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
Brendle, S.: Embedded minimal tori in and the Lawson conjecture. Acta Math. 2(211), 177–190 (2013)
Choe, J., Soret, M.: New minimal surfaces in \(S^3\) desingularizing the Clifford tori. Math. Ann. 364(3–4), 763–776 (2016)
Frankel, T.: On the Fundamental Group of a Compact Minimal Submanifold. Ann. Math. 83, 68–73 (1966)
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)
Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. 131(2), 239–330 (1990). https://doi.org/10.2307/1971494
Kapouleas, N.: Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math. 119(3), 443–518 (1995). https://doi.org/10.1007/BF01245190
Kapouleas, N.: Complete embedded minimal surfaces of finite total curvature. J. Differ. Geom. 45, 95–169 (1997)
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)
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)
Kapouleas, N.: Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I. J. Differ. Geom. 106(3), 393–449 (2017)
Kapouleas, N.: A general desingularization theorem for minimal surfaces in the compact case (2020) (in preparation)
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)
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
Kapouleas, N., Yang, S.D.: Minimal surfaces in the three-sphere by doubling the Clifford torus. Am. J. Math. 132, 257–295 (2010)
Karcher, H.: Embedded Minimal Surfaces Derived from Scherk’s Examples. Manuscr. Math. 62, 83–114 (1988)
Karcher, H.: Construction of minimal surfaces, Surveys in Geometry, pp. 1–96. University of Tokyo, Tokyo (1989)
Karcher, H., Pinkall, U., Sterling, I.: New minimal surfaces in \(S^3\). J. Differ. Geom. 28, 169–185 (1988)
Lawson Jr., H.B.: Complete minimal surfaces in \(S^3\). Ann. Math. 92, 335–374 (1970)
Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. 179(2), 683–782 (2014)
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
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
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)
Pitts, J.T., Rubinstein, J.H.: Equivariant minimax and minimal surfaces in geometric three-manifolds. Bull. Am. Math. Soc. 19(1), 303–309 (1988)
Scherk, H.F.: Bemerkungen über die kleinste Fläche innherhalb gegebener Grenzen. J. Reine Angew. Math. 13, 185–208 (1835)
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
Traizet, M.: Construction de surfaces minimales en recollant des surfaces de Scherk. Ann. Inst. Fourier (Grenoble) 46(5), 1385–1442 (1996)
Wiygul, D.: Minimal surfaces in the 3-sphere by stacking Clifford tori. J. Differ. Geom. 114(3), 467–549 (2020)
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. 383, 119–170 (2022). https://doi.org/10.1007/s00208-021-02169-8
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DOI: https://doi.org/10.1007/s00208-021-02169-8