Pith. sign in

REVIEW 2 major objections 6 minor 44 references

Every Riemannian 3-sphere contains at least two distinct embedded minimal spheres.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-10 05:03 UTC pith:XKAMOSYI

load-bearing objection First general-metric existence of a second embedded minimal sphere in S^{3}, via a carefully engineered iterative relative min-max that closes the uniqueness contradiction. the 2 major comments →

arxiv 2607.08631 v1 pith:XKAMOSYI submitted 2026-07-09 math.DG math.APmath.GT

Existence of two embedded minimal spheres in S³ with an arbitrary metric

Zhichao Wang , Xin Zhou This is my paper
classification math.DG math.APmath.GT MSC 53A1053C4249Q05
keywords embedded minimal spheresSimon-Smith min-maxrelative min-maxmean-convex foliationmultiplicityLusternik-Schnirelmann3-sphere
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

For more than forty years it has been known that any Riemannian metric on the three-sphere produces at least one embedded minimal two-sphere. This paper proves that there are always at least two. The argument assumes for contradiction that a single minimal sphere is unique, extends it to a mean-convex foliation of the whole three-sphere, and then builds multi-parameter families of (piecewise-smooth) spheres that approximate multiple copies of that foliation. Relative min-max theory applied to these families produces a sequence of critical values that must equal successive integer multiples of the area of the unique sphere; for a large enough number of parameters the constructed families have strictly smaller area, yielding a contradiction. The result removes the positive-Ricci or bumpy-metric hypotheses that earlier multiplicity theorems required, so the existence of a second minimal sphere is now unconditional.

Core claim

Every Riemannian three-sphere admits at least two distinct embedded minimal spheres. Under the assumption that only one such sphere exists, an iterative scheme of relative Simon–Smith min-max constructions on carefully built multi-parameter families produces min-max widths equal to successive multiples of that sphere’s area; for sufficiently many parameters the families themselves have strictly smaller total area, a contradiction.

What carries the argument

An iterative scheme of relative min-max constructions on continuous families of (smoothed) spheres over simplices, built by inserting necks of controlled radii between successive leaves of a mean-convex foliation; the resulting area upper bounds and Lusternik–Schnirelmann-type comparisons force the min-max values to equal integer multiples of the unique sphere’s area until a global contradiction appears.

Load-bearing premise

If a unique embedded minimal sphere exists, it must admit an expanding neighborhood on both sides and therefore extend to a mean-convex foliation of the whole three-sphere; every subsequent construction of the families and all area comparisons rest on that foliation.

What would settle it

An explicit Riemannian metric on the three-sphere whose only embedded minimal sphere is unique (or whose second-width min-max produces only a multiple of that sphere) would refute the claim; equivalently, a gap or non-existence in the mean-convex foliation when uniqueness is assumed would collapse the argument.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proves that every Riemannian 3-sphere (S³,g) contains at least two distinct embedded minimal 2-spheres (Theorem A / Theorem 6.3). Assuming uniqueness of a Simon–Smith sphere Σ₀, the authors reduce to the expanding-neighborhood case, obtain a mean-convex foliation {Υ_t}, and construct multi-parameter families Φ_Λ : Δ_k → E of piecewise-smooth spheres by inserting necks of radii controlled by a decay parameter Λ. After smoothing (Appendix A) and an interpolation that preserves area upper bounds, they run an iterative scheme of relative Simon–Smith min-max constructions on faces F_{1,m} of a large simplex. Under uniqueness the relative widths equal m H²(Σ₀), but for large k the constructed family has area strictly less than k H²(Σ₀), a contradiction. The argument relies on relative min-max theorems (Theorems 2.2–2.3), area estimates for necks (Lemmas 3.5–3.6, Proposition 3.7, Lemmas 4.1–4.2), a Lusternik–Schnirelmann-type topological lemma (Lemma 6.1), and a varifold-neighborhood lemma for Caccioppoli sets (Proposition 2.1).

Significance. This is a substantial advance on Yau’s problem of four embedded minimal spheres in S³. Simon–Smith (1982) produced one sphere for arbitrary metrics; subsequent work obtained two or four only under positive Ricci curvature or bumpiness (White, Haslhofer–Ketover, and the authors’ multiplicity-one theorem). The present paper is the first to construct a second embedded minimal sphere for completely general metrics, more than forty years after the first. The iterative relative min-max scheme and the carefully controlled neck construction are new technical contributions that may apply to other problems previously restricted to positive Ricci curvature. The logical skeleton is complete and the estimates are quantitative; the result is therefore of clear interest to geometric analysis and min-max theory.

major comments (2)
  1. [Section 3 / §1.1] Section 3 (opening paragraphs) and §1.1: The entire construction rests on the existence of a mean-convex foliation {Υ_t} with Υ(1/2)=Σ₀. Under uniqueness the authors invoke weak stability from [40, Thm 8.9] and dispose of the contracting-neighborhood case by Song’s cutting argument, then cite Haslhofer–Ketover [17, Thm 3.1]. These reductions should be stated as a single, self-contained lemma at the start of Section 3 (with precise references), so that a reader can see exactly which external results are used before the families Φ_Λ are defined.
  2. [Appendix A / Remark A.5 / §5] Appendix A, Remark A.5 and the passage from Φ'_Λ to ˜Φ_Λ in §5: Proposition A.4 requires a uniform lower bound θ>0 on dihedral angles along the singular circles. Remark A.5 asserts this from transversality of the deformation retract D to the leaves Υ(t). For the interpolated family H(·,t) of Lemma 4.3 and for all Λ≥Λ₂ on the retracted domain Δ^ν_{k₀}, a short quantitative argument (or a compactness citation) confirming that θ can be chosen independently of Λ and of the interpolation parameter would make the application of the mollification fully rigorous.
minor comments (6)
  1. [§2.1] Notation for the spaces E and E (with overline) is introduced in §2.1 but used heavily thereafter; a brief reminder when Φ_Λ lands in E versus E would help the reader.
  2. [Figure I] Figure I is helpful but the caption could explicitly label which pieces are the interior annuli A_i and which are the necks C_i, matching the notation of (3.3)–(3.4).
  3. [Lemma 3.6] In Lemma 3.6 the equality case is stated only for s=t=1/2 and σ=0; a parenthetical remark that this case never occurs for the actual necks of Φ_Λ (where ξ_i>0 on the interior of Δ_k) would clarify that all subsequent inequalities used in Proposition 3.7 and Lemma 4.1 are strict when needed.
  4. [Proposition 2.1] Proposition 2.1: the constant θ₀ is independent of k, which is used crucially in the induction of Proposition 6.2; this independence is proved but could be highlighted in the statement.
  5. [Introduction] Several recent preprints that rely on multiplicity-one under positive Ricci (e.g., the works cited in the introduction) are listed; a sentence on which of those arguments might extend via the present methods would be of interest to readers, though not required.
  6. [Abstract / throughout] Typographical: “an iterat ive scheme” (abstract) and occasional spacing issues around math mode (e.g., “k H2”) should be cleaned in production.

Circularity Check

1 steps flagged

No significant circularity: the iterative relative min-max construction and area estimates are self-contained; prior self-citations supply independent black-box inputs under stronger hypotheses.

specific steps
  1. self citation load bearing [§3 opening paragraph / §1.1]
    "if Σ₀ is the only embedded minimal 2-sphere in (S³, g), then by [40, Theorem 8.9] we know that Σ₀ is weakly stable; that is, the first eigenvalue λ₁(L_Σ₀) of the associated Jacobi operator L_Σ₀ vanishes … Moreover, by our previous work [40], Σ₀ has an expanding neighborhood."

    The uniqueness assumption is converted into weak stability and the expanding-neighborhood property solely by citation of the authors’ own earlier paper. The citation is not circular in the strong sense (the prior theorem is an independent statement proved under stronger curvature/bumpiness hypotheses), but it is the sole justification for the foliation that the entire subsequent construction rests upon; hence a minor self-citation flag.

full rationale

The paper proceeds by contradiction under the assumption of a unique embedded minimal sphere Σ₀. It invokes the authors’ prior multiplicity-one/stability result [40, Thm 8.9] only to conclude weak stability (and hence an expanding neighborhood after the contracting case is cut by Song’s argument), then imports the mean-convex foliation from the external reference Haslhofer–Ketover [17, Thm 3.1]. From that foliation the authors construct, by hand, the multi-parameter families Φ_Λ (Def. 3.2), prove the neck-area comparisons (Lems 3.5–3.6, Prop. 3.7) and the strict area upper bounds on the simplex and its faces (Lems 4.1–4.2), smooth them (App. A), and run an inductive relative Simon–Smith argument (Prop. 6.2 + topological Lem. 6.1) that forces the min-max widths to equal m·H²(Σ₀) while the large-k upper bound is strictly smaller—a genuine contradiction. None of these estimates or the topological step reduces algebraically to the cited prior theorems; the latter are used as ordinary external lemmas. There are no fitted parameters, no self-definitional identities, and no renaming of known empirical patterns. The single minor self-citation of [40] for stability under uniqueness is therefore non-load-bearing for the new construction and does not raise the score above 1.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 2 invented entities

The proof rests on classical GMT and min-max foundations plus two geometric facts about a unique minimal sphere (expanding neighborhood + mean-convex foliation). Free parameters are purely existential (large Λ, large k_{0}, small η) chosen to close inequalities; no data fitting occurs. Invented objects are the multi-parameter families Φ_Λ and the relative widths W_k, which are constructed explicitly rather than postulated.

free parameters (3)
  • Λ (neck-radius decay rate)
    Chosen larger than a constant Λ_{0} depending only on the unique sphere and a fixed η; controls how fast necks open near the boundary of the simplex.
  • k_{0} (simplex dimension)
    Taken large enough so that every point of Δ_{k_{0}} forces at least one short neck of small area (Lemma 4.2).
  • η (area slack)
    Arbitrary small positive constant used to absorb neck-area errors; final contradiction is independent of its precise value.
axioms (5)
  • domain assumption Simon–Smith min-max theorem produces at least one embedded minimal sphere
    Invoked at the outset; the whole argument begins from the existence of Σ_{0}.
  • domain assumption If Σ_{0} is unique then it is weakly stable and admits an expanding neighborhood, hence a mean-convex foliation {Υ_t}
    Cited from [40, Thm 8.9] and [17, Thm 3.1]; used to build the model sweepouts.
  • standard math Smale conjecture (Diff(S^{3}) ≃ O(4))
    Used to identify the space of embedded spheres with RP^{3}, limiting ordinary sweepouts to dimension ≤3.
  • domain assumption Relative Simon–Smith min-max theorems (Theorems 2.2–2.3)
    Proved in the paper by adapting [40]; supply the lower bounds for the widths W_k.
  • standard math Regularity of almost-minimizing varifolds in 3-manifolds
    Classical (Pitts, Schoen–Simon, Colding–De Lellis); used to conclude that critical varifolds are smooth embedded spheres.
invented entities (2)
  • Multi-parameter family Φ_Λ : Δ_k → E of piecewise-smooth spheres with controlled necks no independent evidence
    purpose: Approximate the Gromov–Almgren–Pitts sweepouts by multiples of Σ_{0} while keeping area strictly below k Area(Σ_{0}) for large k
    Explicitly constructed in Definition 3.2 via the recursive radii ξ_i = e^{-Λ d_i}(1-ξ_{i-1}); not an external postulate.
  • Relative min-max widths W_k = L([Φ̃_Λ | ∂F_{1,k}; rel ∂F_{1,k}]) no independent evidence
    purpose: Produce the sequence of critical values that must equal k Area(Σ_{0}) under uniqueness, contradicting the strict area upper bound
    Defined via the relative homotopy classes in Section 6; their evaluation is the content of Proposition 6.2.

pith-pipeline@v1.1.0-grok45 · 36170 in / 2889 out tokens · 32357 ms · 2026-07-10T05:03:00.824639+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric." pith.science (2026). https://pith.science/paper/XKAMOSYI

@misc{pith2026260708631,
  author       = {Pith},
  title        = {Pith review of: Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKAMOSYI}},
  note         = {Machine review of arXiv:2607.08631}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.

Review history (2 revisions) →

discussion (0)

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

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 8 internal anchors

  1. [1]

    MR0146835

    Frederick Justin Almgren Jr., The homotopy groups of the integral cycle groups , Topology 1 (1962), 257–299. MR0146835

    work page 1962

  2. [2]

    , The theory of varifolds , Mimeographed notes (1965)

    work page 1965

  3. [3]

    Rigidity theorems for the area widths of Riemannian manifolds

    Lucas Ambrozio, Fernando C. Marques, and Andr´ e Neves, Rigidity theorems for the area widths of Riemann- ian manifolds (2024), available at arXiv:math/2408.14375

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  4. [4]

    Ricci flow and contractibility of spaces of metrics

    Richard H. Bamler and Bruce Kleiner, Ricci flow and contractibility of spaces of metrics (2019), available at arXiv:math/1909.08710

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  5. [5]

    Hyeong In Choi and Richard Schoen, The space of minimal embeddings of a surface into a three-dim ensional manifold of positive Ricci curvature , Invent. Math. 81 (1985), no. 3, 387–394. MR807063

    work page 1985

  6. [6]

    Adrian Chun-Pong Chu, Minimal surfaces with arbitrary genus in 3-spheres of posit ive ricci curvature (2025), available at arXiv:math/2508.06019

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    Adrian Chun-Pong Chu and Yangyang Li, Existence of 5 minimal tori in 3-spheres of positive ricci cu rvature (2025), available at arXiv:math/2409.09315

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  8. [8]

    Adrian Chun-Pong Chu, Yangyang Li, and Zhihan Wang, An enumerative min-max theorem for minimal surfaces (2026), available at arXiv:math/2601.01736

    work page arXiv 2026

  9. [9]

    , Min-max theory and minimal surfaces with prescribed genus (2026), available at arXiv:math/2507.23239

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    Colding and Camillo De Lellis, The min-max construction of minimal surfaces , Surveys in differ- ential geometry, Vol

    Tobias H. Colding and Camillo De Lellis, The min-max construction of minimal surfaces , Surveys in differ- ential geometry, Vol. VIII (Boston, MA, 2002), 2003, pp. 75– 107. MR2039986

    work page 2002

  11. [11]

    Colding and William P

    Tobias H. Colding and William P. Minicozzi II, Width and finite extinction time of Ricci flow , Geom. Topol. 12 (2008), no. 5, 2537–2586. MR2460871

    work page 2008

  12. [12]

    Salim Deaibes, Minimal 2-spheres and optimal foliations in 3-spheres with arbitrary metric (2021), available at arXiv:math/2001.07695

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  13. [13]

    MR02573 25

    Herbert Federer, Geometric measure theory , Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR02573 25

    work page 1969

  14. [14]

    Renato Ghini Bettiol and Paolo Piccione, Nonplanar minimal spheres in ellipsoids of revolution , Ann. Sc. Norm. Super. Pisa Cl. Sci (May 2024), 32

    work page 2024

  15. [15]

    80, Birkh¨ auser Verlag, Basel, 1984

    Enrico Giusti, Minimal surfaces and functions of bounded variation , Monographs in Mathematics, vol. 80, Birkh¨ auser Verlag, Basel, 1984. MR775682

    work page 1984

  16. [16]

    Mikhael Gromov, Dimension, nonlinear spectra and width, Geometric aspects of functional analysis (1986/87), 1988, pp. 132–184. MR950979

    work page 1986

  17. [17]

    Robert Haslhofer and Daniel Ketover, Minimal 2-spheres in 3-spheres , Duke Math. J. 168 (2019), no. 10, 1929–1975. MR3983295

    work page 2019

  18. [18]

    Hatcher, A proof of the Smale conjecture, Diff( S3) ≃ O(4), Ann

    Allen E. Hatcher, A proof of the Smale conjecture, Diff( S3) ≃ O(4), Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR701256

    work page 1983

  19. [19]

    MR1867354

    , Algebraic topology, Cambridge University Press, Cambridge, 2002. MR1867354

    work page 2002

  20. [20]

    Marques, and Andr´ e Neves, Density of minimal hypersurfaces for generic metrics , Ann

    Kei Irie, Fernando C. Marques, and Andr´ e Neves, Density of minimal hypersurfaces for generic metrics , Ann. of Math. (2) 187 (2018), no. 3, 963–972. MR3779962

    work page 2018

  21. [21]

    Daniel Ketover and Yevgeny Liokumovich, The Smale conjecture and min-max theory , Invent. Math. 241 (2025), no. 1, 1–34. MR4921571

    work page 2025

  22. [22]

    Marques, and Andr´ e Neves, The catenoid estimate and its geometric applica- tions, J

    Daniel Ketover, Fernando C. Marques, and Andr´ e Neves, The catenoid estimate and its geometric applica- tions, J. Differential Geom. 115 (2020), no. 1, 1–26. MR4081930

    work page 2020

  23. [23]

    Reine Angew

    Xingzhe Li, Tongrui Wang, and Xuan Yao, Minimal surfaces with low genus in lens spaces , J. Reine Angew. Math. 828 (2025), 175–218. MR4979239

    work page 2025

  24. [24]

    Xingzhe Li and Zhichao Wang, Existence of embedded minimal tori in three-spheres with po sitive Ricci cur- vature (2024), available at arXiv:math/2409.10391

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  25. [25]

    Marques, and Andr´ e N eves, Weyl law for the volume spectrum , Ann

    Yevgeny Liokumovich, Fernando C. Marques, and Andr´ e N eves, Weyl law for the volume spectrum , Ann. of Math. (2) 187 (2018), no. 3, 933–961. MR3779961

    work page 2018

  26. [26]

    Marques and Andr´ e Neves, Min-max theory and the Willmore conjecture , Ann

    Fernando C. Marques and Andr´ e Neves, Min-max theory and the Willmore conjecture , Ann. of Math. (2) 179 (2014), no. 2, 683–782. MR3152944

    work page 2014

  27. [27]

    , Existence of infinitely many minimal hypersurfaces in posit ive Ricci curvature , Invent. Math. 209 (2017), no. 2, 577–616. MR3674223

    work page 2017

  28. [28]

    , Morse index of multiplicity one min-max minimal hypersurfa ces, Adv. Math. 378 (2021), Paper No. 107527, 58. MR4191255 32 ZHICHAO W ANG AND XIN ZHOU

    work page 2021

  29. [29]

    Marques, Andr´ e Neves, and Antoine Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent

    Fernando C. Marques, Andr´ e Neves, and Antoine Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math. 216 (2019), no. 2, 421–443. MR3953507

    work page 2019

  30. [30]

    Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol

    Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; Univ ersity of Tokyo Press, Tokyo, 1981. MR626027

    work page 1981

  31. [31]

    348, Birkh¨ auser/Springer, Cham, 2023

    Manuel Ritor´ e, Isoperimetric inequalities in riemannian manifolds , Progress in Mathematics, vol. 348, Birkh¨ auser/Springer, Cham, 2023. MR4676392

    work page 2023

  32. [32]

    Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann

    Jonathan Sacks and Karen K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR604040

    work page 1981

  33. [33]

    Lorenzo Sarnataro and Douglas Stryker, Optimal regularity for minimizers of the prescribed mean cu rvature functional over isotopies , Camb. J. Math. 13 (2025), no. 3, 609–706. MR4933237

    work page 2025

  34. [34]

    Pure Appl

    Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces , Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR634285 (82k:49054)

    work page 1981

  35. [35]

    3, Australian Nation al University, Centre for Mathematical Analysis, Canberra, 1983

    Leon Simon, Lectures on geometric measure theory , Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian Nation al University, Centre for Mathematical Analysis, Canberra, 1983. MR756417 (87a:49001)

    work page 1983

  36. [36]

    Smith, On the existence of embedded minimal 2-spheres in the 3-sphe re, endowed with an arbitrary Riemannian metric , Ph.D

    Francis R. Smith, On the existence of embedded minimal 2-spheres in the 3-sphe re, endowed with an arbitrary Riemannian metric , Ph.D. Thesis, 1982

    work page 1982

  37. [37]

    Antoine Song, Existence of infinitely many minimal hypersurfaces in close d manifolds , Ann. of Math. (2) 197 (2023), no. 3, 859–895. MR4564260

    work page 2023

  38. [38]

    Tongrui Wang, Zhichao Wang, and Xin Zhou, Equivariant min-max theory and the spherical bernstein pro blem in S4 (2026), available at arXiv:math/2602.03984

    work page arXiv 2026

  39. [39]

    Improved $C^{1,1}$ regularity for multiple membranes problem

    Zhichao Wang and Xin Zhou, Improved C 1, 1 regularity for multiple membranes problem (2023), available at arXiv:math/2308.00172. [40] , Existence of four minimal spheres in S3 with a bumpy metric (2024), available at arXiv:math/2305.08755

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  40. [40]

    Differential Geom

    , Min-max minimal hypersurfaces with higher multiplicity , J. Differential Geom. 129 (2025), no. 1, 225–260. MR4856595

    work page 2025

  41. [41]

    Brian White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals , Invent. Math. 88 (1987), no. 2, 243–256. MR880951

    work page 1987

  42. [42]

    , The space of minimal submanifolds for varying Riemannian me trics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR1101226 (92i:58028)

    work page 1991

  43. [43]

    Shing-Tung Yau, Problem section, Seminar on Differential Geometry, 1982, pp. 669–706. MR645 762

    work page 1982

  44. [44]

    Xin Zhou, On the Multiplicity One Conjecture in min-max theory , Ann. of Math. (2) 192 (2020), no. 3, 767–820. MR4172621 Shanghai Center for Mathematical Science, 2005 Songhu Road , Fudan University, Shanghai, 200438, China Email address : zhichao@fudan.edu.cn Department of Mathematics, 531 Malott Hall, Cornell Univer sity, Ithaca, NY 14853, USA Email add...

    work page 2020