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Martin Traizet

Publications and source records attributed to Martin Traizet.

At least 19 recordsLinked to original sources
arXiv2026

CmcMesh: a python implementation of the DPW method

Minimal and constant mean curvature (CMC) surfaces in three-dimensional space forms can be constructed with the Dorfmeister--Pedit--Wu (DPW) method. We introduce CmcMesh, a Python implementation covering the whole pipeline from holomorphic potential to rendered surfaces, using Hoffman and Hoffman's Mesh algorithm. We implement the Weierstrass representation for minimal surfaces in $\mathbb{R}^3$, Bryant's representation for CMC-1 surfaces in $\mathbb{H}^3$, and the DPW method for CMC surfaces in $\mathbb{R}^3$ and for minimal and CMC surfaces in $\mathbb{S}^3$. The package is modular, making it easy to add new DPW variants without modifying existing code.

math.DG↗
arXiv2026

The Enclosed Volume for Periodic Constant Mean Curvature Surfaces

We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a Wess-Zumino-Witten-type term, and a newly defined curvature term of the associated family of flat connections, thereby extending the classical Minkowski formula for closed CMC surfaces. Interpreting the volume as a gauge-invariant quantity, we apply the result to a variety of examples and provide explicit computations. As an application, we construct a counterexample to the isoperimetric problem in $\mathbb{T}^2 \times \mathbb{R}$, disproving the conjecture that minimizers are restricted to spheres, cylinders, or pairs of planes.

math.DG↗
arXiv2025

Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere

Building on Hitchin's work of the Wess-Zumino-Witten term for harmonic maps into Lie groups, we derive a formula for the enclosed volume of a compact CMC surface $f$ in $\mathbb S^3$ in terms of a holonomy on the Chern-Simons bundle and the Willmore functional. By construction the enclosed volume only depends on the gauge classes of the associated family of flat connections of $f$. In this paper we show in various examples the effectiveness of this formula, in particular for surfaces of genus $g\geq2.$

math.DG↗
arXiv2024

Minimal surfaces and alternating multiple zetas

In this paper we show for every sufficiently large integer $g$ the existence of a complete family of closed and embedded constant mean curvature (CMC) surfaces deforming the Lawson surfaces $\xi_{1,g}$ parametrized by their conformal type. When specializing to the minimal case, we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), which generalizes the notion of Riemann's zeta values to multiple integer variables. This allows us to extend a new existence proof of the Lawson surfaces $\xi_{1,g}$ to all $g\geq 3$ using complex analytic methods and to give closed form expressions of their area expansion up to order $7$. For example, the third order coefficient is $\tfrac{9}{4}\zeta(3)$ (the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}). As a corollary, we obtain that the area of $\xi_{1,g}$ is monotonically increasing in their genus $g$ for all $g\geq 0.$

math.DG↗
arXiv2022

Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using complex analytic methods. We identify the rescaled limit hyper-K\"ahler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's $\lambda$-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form $\varpi_t$ on $\mathcal M_t$ at $t=0$ which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of $\mathcal M_t$ lead to some identities of certain MPLs which we believe deserve further investigations.

math.DG↗
arXiv2021

Area Estimates for High genus Lawson surfaces via DPW

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 $g=m k \gg1$.

math.DG↗
arXiv2021

Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton)

For every $g \gg 1$, we show the existence of a complete and smooth family of closed constant mean curvature surfaces $f_\varphi^g,$ $ \varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To construct these we use an implicit function theorem argument in the parameter $t= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$ at $t= 0$ explicitly. In particular, we obtain for large genus Lawson surfaces $\xi_{1,g}$ % due to the real analytic dependence of its area and DPW potential on $t,$ a scheme to explicitly compute the coefficients of the power series in $t$ in terms of multiple polylogarithms. Remarkably, the third order coefficient of the area expansion is identified with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}).

math.DG↗
arXiv2021

Gluing Karcher-Scherk saddle towers I: Triply periodic minimal surfaces

We construct minimal surfaces by gluing simply periodic Karcher--Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase differences between the saddle towers. It turns out that, in addition to the previously known horizontal balancing condition, the saddle towers must also be balanced under a subtle vertical interaction. This interaction vanishes in the presence of a horizontal reflection plane, hence was not perceived in previous works. Our construction will be presented in a series of papers. In this first paper of the series, we will explain the background of the project and establish the graph theoretical setup that will be useful for all papers in the series. The main task of the current paper is to glue saddle towers into triply periodic minimal surfaces (TPMSs). Our construction expands many previously known TPMSs into new 5-parameter families, therefore significantly advances our knowledge on the space of TPMSs.

math.DG↗
arXiv2020

Stacking disorder in periodic minimal surfaces

We construct 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in $T \times \mathbb{R}$, where $T$ denotes a flat 2-tori. Each of our families converges to a foliation of $T \times \mathbb{R}$ by $T$. These surfaces then lift to minimal surfaces in $\mathbb{R}^3$ that are periodic in horizontal directions but not periodic in the vertical direction. In the language of crystallography, our construction can be interpreted as disordered stacking of layers of periodically arranged catenoid necks. Our work is motivated by experimental observations of twinning defects in periodic minimal surfaces, which we reproduce as special cases of stacking disorder.

math.DG↗
arXiv2020

Opening nodes in the DPW method: co-planar case

We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

math.DG↗
arXiv2018

Opening nodes and the DPW method

We combine the DPW method and Opening Nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

math.DG↗
arXiv2016

Helicoidal minimal surfaces of prescribed genus

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$.

math.DG↗
arXiv2014

Hollow vortices and minimal surfaces

We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.

math.AP↗
arXiv2013

Helicoidal minimal surfaces of prescribed genus, I

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.

math.DG↗