²×S¹(m)中Karcher构造的单周期极小曲面子族的Morse指数与零空间维数，利用对称性和Dirichlet-to-Neumann映射完成相关指数与维数的公式推导及证明。
Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ ξ_{m,k}$ in terms of their genus $g=m k \gg1$.
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 $\mathcal{T}\times\mathbb{R}$. For this purpose, inspired by the work of Martin Traizet, we open the nodes of singular Riemann surfaces to glue together simply periodic Karcher saddle towers, each placed at a vertex of the tiling in such a way that its wings go along the corresponding edges of the tiling ending at that vertex.
The chiral gyrating H'-T surface family: construction from the dual qtz--qzd nets and existence proof using a toroidal Weierstrass method
本文提出一种手性旋转H'-T表面族，通过双qtz--qzd网构造并利用双扭Weierstrass方法证明其存在性，用于光子材料设计。
A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property
本文构造了 $\mathbb{H}^2\times\mathbb{R}$ 中一族常平均曲率曲面，推广了鞍形塔和 $k$-noids，并证明 Krust 性质对任何 $H>0$ 不成立。
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