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arxiv: 1907.09122 · v1 · pith:KKGM6QP4new · submitted 2019-07-22 · 🧮 math.DG

Elliptic special Weingarten surfaces of minimal type in mathbb{R}³ of finite total curvature

Jos\'e M. Espinar , H\'eber Mesa This is my paper

Pith reviewed 2026-05-24 18:19 UTC · model grok-4.3

classification 🧮 math.DG
keywords elliptic special Weingarten surfacesfinite total curvaturerotational symmetryspecial catenoidsJorge-Meeks formulaminimal surfaces
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The pith

A complete two-ended elliptic special Weingarten surface of minimal type with finite total curvature in three-space is rotationally symmetric.

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

The paper extends the theory of complete minimal surfaces of finite total curvature to elliptic special Weingarten surfaces of minimal type. It extends the Jorge-Meeks formula relating total curvature to topology and uses this to show that planes are the only such surfaces with total curvature less than 4π. It proves that any complete connected surface in this class that is embedded outside a compact set, has finite total curvature, and two ends must be rotationally symmetric, hence a rotational special catenoid. This resolves a question from 1993. It also shows that special catenoids are the only non-flat surfaces in the class with total curvature less than 8π.

Core claim

A complete connected elliptic special Weingarten surface of minimal type in R^3 that is embedded outside a compact set, has finite total curvature, and two ends is rotationally symmetric and therefore one of the rotational special catenoids. The authors extend the Jorge-Meeks formula to this class of surfaces and classify planes as the only ones with total curvature less than 4π.

What carries the argument

The elliptic special Weingarten condition of minimal type, which is a relation between the principal curvatures that is elliptic and allows extension of minimal surface techniques like the Jorge-Meeks formula.

If this is right

  • The Jorge-Meeks formula holds for these surfaces, relating total curvature to the number of ends and genus.
  • Planes are the unique elliptic special Weingarten surfaces of minimal type with total curvature less than 4π.
  • Special catenoids are the only connected non-flat special Weingarten surfaces with total curvature less than 8π.

Where Pith is reading between the lines

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

  • If the rotational symmetry holds, it may be possible to classify surfaces with more than two ends under similar conditions.
  • The extension of the Jorge-Meeks formula could apply to other related classes of surfaces beyond minimal ones.

Load-bearing premise

The surface satisfies the elliptic special Weingarten relation of minimal type between its principal curvatures.

What would settle it

The existence of a non-rotationally symmetric complete connected elliptic special Weingarten surface of minimal type with two ends, finite total curvature, and embedded outside a compact set would falsify the classification.

Figures

Figures reproduced from arXiv: 1907.09122 by H\'eber Mesa, Jos\'e M. Espinar.

Figure 1
Figure 1. Figure 1: Generatrix (left) and complete ESWMT-surface [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Generatrixes of rotational ESWMT-surfaces. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Parametrizing the end E with a conformal parameter. The first claim about the height function u of an end E is that, after a vertical translation of the end E, either u ≥ 0 or u ≤ 0. This means that the end E cannot grow infinitely in both directions ν and −ν simultaneously. Specifically, we are going to prove the follow lemma. Lemma 4 (Growth lemma). Let E be an embedded end of an immersed ESWMT￾surface Σ… view at source ↗
Figure 4
Figure 4. Figure 4: The Tchebychev theorem. The next lemma is due to S. Bernstein and it was used by him to prove the famous Bernstein’s theorem about entire minimal graphs. Lemma 7 (Bernstein, [6]). Let φ be of class C 2 in a connected open set Ω and let φxxφyy − φ 2 xy ≤ 0 in Ω. Assume φ > 0 in Ω and φ = 0 on ∂Ω. Suppose that Ω can be placed in a sector of angle less than π. Let M(r) be the maximum of φ on the part of the c… view at source ↗
Figure 5
Figure 5. Figure 5: The set ∂G in the first quadrant. We must observe that ∂G is symmetric with respect the x-axis and y-axis. Therefore, ∂G is completely determined by the points in the curve solution to |y| = r(r) q0 contained in the first quadrant, see [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The set G. We have obtained that the set G is bounded by two curves L + and L −. The curve L − is the union of the curves solution to |y| = r(r) q0 in the first and the second quadrants and the arc of ∂B(¯r) (in these quadrants) joining these curves. This arc can be described as the set of points (x, y) ∈ ∂B(¯r) such that y ≥ r¯ (¯r) q0 , see the left side of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Assuming that Ω+ is empty [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Decreasing the inclination of the plane. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Assuming that Ω+ 1 = Ω+ 2 . Now we will consider the components of the sets  (x, y) ∈ R 2 \ B(r0) : ψ(x, y) > 0 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The set Ω± i placed in an angle sector. Summarizing, we have proven that ∂G ∩ ∂Ω ± i = {−∞, +∞} for all sets Ω ± i ; therefore Lemma 8 guaranties the existence of the Jordan curves J ± i . This proves Claim E. To finish the proof of the growth lemma, consider for i = 1, 2 a curve in G that joins +∞ and β + i , also consider the segment inside B(r0) joining β + 1 and β + 2 . The closed curve formed by thes… view at source ↗
Figure 11
Figure 11. Figure 11: The point β − 2 cannot be connected to −∞ by the curve J − 2 . As a consequence of the growth lemma we have an obstruction for non￾planar one ended ESWMT-surfaces with finite total curvature in R 3 . This is an evidence about the premise that the theory of ESWMT-surfaces is quite similar to the theory of minimal surfaces, even more if we note that there is not such obstruction for harmonic surfaces of fin… view at source ↗
Figure 12
Figure 12. Figure 12: Representation of an end E of an ESWMT-surface satisfying the hypothesis of Theorem 12. Lemma 11. Let Σ ⊂ R 3 be a complete ESWMT-surface satisfying (1) of finite total curvature embedded outside a compact set and two ends. Then, the ends of Σ are parallel and the surface must grow infinitely in opposite directions. We would like to finish this section by noting that the equality (31) can be seen as a gen… view at source ↗
Figure 13
Figure 13. Figure 13: The Alexandrov function Λ1 is valued as −∞ outside of a compact set. Proof Claim A. If Λ(ρ) ≤ Λ(0) for all ρ > 0, then for an arbitrary t > Λ(0) we have that Λ(ρ) ≤ t for all ρ > 0. By the definition of the Alexandrov function, we infer that E∗ t+ ∩ {p ∈ R 3 : h(p) > ρ} ⊂ W¯ . If E∗ t+ ∩ {p ∈ R 3 : h(p) > ρ} ⊂ W¯ for all t > Λ(0), we will show that Λ(ρ) ≤ Λ(0) for all ρ > 0 reasoning by contradiction. Sup… view at source ↗
Figure 14
Figure 14. Figure 14: Λε 1 attains its maximum value t at q ∈ Dε ∩ Ω such that h ε (q) > hε 0 . Writing z ε for the maximum value of Λε 1 , it follows that (E ε 0 ) ∗ t+ ⊂ W¯ ε 0 for all t ≥ z ε . (42) By letting ε → 0, from (42) we get E ∗ t+ ⊂ W¯ for all t ≥ lim sup ε→0 z ε . (43) The semicontinuity of the Alexandrov function, Lemma 13, and Claim B imply that lim sup ε→0 z ε ≤ Λ(0). Given t ≥ Λ(0), then t ≥ lim sup ε→0 z ε b… view at source ↗
Figure 15
Figure 15. Figure 15: The associated Alexandrov function, Λ, on Σ. [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗
read the original abstract

We extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular, we extend the seminal works of L. Jorge and W. Meeks and R. Schoen. Specifically, we extend the Jorge-Meeks formula relating the total curvature and the topology of the surface and we use it to classify planes as the only elliptic special Weingarten surfaces whose total curvature is less than $4 \pi$. Moreover, we show that a complete (connected), embedded outside a compact set, elliptic special Weingarten surface of minimal type in $\mathbb{R}^3$ of finite total curvature and two ends is rotationally symmetric; in particular, it must be one of the rotational special catenoids described by R. Sa Earp and E. Toubiana. This answers in the positive a question posed in 1993 by R. Sa Earp. We also prove that the special catenoids are the only connected non-flat special Weingarten surfaces whose total curvature is less than $8 \pi$.

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 extends the Jorge-Meeks formula relating total curvature to topology from minimal surfaces to the class of elliptic special Weingarten surfaces of minimal type in R^3. It classifies planes as the only such surfaces with total curvature less than 4π. It proves that any complete connected elliptic special Weingarten surface of minimal type with finite total curvature, embedded outside a compact set, and having exactly two ends must be rotationally symmetric and hence one of the rotational special catenoids of Sa Earp-Toubiana, answering a 1993 question of Sa Earp in the affirmative. It further shows that the special catenoids are the only connected non-flat special Weingarten surfaces with total curvature less than 8π.

Significance. If the extensions of the Jorge-Meeks formula and the symmetry arguments hold, the results generalize classical classification theorems for minimal surfaces of finite total curvature to a broader elliptic Weingarten class while preserving the key analytic properties (maximum principle, Gauss-map control) needed for the arguments. The positive resolution of Sa Earp's question and the sharp total-curvature bound of 8π constitute concrete advances in the geometry of Weingarten surfaces.

minor comments (1)
  1. The abstract and introduction state the main theorems clearly, but the provided materials do not include the full proofs of the extended Jorge-Meeks formula or the symmetry classification, preventing direct verification of the analytic estimates invoked for ellipticity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary of our results and the positive evaluation of their significance. No specific major comments appear in the report, so we have no individual points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; extends external theorems

full rationale

The derivation extends the Jorge-Meeks formula and Schoen's results to the elliptic special Weingarten class using the ellipticity assumption for analytic properties like the maximum principle. Central claims (rotational symmetry for two-ended surfaces, classification of planes and special catenoids) rest on these extensions plus the external Sa Earp-Toubiana rotational examples and the 1993 Sa Earp question, none of which reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The paper is self-contained against external benchmarks with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper extends existing theory without introducing new free parameters or invented entities; it relies on standard axioms from differential geometry and the definition of the surface class.

axioms (2)
  • standard math Gauss-Bonnet theorem and properties of the Gauss map for surfaces in R^3
    Used to relate total curvature to topology, extending the original Jorge-Meeks argument.
  • domain assumption Elliptic condition on the Weingarten relation for surfaces of minimal type
    Defines the class of surfaces to which the extended formula and classifications apply.

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Works this paper leans on

53 extracted references · 53 canonical work pages · 2 internal anchors

  1. [1]

    Achieser

    N. Achieser. Theory of approximation. Dover publications, inc., 1992

    work page 1992

  2. [2]

    L. Ahlfors. Lectures on Quasiconformal Mappings. D. Van Nostrand Com- pany, Inc, 1966

    work page 1966

  3. [3]

    Aledo, J

    J. Aledo, J. Espinar, and J. G´ alvez. The Codazzi equation for surfaces. Advances in Mathematics, 224:2511–2530, 2010

    work page 2010

  4. [4]

    Aleksandrov

    A. Aleksandrov. Uniqueness theorems for surfaces in the large, I. Vestnik Leningrad University: Mathematics , 11:5–17, 1956

    work page 1956

  5. [5]

    Axler, P

    S. Axler, P. Bourdon, and R. Wade. Harmonic Function Theory. Springer- Verlag New York, 2001

    work page 2001

  6. [6]

    Bernstein

    S. Bernstein. Sur une th´ eor` eme de g´ eometrie et ses applications aux ´ equations d´ eriv´ ees partielles du type elliptique.Communications de la Soci´ et´ e Math´ ematique de Kharkow, 15(1):38–45, 1915

    work page 1915

  7. [7]

    Braga and R

    F. Braga and R. Sa Earp. On the structure of certain Weingarten surfaces with boundary a circle. Annales de la facult´ e des sciences de Toulouse , 6(2):243–255, 1997

    work page 1997

  8. [8]

    R. Bryant. Complex analysis and a class of Weingarten surfaces. arXiv: 1105.5589v1 [math.DG]. Preprint, 2011. 38

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  9. [9]

    H. Chan. Nonexistence of nonpositively curved surfaces with one embedded end. Manuscripta Mathematica, 102:177–186, 2000

    work page 2000

  10. [10]

    Chan and A

    H. Chan and A. Treibergs. Nonpositively curved surfaces in R3. Journal of Differential Geometry, 57:389–407, 2001

    work page 2001

  11. [11]

    S. Chern. Some new characterizations of the Euclidean sphere. Duke Math- ematical Journal, 12:279–290, 1945

    work page 1945

  12. [12]

    S. Chern. On special W-surfaces. Proceedings of the American Mathemat- ical Society, 6(5):783–786, 1955

    work page 1955

  13. [13]

    Chern and R

    S. Chern and R. Osserman. Complete minimal surfaces in euclidean n- space. Journal d’Analyse Math´ ematique, 19(1):15–34, 1967

    work page 1967

  14. [14]

    Connor, K

    P. Connor, K. Li, and M. Weber. Complete embedded harmonic surfaces in R3. Experimental Mathematics, 24(2):196–224, 2015

    work page 2015

  15. [15]

    do Carmo

    M. do Carmo. Differential geometry of curves and surfaces . Prentice-Hall, 1976

    work page 1976

  16. [16]

    J. Espinar. La ecuaci´ on de Codazzi en superficies. PhD thesis, Universidad de Granada, 2008

    work page 2008

  17. [17]

    R. Finn. On normal metrics, and a theorem of Cohn-Vossen. Bulletin of the American Mathematical Society, 70:772–773, 1964

    work page 1964

  18. [18]

    R. Finn. On a class of conformal metrics, with applications to differential geometry in the large. Commentarii Mathematici Helvetici , 40(1):1–30, 1965

    work page 1965

  19. [19]

    Gutlyanskii, V

    V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov. The Beltrami Equation. A Geometric Approach . Springer-Verlag New York, 2012

    work page 2012

  20. [20]

    G´ alvez, A

    J. G´ alvez, A. Mart´ ınez, and F. Milan. Linear Weingarten surfaces in R3. Monatshefte f¨ ur Mathematik, 138:133–144, 2003

    work page 2003

  21. [21]

    Complete Surfaces with Ends of Non Positive Curvature

    J. G´ alvez, A. Mart´ ınez, and J. Teruel. Complete surfaces with ends of non positive curvature. arXiv: 1405.0851 [math.DG]. Preprint, 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  22. [22]

    Hartman and A

    P. Hartman and A. Wintner. Umbilical points and W-surfaces. American Journal of Mathematics , 76(3):502–508, 1954

    work page 1954

  23. [23]

    Hoffman and W

    D. Hoffman and W. Meeks III. The strong halfspace theorem for minimal surfaces. Inventiones Mathematicae, 101:373–377, 1990

    work page 1990

  24. [24]

    E. Hopf. Element¨ are bemerkungen ¨ uber die l¨ osungen partieller differen- tialgleichungen zweiter ordnung vom elliptischen typus. Sitzungsberichte Preussische Akademie der Wissenschaften , 19:147–152, 1927

    work page 1927

  25. [25]

    E. Hopf. On S. Bernstein’s theorem on surfaces z(x, y) of nonpositive curvature. Proceedings of the American Mathematical Society , 1:80–85, 1950

    work page 1950

  26. [26]

    E. Hopf. A theorem on the accessibility of boundary parts of an open point set. Proceedings of the American Mathematical Society, 1:76–79, 1950. 39

    work page 1950

  27. [27]

    E. Hopf. A remark on linear elliptic differential equations of second order. Proceedings of the American Mathematical Society, 3(5):791–793, 1952

    work page 1952

  28. [28]

    H. Hopf. Differential geometry in the large , volume 1000. Springer-Verlag Berlin Heidelberg, 1983

    work page 1983

  29. [29]

    A. Huber. On subharmonic functions and differential geometry in the large. Commentarii Mathematici Helvetici , 32:13–72, 1957

    work page 1957

  30. [30]

    Jorge and W

    L. Jorge and W. Meeks III. The topology of complete minimal surfaces of finite total Gaussian curvature. Topology, 22(2):203–221, 1983

    work page 1983

  31. [31]

    Kenmotsu

    K. Kenmotsu. Weierstrass formula for surfaces of prescribed mean curva- ture. Mathematische Annalen, 245:89–99, 1979

    work page 1979

  32. [32]

    Klotz and R

    T. Klotz and R. Osserman. Complete surfaces in E3 with constant mean curvature. Commentarii Mathematici Helvetici , 41:313–318, 1966-1967

    work page 1966

  33. [33]

    Korevaar, R

    N. Korevaar, R. Kusner, and B. Solomon. The structure of complete em- bedded surfaces with constant mean curvature. Journal of Differential Geometry, 30:465–503, 1989

    work page 1989

  34. [34]

    Li and L.-F

    P. Li and L.-F. Tam. Complete surfaces with total finite curvature. Journal of Differentia Geometry, 33:139–168, 1991

    work page 1991

  35. [35]

    Meeks III

    W. Meeks III. The topology and geometry of embedded surfaces of constant mean curvature. Journal of Differential Geometry , 27(3):539–552, 1988

    work page 1988

  36. [36]

    T. Milnor. Abstract Weingarten surfaces. Journal of Differential Geometry, 15:365–380, 1980

    work page 1980

  37. [37]

    A. Mori. On an absolute constant in the theory of quasi-conformal map- pings. Journal of the Mathematical Society of Japan , 8(2):156–166, 1956

    work page 1956

  38. [38]

    Osserman

    R. Osserman. Global properties of minimal surfaces in E3 and En. Annals of Mathematics, 80(2):340–364, 1964

    work page 1964

  39. [39]

    Osserman

    R. Osserman. A survey of minimal surfaces . Van Nostrand Reinhold Com- pany, 1969

    work page 1969

  40. [40]

    J. P´ erez. A new golden age of minimal surfaces. Notices of the AMS , 64(4):347–358, 2017

    work page 2017

  41. [41]

    Pucci and J

    P. Pucci and J. Serrin. The Maximum Principle . Birkh¨ auser Basel, 2007

    work page 2007

  42. [42]

    Rosenberg and R

    H. Rosenberg and R. Sa Earp. The geometry of properly embedded special surfaces in R3; e.g., surfaces satisfying aH + bK = 1, where a and b are positive. Duke Mathematical Journal , 73(2):291–306, 1994

    work page 1994

  43. [43]

    Sa Earp and E

    R. Sa Earp and E. Toubiana. A note on special surfaces in R3. Matem´ atica Contemporˆ anea, 4:108–118, 1993

    work page 1993

  44. [44]

    Sa Earp and E

    R. Sa Earp and E. Toubiana. Sur les surfaces de Weingarten sp´ eciales de type minimal. Boletim da Sociedade Brasileira de Matem´ atica, 26(2):129– 148, 1995. 40

    work page 1995

  45. [45]

    Sa Earp and E

    R. Sa Earp and E. Toubiana. Classification des surfaces de type Delaunay. American Journal of Mathematics , 121(3):671–700, 1999

    work page 1999

  46. [46]

    Sa Earp and E

    R. Sa Earp and E. Toubiana. Symmetry of properly embedded special Weingarten surfaces in H3. Transactions of the American Mathematical Society, 351(12):4693–4711, 1999

    work page 1999

  47. [47]

    R. Schoen. Uniqueness, symmetry, and embeddedness of minimal surfaces. Journal of Differential Geometry , 18:791–809, 1983

    work page 1983

  48. [48]

    Shiohama

    K. Shiohama. Total curvatures and minimal areas of complete open sur- faces. Proceedings of the American Mathematical Society , 94(2):310–316, 1985

    work page 1985

  49. [49]

    Taliaferro

    S. Taliaferro. On the growth of superharmonic functions near isolated sigularity, I. Journal of Differential Equations , 158:28–47, 1999

    work page 1999

  50. [50]

    Taliaferro

    S. Taliaferro. Isolated singularities of nonlinear elliptic inequalities. Indiana University Mathematics Journal , 50(4):1885–1898, 2001

    work page 2001

  51. [51]

    Taliaferro

    S. Taliaferro. Isolated singularities of nonlinear elliptic inequalities II. Asymptotic behavior of solutions. Indiana University Mathematics Jour- nal, 55(6):1791–1812, 2006

    work page 2006

  52. [52]

    B. White. Complete surfaces of finite total curvature. Journal of Differen- tial Geometry, 26:315–326, 1987

    work page 1987

  53. [53]

    Yoon and J

    D. Yoon and J. Lee. Linear Weingarten helicoidal surfaces in isotropic space. Symmetry, 8(11):126, 2016. 41

    work page 2016