pith. sign in

arxiv: 2105.10711 · v3 · submitted 2021-05-22 · 🧮 math.DG

Genus three embedded doubly periodic minimal surfaces with parallel ends

Peter Connor , Shoichi Fujimori , Phillip Marmorino , Toshihiro Shoda This is my paper

Pith reviewed 2026-05-24 13:39 UTC · model grok-4.3

classification 🧮 math.DG
keywords minimal surfacesdoubly periodicgenus threeparallel endsWeierstrass dataperiod problemembedded surfaces
0
0 comments X

The pith

A one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends is constructed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors construct explicit Weierstrass data for a continuous family of minimal surfaces. These surfaces are doubly periodic with genus three and feature four parallel ends. They solve the associated two-dimensional period problem to ensure the surfaces are closed and embedded. This provides concrete new examples of such surfaces in Euclidean three-space. A reader would care because it enlarges the set of known periodic minimal surfaces with these properties.

Core claim

We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved.

What carries the argument

The Weierstrass data, which encode the surface via meromorphic functions on a Riemann surface, together with the solution of the two-dimensional period problem that enforces periodicity and embedding.

Load-bearing premise

The Weierstrass data chosen for the family actually admit a solution to the two-dimensional period problem that produces embedded surfaces rather than immersed or self-intersecting ones.

What would settle it

A numerical check for a specific value of the parameter showing that the resulting surface self-intersects or that the periods fail to close.

Figures

Figures reproduced from arXiv: 2105.10711 by Peter Connor, Phillip Marmorino, Shoichi Fujimori, Toshihiro Shoda.

Figure 1.1
Figure 1.1. Figure 1.1: Mλ with λ = 0.4 (left), λ = 0.9 (center), and λ = 0.99 (right). Connor and Weber showed the existence of these surfaces in a small neighborhood of their limit as a foliation of R 3 by vertical planes. This result significantly expands the known parameters for which these surfaces exist. As λ → 0, Mλ limits as a foliation of vertical planes, the Connor-Weber limit. As λ → 1, Mλ limits as two copies of a g… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Left: Triply periodic minimal surface of genus 3, called Schwarz CLP. Center: Weber’s triply periodic minimal surface of genus 5. Right: A limit of the Weber’s surface. 2. Weierstrass Representation We use the Weierstrass Representation of minimal surfaces. Let M be a minimal surface and R the underlying Riemann surface of M. Then M can be expressed by X : R → R 3 , (2.1) X(z) = Re Z z z0 (φ1, φ2, φ3), w… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Signs of ξ1 and ξ2. in the parallelogram so that an endpoint of c2 lies on BC and the other endpoint lies on DE. c1 divides the parallelogram into two regions, one includes BC and the other includes DE. On the other hand, the endpoints of c2 lie on BC and DE. Therefore c2 must intersect to c1. The intersection (λ1, λ2) satisfies ξ1(λ1, λ2) = ξ2(λ1, λ2) = 0. Thus the minimal surface exists for that λ, and… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Fundamental piece Ω1 (left) and its conjugate Ω∗ 1 (right) We will show that Ω1 is a graph over a domain in the (x2, x3)-plane. By Lemma 2.3 and Fact 4.2, the boundary of Ω1 consists of seven planar symmetry curves: (1) s1 is the real interval [λ, a) and S1 := X(s1) lies in a plane parallel to the (x2, x3)-plane; (2) s2 is the real interval [0, λ] and S2 := X(s2) lies in a plane parallel to the (x1, x3)-… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Ω ∗ 2 (left), and their projections onto x2x3-plane (right) We denote by Ω∗ 2 the resulting surface, which is two copies of Ω∗ 1 . We also denote by Sˆ∗ 2 , Sˆ∗ 4 , Sˆ∗ 6 the copies of S ∗ 2 , S ∗ 4 , S ∗ 6 , respectively. Let π : R 3 → R 2 be the projection from R 3 into the (x2, x3)- plane, that is, π(x1, x2, x3) = (x2, x3). Let Ck (k = 1, 2, 3, 4) be straight segments in (x2, x3)-plane as C1 = π(Sˆ∗ 6… view at source ↗
read the original abstract

We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper constructs a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface in the family are explicitly given on a suitable Riemann surface, the two-dimensional period problem is reduced to a pair of real equations in one real parameter, a solution is exhibited via a sign-change argument, and the resulting Gauss map is verified to have the correct degree with no extraneous branch points, yielding embedded surfaces by standard criteria for such minimal surfaces.

Significance. If the result holds, the work supplies an explicit new family of examples in the theory of minimal surfaces, which are scarce for genus three with parallel ends. The reduction of the period problem to a one-parameter setting with an intermediate-value argument, together with the verification of embedding criteria, provides a concrete and falsifiable construction that advances the understanding of doubly periodic minimal surfaces.

minor comments (1)
  1. The abstract and introduction could more explicitly reference the specific Riemann surface on which the Weierstrass data are defined (e.g., by its equation or branch points) to aid readers in reproducing the setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs explicit Weierstrass data on a Riemann surface for a one-parameter family and reduces the two-dimensional period problem to a pair of real equations in one parameter. It then locates a parameter value at which the period integrals vanish (via sign-change or intermediate-value argument) and verifies the Gauss map degree and absence of extraneous branch points. This is a standard, independent existence argument for minimal surfaces; the period integrals are not defined in terms of the target surface, no parameter is fitted to the output, and no load-bearing step reduces by the paper's own equations to a self-citation or ansatz smuggled from prior work by the same authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The result rests on the standard Weierstrass representation for minimal surfaces and the solvability of the period problem for the chosen data; the one free parameter parametrizes the family.

free parameters (1)
  • one-parameter family parameter
    The family is indexed by a single real parameter whose value is chosen so that the period conditions hold.
axioms (1)
  • standard math Weierstrass representation theorem for minimal surfaces in R^3
    Invoked to define the surfaces from meromorphic data on a compact Riemann surface.

pith-pipeline@v0.9.0 · 5550 in / 1154 out tokens · 20968 ms · 2026-05-24T13:39:04.758564+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Connor and M

    P. Connor and M. Weber, The construction of doubly periodic minimal surfaces via balance equations, Amer. J. Math., 134 (2012), 1275--1301

    work page 2012

  2. [2]

    Dorff, Minimal graphs in R ^3 over convex domains , Proc

    M. Dorff, Minimal graphs in R ^3 over convex domains , Proc. Amer. Math. Soc., 132 (2004), 491--498

    work page 2004

  3. [3]

    Jenkins and J

    H. Jenkins and J. Serrin, Variational problems of minimal surface type, II, Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal., 21 (1966), 321--342

    work page 1966

  4. [4]

    Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83--114

    H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83--114

    work page 1988

  5. [5]

    , Construction of Minimal Surfaces, Surveys in Geometry, 1--96, University of Tokyo, 1989

    work page 1989

  6. [6]

    W. H. Meeks III and H. Rosenberg, The Global Theory of Doubly Periodic Minimal Surfaces, Invent. Math., 97 (1989), 351--379

    work page 1989

  7. [7]

    Rossman, E

    W. Rossman, E. C. Thayer, and M. Wohlgemuth Embedded, doubly periodic minimal surfaces, Experiment. Math., 9 (2000), 197--219

    work page 2000

  8. [8]

    u ber die kleinste F l\

    H. F. Scherk, Bemerkungen \" u ber die kleinste F l\" a che I nnerhalb gegebener G renzen , J. Reine Angew. Math., 13 (1835), 185--208

    work page

  9. [9]

    Weber, Minimal Surface Repository, https://minimalsurfaces.blog

    M. Weber, Minimal Surface Repository, https://minimalsurfaces.blog

    work page

  10. [10]

    Weber and M

    M. Weber and M. Wolf,

    work page

  11. [11]

    Wei, Some existence and uniqueness theorems for doubly periodic minimal surfaces, Invent

    F. Wei, Some existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math., 109 (1992), 113--136

    work page 1992