Francisco Martin, Valerio Ramos-Batista
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 result concerning asymmetric complete minimal submanifolds of Euclidean space with finite total curvature.
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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.
Inspired by an argument of Ros [15] -- we use the López-Ros deformation to give another proof of the fact -- due to Meeks and Wolf [13] -- that the only smooth, connected, singly-periodic minimal surfaces in $\Real^3$ with the area growth of two planes are the singly-periodic Scherk surfaces.
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。
Catenoid limits of singly periodic minimal surfaces with Scherk-type ends
本文构造了任意亏格和偶数个Scherk端的单周期极小曲面族，其极限为带节点的多层垂直平面，节点打开条件由Stieltjes多项式根的平衡方程决定。
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 …
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…
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 surface for each genus. We illustrate multiple new examples for each genus g>2. In the parallel ends case, the known examples limit as a foliation of parallel planes with nodes. We construct a new example for each genus g>2 that limit as g-1 singly periodic Scherk surfaces glued between two doubly periodic Scherk surfaces and also as a singly periodic surface wit…
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.
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…
We describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfaces and the singly periodic Scherk minimal surface of angle $π/2$.
Inspired by the Finn-Osserman (1964), Chern (1969), do Carmo-Peng (1979) proofs of the Bernstein theorem, which characterizes flat planes as the only entire minimal graphs, we prove a new rigidity theorem for associate families connecting the doubly periodic Scherk graphs and the singly periodic Scherk towers. Our characterization of Scherk's surfaces discovers a new idea from the original Finn-Osserman curvature estimate. Combining two generically independent flat structures introduced by Chern and Ricci, we shall construct geometric harmonic functions on minimal surfaces, and establish that …
We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.
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 space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1)$-periodic embedded minimal hypersurfaces with four hyperplanar ends. The moduli space of these hypersurfaces forms a 1-dimensional fibration over the moduli space of flat tori in ${\R}^{n-1}$. A partial description of the boundary of this moduli space is also given.
In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in $\mathbb R^3$. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol3 with a one-parameter family of non-isometric metrics.
In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems involving the support function. We give a practical application by proving existence of the sheared Scherk-Karcher family of surfaces numerically described by Wei. Moreover, we show that this family is continuous, and both of its limit-members are the singly periodic genus-one helicoid.
We carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and desingularizing the intersection circle using an appropriately modified singly periodic Scherk surface, called the core. Using an inverse function theorem, we show that for small boundary conditions on the core, there is an embedded surface close to the core that is a solution of the equation for self-similar surfaces. This provides us with an adequate central…
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 the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an…
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