Pith. sign in

REVIEW 3 major objections 4 minor 56 references

A new family of minimal surfaces of even genus in the three-dimensional sphere

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For each integer n≥2, a closed embedded minimal surface Γ_n of even genus exists in the round 3-sphere, with controlled area, prescribed symmetry, and Morse index at least 4n−1.

desk verdict A genuinely new min-max family of minimal surfaces in S^3, but the existence proof leans on an unpublished theorem whose hypotheses are not checked in the paper. read the letter →

arxiv 2507.22531 v1 pith:CUBMLW7B submitted 2025-07-30 math.DG

Mario B. Schulz , David Wiygul This is my paper
classification math.DG MSC 53A1049Q0553C42
keywords minimalsurfacesthree-dimensionalsphereequivariantmin-maxsweepoutsCliffordtorusMorseindexsymmetrygroupsLawson
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to prove that the round three-sphere contains a new infinite family of closed, embedded minimal surfaces, one for every integer n≥2, filling in unexplored even genera. The construction starts from a 1-parameter sweepout built by fusing the equatorial sphere with a quarter of the Clifford torus, then applies equivariant min-max to extract a minimal surface Γ_n. The paper claims each Γ_n has genus 2n or 2n−2, area strictly between 2π² and 2π²+4π+1/25, contains the equator and n equally spaced meridians, has full symmetry group of order 8n (for n≥4), and has Morse index at least 4n−1. It also claims the new surfaces are geometrically distinct from every previously known embedded minimal surface in $S^{3}$ (for n≥3), including the Lawson surfaces and the Karcher–Pinkall–Sterling genus-six surface.

What carries the argument

The machinery is a G_n-equivariant 1-parameter sweepout whose middle surfaces are formed by fusing the equatorial sphere $S^{2}$ with the Clifford torus $T^{2}$: each Σ_t combines a quarter of a constant-mean-curvature torus C_t with alternating meridional patches σ_k and ζ_k of $S^{2}$, then is symmetrized by reflections through the prescribed group G_n of order 8n. The min-max width of the saturation lies strictly between the area of the equator sphere and the target bound 2π²+4π+1/25, so an equivariant min-max theorem yields a multiplicity-one limit surface. Topological control comes from Lemma 2.2, which constrains the genus by counting intersections with the perpendicular circle $S^{1}$_⊥; index control comes from partitioning by the pyramidal subgroup Y_n and applying the Montiel–Ros method.

What would settle it

Verify the hypotheses of the unpublished equivariant min-max theorem for the G_n-action, especially the singular orbit types ∗11, ∗22 and ∗nn that the paper lists; alternatively, run a numerical min-max computation for n=2 and check whether any G_2-sweepout of width at or below 4π admits a smooth embedded critical point. If the width bound W>4π fails, the construction has no limit surface.

Watch

Extended reading notes

Core claim

The paper's central discovery is that for each integer n≥2 the round three-sphere admits a closed, embedded, G_n-equivariant minimal surface Γ_n whose genus is either 2n or 2n−2, whose area lies strictly between 2π² and 2π²+4π+1/25, and which contains the equator $S^{1}$ together with n equally spaced meridians. The full isometry group of Γ_n is exactly the prescribed group G_n of order 8n for every n≥4, so the surface carries no accidental symmetries; and its Morse index is at least 4n−1. The paper additionally establishes that Γ_n is not congruent to any previously known embedded minimal surface in $S^{3}$ for n≥3, ruling out the Lawson surfaces, the Karcher–Pinkall–Sterling genus-six surface, gluing doublings, and the other families surveyed.

Load-bearing premise

The whole construction rests on an unpublished equivariant min-max theorem from a companion preprint, whose applicability is checked only by listing singular orbit types; if that theorem has unstated hypotheses or a hidden flaw, the surface Γ_n might not exist as claimed.

Editorial extensions

If this is right

  • For every n≥2, there exists a new embedded minimal surface of even genus in S^3, including the first rigorous low-genus examples beyond the equatorial sphere and the Clifford torus.
  • Each Γ_n carries exactly the prescribed symmetry group G_n for n≥4, so equivariant min-max can in this case determine the full isometry group of the limit surface.
  • The Morse index of Γ_n grows at least linearly in n, with a conjectured exact value of 4n+5.
  • The surfaces are geometrically distinct from all previously known embedded minimal surfaces in S^3 for n≥3, including Lawson surfaces, gluing doublings, and the Karcher–Pinkall–Sterling genus-six surface.
  • The genus is either 2n or 2n−2, and the paper conjectures that the topological degeneration to 2n−2 never occurs, so Γ_n should have genus 2n for every n≥2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the genus-2n case always occurs, the Γ_n provide a sequence of embedded minimal surfaces with genus growing linearly while area stays within 4π+1/25 of the equatorial sphere, so their total mean-curvature energy is close to that of the round sphere—this could be checked numerically for small n.
  • The same fusing construction with the reflection Z in place of S^1 yields H_n-equivariant surfaces L_n that the paper suggests may agree with Lawson or Karcher–Pinkall–Sterling examples; testing L_n would clarify how the chosen symmetry subgroup controls which minimal surface the min-max procedure selects.
  • The index lower bound obtained by pyramidal symmetry may extend to other equivariant min-max constructions, giving index growth for free-boundary analogues or for desingularizations built from the same sweepout template.
  • A direct numerical simulation of the min-max limit for n=3 could distinguish whether Γ_3 is the expected new surface or accidentally congruent to the Karcher–Pinkall–Sterling genus-six surface, which the paper argues is impossible via umbilic counting.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constructs, for each integer n at least 2, a closed embedded G_n-equivariant minimal surface Γ_n in the round sphere S^3. The construction is by equivariant min-max applied to a new one-parameter G_n-sweepout obtained by fusing the equatorial sphere with the Clifford torus. The main theorem asserts that Γ_n has genus 2n or 2n−2, area strictly between 2π^2 and 2π^2+4π+1/25, contains the equator and n equally spaced meridians, has full symmetry group G_n for n≥4, and has Morse index at least 4n−1. The paper also proves geometric distinctness from previously known examples for n≥3 and gives evidence for the case n=2. The proof combines a topological classification of equivariant surfaces (Lemma 2.2 and Corollary 2.3), a width estimate based on a stability statement for the isoperimetric inequality (Lemma 4.1 and Lemma 4.2), and a Montiel–Ros type index estimate (Proposition 5.6).

Significance. If correct, the result would be a genuinely new infinite family of closed embedded minimal surfaces in S^3, including new low-genus examples beyond the classical equatorial sphere and Clifford torus. The explicit sweepout construction and the symmetry/index analysis are original and go beyond a straightforward application of known methods. The paper is also careful to state what is conjectural and what is proved, and it gives a detailed distinctness discussion. The main reservations concern not the internal consistency of the arguments but the heavy reliance on an unpublished, co-authored equivariant min-max theorem and on a stability lemma that is asserted without proof; these are load-bearing for the central existence and area claims.

major comments (3)
  1. [Section 5.1, proof of Theorem 1.1(i)–(iii), around Eq. (15)] The existence of Γ_n, the reduction to multiplicity m=1, and the genus upper bound all rest on [21, Theorem 2.14] and [40, Theorem 3.2(iii)], but neither theorem is stated in the paper. In particular, [21] is an unpublished preprint co-authored by the first author, and the only verification offered is the sentence that the singular locus of the G_n-action consists of great circles and great spheres of types *11, *22 and *nn. It is not demonstrated that the G_n-sweepout of Lemma 3.1 satisfies every hypothesis of [21, Theorem 2.14], nor that the odd-multiplicity conclusion of [40, Theorem 3.2(iii)] applies to a reflection group such as G_n. These are not cosmetic checks: if the min-max limit had even multiplicity or a genus bound larger than 2n, the area estimate and the classification by Lemma 2.2 would fail. The paper should state the needed theorem (e.g., in an appendix) with all hypotheses, verify them for the present action and sweepout, or replace it by a published and checked result.
  2. [Lemma 4.1] Lemma 4.1 is stated without proof; the text says only that the proof is analogous to [10, Lemma 4.7]. This lemma is the only ingredient that yields the strict width lower bound W_Π > H^2(S^2) in Lemma 4.2, which is required before the equivariant min-max theorem can be applied. The analogy is not exact: here the symmetry group is G_n rather than the group in [10], the ambient manifold is S^3 rather than a ball, and the notion of equivariant set is the relaxed one of Definition 2.1 (up to negligible sets). In particular, one must justify that the approximate hemisphere produced by the quantitative isoperimetric inequality can be chosen G_n-equivariant and that only the finitely many hemispheres bounded by the great spheres of Lemma 4.3 can occur. A complete proof or a precise statement with the exact hypotheses used should be supplied.
  3. [Lemma 3.1] The construction of the sweepout depends on the assertion that the piecewise smooth surface S′_t admits an equivariant perturbation Σ′_t with properties (a)–(d), but no construction or smoothing argument is given. Since the entire min-max input relies on this perturbation preserving smoothness, embeddedness, G_n-equivariance, the prescribed boundary behavior along S^1, and the area bound, this step should be justified in detail. In particular, the behavior near the seams between the Clifford-torus pieces and the meridian segments, and near S^1, must be shown to be compatible with the reflections that generate G_n.
minor comments (4)
  1. [Corollary 2.3, final sentence] The expression involving the union of the arcs α_j and the reflection ξ_{i/n} is malformed; it should read something like ⋃_{j=1}^k (α_j ∪ ξ_{i/n}(α_j)).
  2. [Proposition 5.16] There is a duplicated word in the first sentence: “has has exactly20 umbilics” should be “has exactly 20 umbilics.”
  3. [Author information] The email address for David Wiygul is listed as mario.schulz@unitn.it, which is presumably the email of the first author; the second author's contact address should be corrected.
  4. [Section 5.4] The distinctness argument is presented as a series of “ad hoc observations” and is necessarily conditional on the completeness of the list of previously known examples. The authors may wish to state more explicitly that the enumeration of known examples is intended to be exhaustive, or to qualify the claim accordingly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the construction, width estimate, genus control, symmetry analysis and Morse-index bound are either self-contained or rest on parameter-free external theorems; the self-citations, though present, do not make the central claim reduce to its inputs.

full rationale

The derivation chain is not circular. The sweepout in Lemma 3.1 is constructed explicitly from Clifford tori and equatorial sectors, with area and genus statements derived from explicit formulas and Lemma 2.2; the genus assertion is not obtained by fitting or by assuming Theorem 1.1. The width estimate in Lemma 4.2 is proven from the quantitative isoperimetric inequality and the self-contained Lemma 4.3 classifying An-equivariant great spheres. The min-max existence step invokes [21, Theorem 2.14], which is a preprint by an overlapping author and is load-bearing for the existence and genus upper bound of Gamma_n. However, it is a general equivariant min-max theorem whose stated hypotheses concern the group action and the sweepout, not the target surface Gamma_n itself; under the review rules, a parameter-free external theorem with assumptions that do not include the target result counts as independent support, so this is not a circular reduction. The odd-multiplicity conclusion is taken from [40, Theorem 3.2(iii)], also an external min-max tool, and the area lower bound follows from standard references [46,47] rather than from the construction. The full symmetry group and Morse index arguments are based on the already-established geometric properties of Gamma_n together with external spectral estimates [13,52]; these are not restatements of Theorem 1.1. The distinctness proofs compare Gamma_n with known families using ad-hoc but independent geometric invariants. The only serious concern is that the unpublished theorem [21, Theorem 2.14] is not stated or fully verified in the paper, so the existence of Gamma_n depends on external machinery whose correctness cannot be checked from this text alone. That is a verification gap and a robustness risk, not a circularity. Because the paper does rely on several self-citations, a score of 2 is appropriate; the central derivation is not equivalent to its inputs by definition or by fitted quantities.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard geometric analysis and on a small number of unproved or preprint-level technical inputs, most notably the equivariant min-max theorem [21] and the stability lemma 4.1. There are no fitted numerical parameters, and no exotic entities are posited.

assumptions (5)
  • domain assumption Equivariant min-max theorem [21, Theorem 2.14] with genus bounds and odd multiplicity applies to the constructed G_n-sweepout.
    Invoked in Section 5.1 to pass from the sweepout to the minimal surface Gamma_n; the theorem is a preprint by one of the authors and its hypotheses are only briefly checked.
  • domain assumption Stability of the isoperimetric inequality for G_n-equivariant volume-half sets (Lemma 4.1).
    Used in Lemma 4.2 to prove W_Pi > 4*pi, which is required for min-max to yield a nontrivial surface; the proof is only stated as analogous to [10, Lemma 4.7].
  • domain assumption Existence of a smooth G_n-equivariant perturbation Sigma'_t of the piecewise-smooth set S'_t with the stated properties (Lemma 3.1).
    The construction of the sweepout depends on this smoothing step, which is asserted without an explicit proof.
  • standard math Riemann-Hurwitz and Gauss-Bonnet formulas for orbifold quotients of surfaces under finite group actions.
    Used in Lemma 2.2 and Corollary 2.3 to derive the allowed genera from symmetry and intersection counts.
  • standard math Lawson's genus bound on number of umbilics [44, Proposition 1.5].
    Used in Lemma 5.4 to bound the number of umbilics of Gamma_n.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A new family of minimal surfaces of even genus in the three-dimensional sphere." pith.science (2026). https://pith.science/paper/CUBMLW7B

@misc{pith2026250722531,
  author       = {Pith},
  title        = {Pith review of: A new family of minimal surfaces of even genus in the three-dimensional sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUBMLW7B}},
  note         = {Machine review of arXiv:2507.22531}
}
read the original abstract

We discover a family of closed, embedded minimal surfaces in the three-dimensional round sphere which includes new examples with low genus. The existence proof relies on an equivariant min-max procedure applied to a novel sweepout which is constructed by fusing the equatorial sphere with the Clifford torus. We determine the full symmetry groups of our surfaces, prove lower bounds on their Morse indices, and show that they are geometrically distinct from all previously known examples.

Figures

Figures reproduced from arXiv: 2507.22531 by the authors.

Figure 1
Figure 1. Stereographic projection of Γn for n = 3 (top image) and n = 12 (bottom image, cross￾section), containing the equator S 1 and the n meridians ξk/n. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The submanifolds S 2 , S 1 , T 2 , ξ i n , Ξ 1 2n , Z ⊂ S 3 in spherical coordinates θ1, θ2, θ3 for n = 3. Using this convention, we understand the equatorial sphere S 2 , its equator S 1 , and the Clifford torus T 2 as follows (see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Left: Intersection of a Gn-equivariant surface of genus 2n with the spherical lune V ⊂ S 3 . Right: Intersection of a Gn-equivariant surface of genus 2n − 2 with the spherical lune V . (The right image displays a nonminimal surface in S 3 .) Proof. We use arguments similar to ones used at the beginning of [32, § 4]. Let V be the closure of the connected component of S 3 \ S i∈Z Ξ(2i+1)/(2n) containing (1, 0, 0, 0); … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The subsets σk, ζk ⊂ S 2 ⊂ S 3 defined in (10) using spherical coordinates θ1, θ2, θ3. Remark 3.2. Consider the group Hn := ⟨An, Z⟩ in place of Gn = ⟨An, S 1 ⟩. The same construction as for Lemma 3.1 yields an Hn-sweepout of S 3 if we replace the second assertion in (1…
Figure 5
Figure 5. Figure 5: Left image: Stereographic projection of Γ2 containing the great circles S 1 , ξ0, ξ1/2 . Right image: Typical stereographic projection of ξ2,2 and its 144 fundamental domains. Proof. By Theorem 1.1 (i) the surface Γ2 has either genus 4 or genus 2; thus it suffices to p…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

56 extracted references · 52 canonical work pages

  1. [21]

    Genus one critical catenoid

    G. Franz, D. Ketover, and M. B. Schulz,Genus one critical catenoid, preprint, available atarXiv:2409.12588

    work page Pith review arXiv

  2. [3]

    A. I. Bobenko, S. Heller, and N. Schmitt,Minimal reflection surfaces inS3. Curvature line foliations and new examples based on fundamental pentagons, J. Geom. Phys.209 (2025)

    work page 2025

  3. [40]

    , Free boundary minimal surfaces of unbounded genus, preprint, available atarXiv:1612.08691

    arXiv

  4. [10]

    Carlotto, G

    A. Carlotto, G. Franz, and M. Schulz,Free boundary minimal surfaces with connected boundary and arbitrary genus, Camb. J. Math.10 (2022), no. 4, 835–857

    work page 2022

  5. [8]

    Buzano, H

    R. Buzano, H. T. Nguyen, and M. B. Schulz,Noncompact self-shrinkers for mean curvature flow with arbitrary genus, J. Reine Angew. Math.818 (2025), 35–52

    work page 2025

  6. [1]

    F. J. Almgren Jr.,Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2)84 (1966), 277–292

    work page 1966

  7. [2]

    S. Bai, C. Wang, and S. Wang,Minimal surfaces in the three-dimensional sphere with high symmetry, Journal of Topology and Analysis13 (2021), no. 02, 289–317

    work page 2021

  8. [4]

    A. I. Bobenko, S. Heller, and N. Schmitt,Constant mean curvature surfaces based on fundamental quadrilaterals, Math. Phys. Anal. Geom.24 (2021)

    work page 2021

Show all 56 references
  1. [5]

    Bögelein, F

    V. Bögelein, F. Duzaar, and N. Fusco,A quantitative isoperimetric inequality on the sphere, Adv. Calc. Var.10 (2017), no. 3, 223–265

    2017

  2. [6]

    Brendle,Embedded minimal tori inS3 and the Lawson conjecture, Acta Math.211 (2013), no

    S. Brendle,Embedded minimal tori inS3 and the Lawson conjecture, Acta Math.211 (2013), no. 2, 177–190

    2013

  3. [7]

    , Minimal surfaces inS3: a survey of recent results, Bull. Math. Sci.3 (2013), no. 1, 133–171

    2013

  4. [9]

    Calabi,Minimal immersions of surfaces in Euclidean spheres, J

    E. Calabi,Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry1 (1967), 111–125

    1967

  5. [11]

    Carlotto, M

    A. Carlotto, M. B. Schulz, and D. Wiygul,Disc stackings and their Morse index, Adv. Nonlinear Stud. (to appear), available atarXiv:2308.07138

    arXiv

  6. [12]

    , Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group, Mem. Amer. Math. Soc. (to appear), available atarXiv:2205.04861

    arXiv

  7. [13]

    PDE 18 (2025), no

    , Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods, Anal. PDE 18 (2025), no. 7, 1715–1768

    2025

  8. [14]

    Choe, Index, vision number and stability of complete minimal surfaces, Arch

    J. Choe, Index, vision number and stability of complete minimal surfaces, Arch. Ration. Mech.109 (1990), 195–212

    1990

  9. [15]

    Choe and M

    J. Choe and M. Soret,New minimal surfaces inS3 desingularizing the Clifford tori, Math. Ann.364 (2016), no. 3-4, 763–776. 20 M. B. Schulz, D. Wiygul A new family of minimal surfaces of even genus in the three-dimensional sphere

    2016

  10. [16]

    C. J. Costa,Uniqueness of minimal surfaces embedded inR3, with total curvature12π, J. Differential Geom.30 (1989), no. 3, 597 –618

    1989

  11. [17]

    Dorfmeister, F

    J. Dorfmeister, F. Pedit, and H. Wu,Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom.6 (1998), no. 4, 633–668

    1998

  12. [18]

    Ejiri and M

    N. Ejiri and M. Micallef,Comparison between second variation of area and second variation of energy of a minimal surface, Adv. Calc. Var.1 (2008), no. 3, 223–239

    2008

  13. [19]

    Frankel,On the fundamental group of a compact minimal submanifold, Ann

    T. Frankel,On the fundamental group of a compact minimal submanifold, Ann. of Math. (2)83 (1966), 68–73

    1966

  14. [20]

    Franz,Equivariant index bound for min-max free boundary minimal surfaces, Calc

    G. Franz,Equivariant index bound for min-max free boundary minimal surfaces, Calc. Var. Partial Differential Equations 62 (2023), no. 7, Paper No. 201, 28

    2023

  15. [22]

    Franz and M

    G. Franz and M. B. Schulz,Toplogical control for free boundary minimal surfaces, preprint, available atarXiv: 2307.00941

  16. [23]

    Fusco,The quantitative isoperimetric inequality and related topics, Bull

    N. Fusco,The quantitative isoperimetric inequality and related topics, Bull. Math. Sci.5 (2015), no. 3, 517–607

    2015

  17. [24]

    Heller, S

    L. Heller, S. Heller, and M. Traizet,Area estimates for high genus Lawson surfaces via DPW, J. Differential Geom. 124 (2023), no. 1, 1–35

    2023

  18. [25]

    Hoffman and W

    D. Hoffman and W. H. Meeks III,Embedded minimal surfaces of finite topology, Ann. of Math. (2)131 (1990), no. 1, 1–34

    1990

  19. [26]

    Kapouleas,Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I, J

    N. Kapouleas,Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I, J. Differential Geom. 106 (2017), 393–449

    2017

  20. [27]

    Kapouleas and P

    N. Kapouleas and P. McGrath,New minimal surface doublings of the Clifford torus and contributions to questions of Yau, preprint, available atarXiv:2411.00613

    arXiv

  21. [28]

    Pure Appl

    , Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, II, Comm. Pure Appl. Math. 72 (2019), no. 10, 2121–2195

    2019

  22. [29]

    , Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self- shrinkers, Camb. J. Math.11 (2023), no. 2, 299–439

    2023

  23. [30]

    Kapouleas and D

    N. Kapouleas and D. Wiygul,New low-genus desingularizations of three Clifford tori and related characterizations, preprint, available atarXiv:2306.09159v2

    arXiv

  24. [31]

    , The index and nullity of the Lawson surfacesξg,1, Camb. J. Math.8 (2020), no. 2, 363–405

    2020

  25. [32]

    Reine Angew

    , The Lawson surfaces are determined by their symmetries and topology, J. Reine Angew. Math.2022 (2022), no. 786, 155–173

    2022

  26. [33]

    Ann.383 (2022), 119–170

    , Minimal surfaces in the three-sphere by desingularizing intersecting Clifford tori, Math. Ann.383 (2022), 119–170

    2022

  27. [34]

    Kapouleas and S.-D

    N. Kapouleas and S.-D. Yang,Minimal surfaces in the three-sphere by doubling the Clifford torus, Am. J. Math. 132 (2010), 257–295

    2010

  28. [35]

    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

    1988

  29. [36]

    Karcher, U

    H. Karcher, U. Pinkall, and I. Sterling,New minimal surfaces inS3, J. Differential Geom.28 (1988), 169–185

    1988

  30. [37]

    Karpukhin, R

    M. Karpukhin, R. Kusner, P. McGrath, and D. Stern,Embedded minimal surfaces inS3 and B3 via equivariant eigenvalue optimization, preprint, available atarXiv:2402.13121

    arXiv

  31. [38]

    Ketover,Equivariant min-max theory, preprint, available atarXiv:1612.08692

    D. Ketover,Equivariant min-max theory, preprint, available atarXiv:1612.08692

    arXiv

  32. [39]

    , Flipping Heegaard splittings and minimal surfaces, preprint, available atarXiv:2211.03745

    arXiv

  33. [41]

    Differential Geom.112 (2019), no

    , Genus bounds for min-max minimal surfaces, J. Differential Geom.112 (2019), no. 3, 555–590

    2019

  34. [42]

    Ketover, F

    D. Ketover, F. C. Marques, and A. Neves,The catenoid estimate and its geometric applications, J. Differential Geom. 115 (2020), no. 1, 1–26

    2020

  35. [43]

    Kusner, Y

    R. Kusner, Y. Lü, and P. Wang,The Willmore problem for surfaces with symmetry, preprint, available at arXiv:2410.12582. 21 M. B. Schulz, D. Wiygul A new family of minimal surfaces of even genus in the three-dimensional sphere

  36. [44]

    H. B. Lawson Jr.,Complete minimal surfaces inS3, Ann. of Math. (2)92 (1970), 335–374

    1970

  37. [45]

    Math.11 (1970), 183–187

    , The unknottedness of minimal embeddings, Invent. Math.11 (1970), 183–187

    1970

  38. [46]

    Li and S

    P. Li and S. T. Yau,A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math.69 (1982), no. 2, 269–291

    1982

  39. [47]

    F. C. Marques and A. Neves,Min-Max theory and the Willmore conjecture, Ann. of Math. (2)179 (2014), no. 2, 683–782

    2014

  40. [48]

    Montiel and A

    S. Montiel and A. Ros,Schrödinger operators associated to a holomorphic map, Global Differential Geometry and Global Analysis (Berlin, 1990), 1991, pp. 147–174

    1990

  41. [49]

    Morabito,Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces, Indiana Univ

    F. Morabito,Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces, Indiana Univ. Math. J. 58 (2009), no. 2, 677–707

    2009

  42. [50]

    Nayatani,Morse index and Gauss maps of complete minimal surfaces in Euclidean3-space, Comment

    S. Nayatani,Morse index and Gauss maps of complete minimal surfaces in Euclidean3-space, Comment. Math. Helv. 68 (1993), no. 4, 511–537

    1993

  43. [51]

    J. T. Pitts and J. H. Rubinstein,Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.)19 (1988), no. 1, 303–309

    1988

  44. [52]

    Savo,Index bounds for minimal hypersurfaces of the sphere, Indiana Univ

    A. Savo,Index bounds for minimal hypersurfaces of the sphere, Indiana Univ. Math. J.59 (2010), no. 3, 823–837

    2010

  45. [53]

    H. F. Scherk,Bemerkungen über die kleinste Fläche innherhalb gegebener Grenzen, J. Reine Angew. Math.13 (1835), 185–208

  46. [54]

    W. P. Thurston, The geometry and topology of three-manifolds. Vol. IV, American Mathematical Society, Providence, RI, 2022

    2022

  47. [55]

    Urbano,Minimal surfaces with low index in the three-dimensional sphere, Proc

    F. Urbano,Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc.108 (1990), no. 4, 989–992

    1990

  48. [56]

    Wiygul,Minimal surfaces in the 3-sphere by stacking Clifford tori, J

    D. Wiygul,Minimal surfaces in the 3-sphere by stacking Clifford tori, J. Differential Geom.114 (2020), no. 3, 467–549. 30 July 2025 Mario B. Schulz Università di Trento, Dipartimento di Matematica, via Sommarive 14, 38123 Povo, Italy E-mail address: mario.schulz@unitn.it David...

    2020

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.