The doubly periodic Scherk-Costa surfaces

A Jenkins-Serrin problem on the strip

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 $π/2$.

Uniqueness of minimal surfaces, Jacobi fields, and flat structures

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 …

双周期极小曲面的端到端构造

An end-to-end construction of doubly periodic minimal surfaces

本文基于Traizet再生方法，构造了亏格为2n-1且具有4个平行端的双周期极小曲面族，证明其光滑收敛于2n份Scherk曲面，填补了亏格大于3的该类曲面研究空白。

The classification of doubly periodic minimal tori with parallel ends

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. The degenerate limits of surfaces in $\mathcal{K}$ are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examp…

Scherk-like Translators for Mean Curvature Flow

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.

The space of doubly periodic minimal tori with parallel ends: Standard Examples

We describe a 3-parametric family $\mathcal{K}$ of properly embedded minimal tori with four parallel ends in quotients of $\mathbb{R}^3$ by two independent translations, which we will call the \textit{Standard Examples.} These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. $\mathcal{K}$ can be endowed with a natural structure of a self-conjugated 3-dimensional real analytic manifold diffeomorphic to $\mathbb{R}\times(\mathbb{R}^2-\{(\pm 1,0)\})$ whose degenerate limits are the catenoid, the helicoid, the simply and doubly periodic Scherk minimal s…

Periodic minimal surfaces in semidirect products

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.

Minimal Surface Linear Combinatoin Theorem

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surface to Scherk's singularly periodic surface, Sckerk's doubly periodic surface to the catenoid, and the 4-Enneper surface to the 4-noid.

Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space

The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known Lopez-Ros deformation of minimal surfaces is a special case of this transformation. We express…

手性旋转H'-T表面族：基于双qtz--qzd网的构造及利用双扭Weierstrass方法的存在性证明

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。