Minimal surfaces in the three-sphere by desingularizing intersecting Clifford tori

Abstract

For each integer \(k \ge 2\) we apply PDE gluing methods to desingularize certain collections of intersecting Clifford tori, thus producing sequences of minimal surfaces embedded in the round three-sphere. The collections of the Clifford tori we use consist of either k Clifford tori intersecting with maximal symmetry along two orthogonal great circles (lying on orthogonally complementary two-planes) or of the same k Clifford tori supplemented by an additional Clifford torus equidistant from the original two circles of intersection so that the latter torus orthogonally intersects each of the former k tori along a pair of disjoint orthogonal circles. The former two circles get desingularized by using singly periodic Karcher–Scherk towers of order k as models, so that after rescaling the sequences of minimal surfaces converge smoothly on compact subsets to the Karcher–Scherk tower of order k. Near the other 2k circles (in the latter case) the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type, where the number of handles desingularizing each circle is the same, resemble surfaces constructed by Choe and Soret (Math Ann 364(3–4):763–776, 2016) by different methods. There are many new examples which are more complicated and on which the numbers of handles for the two circles differ. All examples of the latter type are new as well.

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References

  1. 1.

    Brendle, S.: Embedded minimal tori in and the Lawson conjecture. Acta Math. 2(211), 177–190 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Choe, J., Soret, M.: New minimal surfaces in \(S^3\) desingularizing the Clifford tori. Math. Ann. 364(3–4), 763–776 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Frankel, T.: On the Fundamental Group of a Compact Minimal Submanifold. Ann. Math. 83, 68–73 (1966)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jenkins, H., Serrin, J.: Some variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation. Arch. Ration. Mech. Anal. 21, 321–342 (1966)

  5. 5.

    Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. 131(2), 239–330 (1990). https://doi.org/10.2307/1971494

  6. 6.

    Kapouleas, N.: Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math. 119(3), 443–518 (1995). https://doi.org/10.1007/BF01245190

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Kapouleas, N.: Complete embedded minimal surfaces of finite total curvature. J. Differ. Geom. 45, 95–169 (1997)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Kapouleas, N.: Constructions of minimal surfaces by gluing minimal immersions, Global Theory of Minimal Surfaces, Clay Mathematics Proceedings, vol. 2, pp. 489–524. American Mathematical Society, Providence, RI (2005)

    MATH  Google Scholar 

  9. 9.

    Kapouleas, N.: Doubling and desingularization constructions for minimal surfaces. In: Surveys in geometric analysis and relativity. Adv. Lect. Math. (ALM), vol. 20, pp. 281–325. Int. Press, Somerville (2011)

  10. 10.

    Kapouleas, N.: Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I. J. Differ. Geom. 106(3), 393–449 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kapouleas, N.: A general desingularization theorem for minimal surfaces in the compact case (2020) (in preparation)

  12. 12.

    Kapouleas, N., Kleene, S.J., Møller, N.M.: Mean curvature self-shrinkers of high genus: non-compact examples. J. Reine Angew. Math. 739, 1–39 (2018)

  13. 13.

    Kapouleas, N., Li, M.M.: Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disk (2020). arXiv:1709.08556

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Karcher, H.: Embedded Minimal Surfaces Derived from Scherk’s Examples. Manuscr. Math. 62, 83–114 (1988)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Karcher, H.: Construction of minimal surfaces, Surveys in Geometry, pp. 1–96. University of Tokyo, Tokyo (1989)

  17. 17.

    Karcher, H., Pinkall, U., Sterling, I.: New minimal surfaces in \(S^3\). J. Differ. Geom. 28, 169–185 (1988)

    Article  Google Scholar 

  18. 18.

    Lawson Jr., H.B.: Complete minimal surfaces in \(S^3\). Ann. Math. 92, 335–374 (1970)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. 179(2), 683–782 (2014)

  20. 20.

    Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow, Part III. Duke Math. J. 163(11), 2023–2056 (2014). https://doi.org/10.1215/00127094-2795108

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Montiel, S., Ros, A.: Schrödinger operators associated to a holomorphic map. In: Global Differential Geometry and Global Analysis. Lecture Notes in Math, vol. 1481. Springer, Berlin, pp. 147–174 (1991). https://doi.org/10.1007/BFb0083639

  22. 22.

    Pérez, J., Traizet, M.: The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends. Trans. Am. Math. Soc. 359(3), 965–990 (2007)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Pitts, J.T., Rubinstein, J.H.: Equivariant minimax and minimal surfaces in geometric three-manifolds. Bull. Am. Math. Soc. 19(1), 303–309 (1988)

    MathSciNet  Article  Google Scholar 

  24. 24.

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

    MathSciNet  Google Scholar 

  25. 25.

    Schoen, R.M.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math. 41(3), 317–392 (1988). https://doi.org/10.1002/cpa.3160410305

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Traizet, M.: Construction de surfaces minimales en recollant des surfaces de Scherk. Ann. Inst. Fourier (Grenoble) 46(5), 1385–1442 (1996)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wiygul, D.: Minimal surfaces in the 3-sphere by stacking Clifford tori. J. Differ. Geom. 114(3), 467–549 (2020)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank Richard Schoen for his continuous support and interest in the results of this article. N.K. was partially supported by NSF grants DMS-1105371 and DMS-1405537. The authors would like to thank the referee for carefully reading the manuscript and making many valuable suggestions.

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Kapouleas, N., Wiygul, D. Minimal surfaces in the three-sphere by desingularizing intersecting Clifford tori. Math. Ann. (2021). https://doi.org/10.1007/s00208-021-02169-8

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