Minimal surfaces in $R^3$ with dihedral symmetry

The periodic Plateau problem and its application

Given a noncompact disconnected complete periodic curve $Γ$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $Γ$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\partial T$ along their boundary. It is also proved that every Platonic solid of $\mathbb R^3$ contains five types of free boundary embedded minimal surfaces of genus zero.

Surfaces with constant mean curvature 1/2 and genus one in H2xR

We construct new constant mean curvature surfaces in H2xR. They arise as sister surfaces of Plateau solutions. It is a family of MC 1/2 surfaces with k ends, genus 1 and k-fold dihedral symmetry, k greater 2. The surfaces are Alexandrov- embedded.

一类包含 Costa-Hoffman-Meeks 曲面的高亏格完备极小曲面族

A family of higher genus complete minimal surfaces that includes the Costa-Hoffman-Meeks one

本文构造了一类具有任意高亏格、三个端和有限全曲率的完备浸入极小曲面族，其对称群为二面体群，并在参数满足特定条件时退化为 Costa-Hoffman-Meeks 嵌入极小曲面。

极小曲面的指标与零化度：表示论方法

Index and nullity of minimal surfaces via representation theory

本文提出一种基于表示论的方法，仅利用拓扑与对称群计算嵌入极小曲面的Morse指标与零化度，并应用于自由边界极小曲面及Lawson曲面，结果不依赖具体构造。

Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$

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…

Minimal surfaces in the 3-sphere by stacking Clifford tori

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by $k\ell m^2 (N-1)$ small catenoidal tunnels, with $k \ell m^2$ tunnels join…

Generalized Henneberg stable minimal surfaces

We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface $H_1$ is characterized as the unique example in the subfamily of the simplest complexity $m=1$, while for $m\geq 2$ multiparameter families are given. The isometry group of the most symmetric example $H_m$ with a given complexity $m\i…

On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball

We develop new methods to compare the span $\mathcal{C}(Σ)$ of the coordinate functions on a free boundary minimal submanifold $Σ$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace $\mathcal{E}_{σ_1}(Σ)$. Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{σ_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\mathcal{C}=\mathcal{E}_{σ_1}$ for any free boundary minimal surface embedded in $\mathbb{B}^3$ with the …

Constructing mean curvature 1 surfaces in $H^3$ with irregular ends

With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonali…

Sufficient symmetry conditions for free boundary minimal annuli to be the critical catenoid

We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball. Finally, we obtain several uniqueness results for the critical catenoid under symmetry conditions on the boundary. For example, we show that if an embedded free boundary minimal annulus whose boundary consists of two congruent components and has a reflection symmetry by a plane, then it is congruent to the critical catenoid.

Maximum Principle and Symmetry for Minimal Hypersurfaces in H^nxR

The aim of this work is to study how the asymptotic boundary of a minimal hypersurface in H^nxR determines the behavior of the hypersurface at finite points, in several geometric situations.

Minimal surfaces associated with nonpolynomial contact symmetry flows

Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.

Minimal surfaces with the area growth of two planes; the case of infinite symmetry

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.

Triply Periodic Minimal Surfaces Bounded by Vertical Symmetry Planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.

具有相同拓扑结构与对称群的无穷多对自由边界极小曲面

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

该研究指出三维欧氏单位球中自由边界极小曲面的拓扑与对称群无法唯一确定曲面，构造出无穷多对非等距、具任意充分大亏格、三个边界分支及对应阶反棱柱对称群的此类曲面对。

Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities

By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.