Classification and existence of doubly-periodic instantons

The doubly periodic Scherk-Costa surfaces

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.

From spatially periodic instantons to singular monopoles

The main result is a computation of the Nahm transform of a SU(2)-instanton over RxT^3, called spatially-periodic instanton. It is a singular monopole over T^3, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.

Spatially periodic instantons: Nahm transform and moduli

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-Mukai transform via Kobayashi-Hitchin correspondences. We also prove existence and non-existence results.

Handle Addition for doubly-periodic Scherk Surfaces

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 proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.

Embedded, Doubly Periodic Minimal Surfaces

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 the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.

A note on balance equations for doubly periodic minimal surfaces

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a corresponding three-parameter family of doubly periodic minimal surfaces. In this note we derive a differential equation that is equivalent to the balance equations for doubly periodic minimal surfaces. This allows for the generation of many more solutions to the balance equations, enabl…

亏格为三的嵌入双周期极小曲面及其平行端口

Genus three embedded doubly periodic minimal surfaces with parallel ends

本文构造了一族亏格为三的嵌入双周期极小曲面，具有四个平行端口，并解决了二维周期问题。

Higher genus doubly periodic minimal surfaces

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…

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。

将双周期Scherk曲面粘合成极小曲面

Gluing doubly periodic Scherk surfaces into minimal surfaces

本文研究通过堆叠并粘合多个双周期Scherk曲面构造极小曲面的可行性，证明除三角形水平格子这一特殊情况外，粘合构造只能得到平凡Scherk曲面、KMR例子或Meeks曲面。

The construction of doubly periodic minimal surfaces via balance equations

Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann surfaces, whose components are copies of C* with finitely many nodes. We derive the balance equations for the location of the nodes and exhibit solutions that allow for surfaces of arbitrarily large genus and number of ends in the quotient.

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…

Doubly periodic self-translating surfaces for the mean curvature flow

We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the ideas of the construction and here prove only that the periodic configuration has the necessary flexibility. These examples show that self-translating surfaces do not necessarily have quadratic volume growth rate in contrast to self-shrinking surfaces.

A Compact Expression for Periodic Instantons

Instantons on various spaces can be constructed via a generalization of the Fourier transform called the ADHM-Nahm transform. An explicit use of this construction, however, involves rather tedious calculations. Here we derive a simple formula for instantons on a space with one periodic direction. It simplifies the ADHM-Nahm machinery and can be generalized to other spaces.

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…

Instanton approximation, periodic ASD connections, and mean dimension

We study a moduli space of ASD connections over $S^3\times \mathbb{R}$. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the mean dimension by using a "Runge-approximation" for ASD connections, and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic ASD connections.

On doubly periodic minimal surfaces in $\mathbb H^2 \times \mathbb R$ with finite total curvature in the quotient space

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in $\mathbb H^2$ or a hyperbolic translation along a geodesic in $\mathbb H^2.$ In fact, we prove that if a properly immersed minimal surface in $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2π,$ and…

Doubly periodic monopoles and $q$-difference modules

An interesting theme in complex differential geometry is to find a correspondence between algebraic objects and differential geometric objects. One of the most attractive is the non-abelian Hodge theory of Simpson. In this paper, pursuing an analogue of the non-abelian Hodge theory in the context of $q$-difference modules, we study Kobayashi-Hitchin correspondences between doubly periodic monopoles and parabolic $q$-difference modules, depending on twistor parameters.

单周期极大图在洛伦兹-闵可夫斯基3空间中的孤立奇点

Singly periodic maximal graphs with isolated singularities in Lorentz-Minkowski 3-space

利用Weierstrass表示法构造了具有孤立锥状奇点的单周期极大曲面，探讨了其在洛伦兹-闵可夫斯基3空间中的性质。

Hyperkahler Metrics from Periodic Monopoles

Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yields interesting examples of metrics with self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described.