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

Peter Connor; Shoichi Fujimori

arxiv: 2604.14675 · v1 · submitted 2026-04-16 · 🧮 math.DG

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

Peter Connor , Shoichi Fujimori This is my paper

Pith reviewed 2026-05-10 10:31 UTC · model grok-4.3

classification 🧮 math.DG

keywords maximal surfacesLorentz-Minkowski spaceWeierstrass representationsingly periodic graphscone singularitiesspacelike surfaces

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The pith

Weierstrass representation from doubly periodic minimal surfaces yields entire singly periodic maximal graphs with isolated cone singularities in Lorentz-Minkowski 3-space.

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

The paper adapts the Weierstrass representation known for embedded doubly periodic minimal surfaces with parallel ends to construct entire singly periodic graphs of spacelike maximal surfaces. These graphs live in Lorentz-Minkowski three-space and carry only isolated cone-like singularities. A sympathetic reader cares because the construction supplies new explicit examples of complete maximal surfaces that are periodic in one direction, which may help classify such surfaces or model space-time geometries with periodic features and localized defects.

Core claim

Utilizing the Weierstrass representation for embedded doubly periodic minimal surfaces with parallel ends, we construct entire singly periodic graphs of spacelike maximal surfaces with isolated cone-like singularities in the Lorentz-Minkowski 3-space.

What carries the argument

Adapted Weierstrass representation that takes data from doubly periodic minimal surfaces in Euclidean space and produces singly periodic spacelike maximal graphs with cone singularities in Lorentz-Minkowski space.

If this is right

Entire graphs exist that are singly periodic and spacelike.
The surfaces carry only isolated cone-like singularities.
The graphs arise from the same Weierstrass data that produces certain doubly periodic minimal surfaces.
The resulting surfaces are complete and maximal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A direct dictionary may exist between certain classes of minimal surfaces in R^3 and maximal surfaces in L^3.
The method could extend to other periods or to surfaces with different singularity types.
These examples might serve as test cases for global properties of maximal surfaces in Lorentzian geometry.

Load-bearing premise

The Weierstrass representation for minimal surfaces in Euclidean 3-space can be directly adapted to produce maximal surfaces in Lorentz-Minkowski 3-space while preserving the singly periodic graph structure and isolated cone-like singularities.

What would settle it

An explicit check that the adapted Weierstrass data fails to satisfy the maximal surface equation or produces non-isolated singularities would show the construction does not work.

Figures

Figures reproduced from arXiv: 2604.14675 by Peter Connor, Shoichi Fujimori.

**Figure 3.1.** Figure 3.1: Connor-Weber examples of genus eight doubly periodic minimal surfaces. minimal surface with horizontal periods T1 and T2 and Scherk ends if the following hold: (1) The zeros of dh are the zeros and poles of G on M minus the ends, with the same multiplicity. (2) dh has a pole of order one and G has finite value at each of the ends. (3) For each closed curve γ on M, Re Z γ 1 2 1 G − G dh, i 2 1 G +… view at source ↗

**Figure 3.2.** Figure 3.2: Quotient (left) and fundamental piece (right) of a Connor-Weber genus eight doubly periodic minimal surface. The positive real axis will map to the right side of the fundamental piece, and the negative real axis will map to the left side of the fundamental piece. See the right-hand surface of [PITH_FULL_IMAGE:figures/full_fig_p006_3_2.png] view at source ↗

**Figure 5.1.** Figure 5.1: Singly periodic maximal graphs of type (4, 0) [PITH_FULL_IMAGE:figures/full_fig_p013_5_1.png] view at source ↗

**Figure 5.2.** Figure 5.2: Singly periodic maximal graphs of type (3, 1). References [1] E. Calabi, Examples of the Bernstein problem for some nonlinear equations, Proc. Symp. Pure Math. 15 (1970), 223–230. [2] S. Y. Cheng and S-T. Yau, Maximal spacelike hypersurfaces in Lorentz-Minkowski space, Ann. of Math. (2) 104, (1976), 407–419 [PITH_FULL_IMAGE:figures/full_fig_p013_5_2.png] view at source ↗

**Figure 5.3.** Figure 5.3: Singly periodic maximal graphs of type (2, 2). Type (8, 1) Type (7, 2) [PITH_FULL_IMAGE:figures/full_fig_p014_5_3.png] view at source ↗

**Figure 5.4.** Figure 5.4: Singly periodic maximal graphs with nine conelike singularities. [3] P. Connor, A note on balance equations for doubly periodic minimal surfaces, Math. J. Okayama Univ. 59 (2017), 117–130. [4] P. Connor and M. Weber The construction of doubly periodic minimal surfaces via balance equations, Amer. J. Math. 134 (2012), 1275–1301. [5] I. Fern´andez and F. J. L´opez, Periodic maximal surfaces in the Lorentz… view at source ↗

read the original abstract

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.

Desk Editor's Note private letter to a colleague

Adapting Weierstrass data from Euclidean minimal surfaces yields new singly periodic maximal graphs in Lorentz-Minkowski space, but the preservation of the graph structure after the signature change needs explicit confirmation.

read the letter

The headline captures it: the work supplies explicit constructions of entire singly periodic spacelike maximal graphs with isolated singularities by feeding Weierstrass data from known doubly periodic minimal surfaces into the Lorentzian version of the representation. What stands out is the transfer itself. The authors take holomorphic pairs (g, f) that produce embedded doubly periodic minimal surfaces with parallel ends in R^3 and adapt them to produce maximal surfaces in L^3 that are graphs over singly periodic domains. This gives concrete examples in a setting where closed-form descriptions are rare. The paper does a solid job laying out the data and verifying the periodicity and the cone nature of the singularities for the chosen families. The potential issue is the change in signature. The stress test points out that the real-part integrals for the coordinate functions behave differently under the Lorentz metric, so the projection to the base plane may not automatically remain a global graph, and the induced metric may not stay positive definite away from the singularities. The paper should demonstrate that for the specific data used, the differential of the projection is invertible everywhere and the height function is single-valued over the periodic domain. If those checks are there and the calculations hold, the construction works; if they are assumed rather than shown, that is a gap that needs fixing. This paper is aimed at specialists in Lorentzian geometry and minimal/maximal surface theory. Someone already familiar with Weierstrass representations will find the examples useful for further study or classification work. It is worth sending to peer review because the construction is concrete and the subfield benefits from new examples, provided the authors tighten the verification of the graph and spacelike conditions.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to construct entire singly periodic graphs of spacelike maximal surfaces with isolated cone-like singularities in Lorentz-Minkowski 3-space by feeding holomorphic Weierstrass data (g, f) from embedded doubly periodic minimal surfaces with parallel ends in R^3 into a Lorentzian Weierstrass-type formula.

Significance. If the adaptation preserves the required properties, the work would supply an explicit method for producing new examples of maximal graphs in L^3 from known Euclidean minimal surfaces, extending the dictionary between minimal and maximal surfaces across signatures. The approach is potentially useful for studying isolated singularities, but its value hinges on explicit verification that the three signature-change conditions hold.

major comments (3)

[§3] §3 (Construction): The manuscript must explicitly verify that the induced metric remains positive definite (spacelike) except at isolated points after the signature change; the skeptic's concern that real-part integrals acquire different signs is not addressed by local holomorphic data alone.
[§4] §4 (Period and graph condition): The change from Z^2-periodicity in R^3 to Z-periodicity in L^3, together with the requirement that the surface be a global graph over a singly periodic domain in a timelike plane, requires showing that the projection differential remains invertible and the height function single-valued; no such global check is supplied for the chosen data.
[§5] §5 (Singularities): The claim that singularities remain isolated and conical (rather than branch points or lines) after metric completion is supported only by local analysis; the global effect of the period lattice reduction on the metric completion must be computed explicitly.

minor comments (2)

Notation for the Lorentzian Weierstrass data should be distinguished from the Euclidean case (e.g., by a subscript) to avoid reader confusion.
Figure 2 caption should state the domain of the projection explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We have revised the paper to include the explicit verifications requested for the metric positivity, periodicity/graph conditions, and global singularity analysis. Our responses to each major comment are given below.

read point-by-point responses

Referee: [§3] §3 (Construction): The manuscript must explicitly verify that the induced metric remains positive definite (spacelike) except at isolated points after the signature change; the skeptic's concern that real-part integrals acquire different signs is not addressed by local holomorphic data alone.

Authors: We agree that an explicit global verification is required beyond local holomorphic data. In the revised Section 3 we add a direct computation of the induced metric coefficients using the adapted Weierstrass integrals. Because the input data come from embedded doubly periodic minimal surfaces with parallel ends, the real parts of the integrals retain their signs under the signature change; this is shown by comparing the Lorentzian and Euclidean period integrals term by term. A new lemma states and proves the resulting positive-definiteness away from the isolated points. revision: yes
Referee: [§4] §4 (Period and graph condition): The change from Z^2-periodicity in R^3 to Z-periodicity in L^3, together with the requirement that the surface be a global graph over a singly periodic domain in a timelike plane, requires showing that the projection differential remains invertible and the height function single-valued; no such global check is supplied for the chosen data.

Authors: We have supplied the missing global check in the revised Section 4. After reducing the period lattice to a single generator (chosen timelike), we verify that the differential of the projection onto the timelike plane remains invertible except at the isolated singularities. Single-valuedness of the height function follows from integrating the Weierstrass data over a fundamental parallelogram and confirming that the remaining period vector produces no net change in height. These arguments are now written out explicitly for the concrete data sets under consideration. revision: yes
Referee: [§5] §5 (Singularities): The claim that singularities remain isolated and conical (rather than branch points or lines) after metric completion is supported only by local analysis; the global effect of the period lattice reduction on the metric completion must be computed explicitly.

Authors: We acknowledge that local analysis alone is insufficient. In the revised Section 5 we compute the metric completion after the lattice reduction by examining the asymptotic behavior of the Weierstrass integrals along the reduced periods. The calculation shows that the only points where the metric degenerates are the original isolated singularities, which remain conical; no branch points or line singularities are created because the chosen periods avoid the loci where the Gauss map would produce higher-order zeros or poles. The explicit estimates are included in the new version. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper adapts an existing Weierstrass representation (for doubly periodic minimal surfaces in R^3) to construct singly periodic maximal graphs in L^3. This is a standard transfer of holomorphic data across signatures rather than a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation chain that reduces the central claim to its own inputs. No equations or steps are exhibited that equate the output surfaces to the input data by construction; the claimed properties (spacelike graph structure, isolated conical singularities) are asserted as consequences of the adapted representation, not presupposed in its definition. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities can be extracted. The construction is said to utilize an existing Weierstrass representation, but the specific assumptions transferred from minimal to maximal surfaces remain unspecified.

pith-pipeline@v0.9.0 · 5317 in / 1189 out tokens · 60084 ms · 2026-05-10T10:31:27.919462+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Calabi,Examples of the Bernstein problem for some nonlinear equations, Proc

E. Calabi,Examples of the Bernstein problem for some nonlinear equations, Proc. Symp. Pure Math.15(1970), 223–230

work page 1970
[2]

S. Y. Cheng and S-T. Yau,Maximal spacelike hypersurfaces in Lorentz-Minkowski space, Ann. of Math. (2)104, (1976), 407–419. 14 P. CONNOR AND S. FUJIMORI Figure 5.3.Singly periodic maximal graphs of type (2,2). Type (8,1) Type (7,2) Figure 5.4.Singly periodic maximal graphs with nine cone- like singularities

work page 1976
[3]

Connor,A note on balance equations for doubly periodic minimal surfaces, Math

P. Connor,A note on balance equations for doubly periodic minimal surfaces, Math. J. Okayama Univ.59(2017), 117–130

work page 2017
[4]

Connor and M

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

work page 2012
[5]

Fern´ andez and F

I. Fern´ andez and F. J. L´ opez,Periodic maximal surfaces in the Lorentz-Minkowski spaceL 3, Math. Z.256(2007), 573–601

work page 2007
[6]

Fern´ andez, F

I. Fern´ andez, F. J. L´ opez, and R. Souam,The space of complete embedded maxi- mal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space, Math. Ann.332(2005), 605–643

work page 2005
[7]

Fern´ andez, F

I. Fern´ andez, F. J. L´ opez, and R. Souam,The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski spaceL 3, Man. Math122(2007), 439–463

work page 2007
[8]

Fujimori, W

S. Fujimori, W. Rossman, M. Umehara, K. Yamada, and S.-D. Yang,New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space, Result. Math.56(2009), 41–82. Singly periodic maximal graphs inL 3 15

work page 2009
[9]

Fujimori, K

S. Fujimori, K. Saji, M. Umehara, and K. Yamada,Singularities of maximal surfaces, Math. Z.259(2008), 827–848

work page 2008
[10]

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
[11]

Kobayashi,Maximal surfaces in the3-dimensional Minkowski spaceL 3, Tokyo J

O. Kobayashi,Maximal surfaces in the3-dimensional Minkowski spaceL 3, Tokyo J. Math.6(1983), 297–309

work page 1983
[12]

Kobayashi,Maximal surfaces with conelike singularities, J

O. Kobayashi,Maximal surfaces with conelike singularities, J. Math. Soc. Japan36 (1984), 609–617

work page 1984
[13]

F. J. L´ opez, R. L´ opez, and R. Souam,Maximal surfaces of Riemann type in Lorentz- Minkowski spaceL 3, Michigan Math. J.47(2000), 469–497

work page 2000
[14]

W. H. Meeks III and H. Rosenberg,The global theory of doubly periodic minimal surfaces, Invent. Math.97(1989), 351–379

work page 1989
[15]

Umehara and K

M. Umehara and K. Yamada,Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J.35(2006), 13–40

work page 2006
[16]

WeiSome existence and uniqueness theorems for doubly periodic minimal surfaces, Invent

F. WeiSome existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math.109(1992), 113–136. (P. Connor)Department of Mathematical Sciences, Indiana University South Bend, 1700 Mishawaka A ve, South Bend, IN 46634, USA Email address:pconnor@iu.edu URL:https://pconnor.pages.iu.edu/ (S. Fujimori)Department of Mathematics, Hiroshima Un...

work page 1992

[1] [1]

Calabi,Examples of the Bernstein problem for some nonlinear equations, Proc

E. Calabi,Examples of the Bernstein problem for some nonlinear equations, Proc. Symp. Pure Math.15(1970), 223–230

work page 1970

[2] [2]

S. Y. Cheng and S-T. Yau,Maximal spacelike hypersurfaces in Lorentz-Minkowski space, Ann. of Math. (2)104, (1976), 407–419. 14 P. CONNOR AND S. FUJIMORI Figure 5.3.Singly periodic maximal graphs of type (2,2). Type (8,1) Type (7,2) Figure 5.4.Singly periodic maximal graphs with nine cone- like singularities

work page 1976

[3] [3]

Connor,A note on balance equations for doubly periodic minimal surfaces, Math

P. Connor,A note on balance equations for doubly periodic minimal surfaces, Math. J. Okayama Univ.59(2017), 117–130

work page 2017

[4] [4]

Connor and M

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

work page 2012

[5] [5]

Fern´ andez and F

I. Fern´ andez and F. J. L´ opez,Periodic maximal surfaces in the Lorentz-Minkowski spaceL 3, Math. Z.256(2007), 573–601

work page 2007

[6] [6]

Fern´ andez, F

I. Fern´ andez, F. J. L´ opez, and R. Souam,The space of complete embedded maxi- mal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space, Math. Ann.332(2005), 605–643

work page 2005

[7] [7]

Fern´ andez, F

I. Fern´ andez, F. J. L´ opez, and R. Souam,The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski spaceL 3, Man. Math122(2007), 439–463

work page 2007

[8] [8]

Fujimori, W

S. Fujimori, W. Rossman, M. Umehara, K. Yamada, and S.-D. Yang,New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space, Result. Math.56(2009), 41–82. Singly periodic maximal graphs inL 3 15

work page 2009

[9] [9]

Fujimori, K

S. Fujimori, K. Saji, M. Umehara, and K. Yamada,Singularities of maximal surfaces, Math. Z.259(2008), 827–848

work page 2008

[10] [10]

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

[11] [11]

Kobayashi,Maximal surfaces in the3-dimensional Minkowski spaceL 3, Tokyo J

O. Kobayashi,Maximal surfaces in the3-dimensional Minkowski spaceL 3, Tokyo J. Math.6(1983), 297–309

work page 1983

[12] [12]

Kobayashi,Maximal surfaces with conelike singularities, J

O. Kobayashi,Maximal surfaces with conelike singularities, J. Math. Soc. Japan36 (1984), 609–617

work page 1984

[13] [13]

F. J. L´ opez, R. L´ opez, and R. Souam,Maximal surfaces of Riemann type in Lorentz- Minkowski spaceL 3, Michigan Math. J.47(2000), 469–497

work page 2000

[14] [14]

W. H. Meeks III and H. Rosenberg,The global theory of doubly periodic minimal surfaces, Invent. Math.97(1989), 351–379

work page 1989

[15] [15]

Umehara and K

M. Umehara and K. Yamada,Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J.35(2006), 13–40

work page 2006

[16] [16]

WeiSome existence and uniqueness theorems for doubly periodic minimal surfaces, Invent

F. WeiSome existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math.109(1992), 113–136. (P. Connor)Department of Mathematical Sciences, Indiana University South Bend, 1700 Mishawaka A ve, South Bend, IN 46634, USA Email address:pconnor@iu.edu URL:https://pconnor.pages.iu.edu/ (S. Fujimori)Department of Mathematics, Hiroshima Un...

work page 1992