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Embedded minimal tori in S 3 and the Lawson conjecture

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Acta Mathematica

Abstract

We show that any embedded minimal torus in S 3 is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrews B.: Noncollapsing in mean-convex mean curvature flow. Geom. Topol., 16, 1413–1418 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aronszajn N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl., 36, 235–249 (1957)

    MATH  MathSciNet  Google Scholar 

  4. Bony, J. M., Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble), 19:1 (1969), 277–304, xii.

    Google Scholar 

  5. Brendle, S., Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics, 111. Amer. Math. Soc., Providence, RI, 2010.

  6. Choe, J., Minimal surfaces in \({\mathbb{S}^3}\) and Yau’s conjecture, in Proceedings of the Tenth International Workshop on Differential Geometry, pp. 183–188. Kyungpook Nat. Univ., Taegu, 2006.

  7. ChoeJ. Soret M.: First eigenvalue of symmetric minimal surfaces in \({\mathbb{S}^3}\) . Indiana Univ. Math. J., 58, 269–281 (2009)

    Article  MathSciNet  Google Scholar 

  8. ChoiH.I. Wang A.N.: A first eigenvalue estimate for minimal hypersurfaces. J. Differential Geom., 18, 559–562 (1983)

    MathSciNet  Google Scholar 

  9. Colding, T. H. & De Lellis, C., The min-max construction of minimal surfaces, in Surveys in Differential Geometry, Vol. VIII (Boston, MA, 2002), Surv. Differ. Geom., VIII, pp. 75–107. International Press, Somerville, MA, 2003.

  10. Grayson M.A.: Shortening embedded curves. Ann. of Math., 129, 71–111 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. HsiangW.-Y. Lawson H.B. Jr.: Minimal submanifolds of low cohomogeneity. J. Differential Geom., 5, 1–38 (1971)

    Google Scholar 

  12. Huisken G.: A distance comparison principle for evolving curves. Asian J. Math., 2, 127–133 (1998)

    MATH  MathSciNet  Google Scholar 

  13. Kapouleas, N., Doubling and desingularization constructions for minimal surfaces, in Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics, 20, pp. 281–325. International Press, Somerville, MA, 2011.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Lawson H.B. Jr.: Local rigidity theorems for minimal hypersurfaces. Ann. of Math., 89, 187–197 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Lawson H.B. Jr.: The unknottedness of minimal embeddings. Invent. Math., 11, 183–187 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  19. Marques, F. C. & Neves, A., Min-max theory and the Willmore conjecture. To appear in Ann. of Math.

  20. Montiel S., Ros A.: Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math., 83, 153–166 (1985)

    Article  MathSciNet  Google Scholar 

  21. Pitts, J. T., Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Mathematical Notes, 27. Princeton Univ. Press, Princeton, NJ, 1981.

  22. Ros A.: A two-piece property for compact minimal surfaces in a three-sphere. Indiana Univ. Math. J., 44, 841–849 (1995)

    MATH  MathSciNet  Google Scholar 

  23. Simons J.: Minimal varieties in riemannian manifolds. Ann. of Math., 88, 62–105 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  24. Urbano F.: Minimal surfaces with low index in the three-dimensional sphere. Proc. Amer. Math. Soc., 108, 989–992 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  25. White B.: The size of the singular set in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 13, 665–695 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. White, B., The nature of singularities in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 16 (2003), 123–138 (electronic).

    Google Scholar 

  27. White, B., Subsequent singularities in mean-convex mean curvature flow. Preprint, 2011. arXiv:1103.1469 [math.DG].

  28. Yau, S.-T., Problem section, in Seminar on Differential Geometry, Annals of Mathematics Studies, 102, pp. 669–706. Princeton Univ. Press, Princeton, NJ, 1982.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, U.S.A.

    Simon Brendle

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  1. Simon Brendle
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Correspondence to Simon Brendle.

Additional information

The author was supported in part by the National Science Foundation under grants DMS-0905628 and DMS-1201924.

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Brendle, S. Embedded minimal tori in S 3 and the Lawson conjecture. Acta Math 211, 177–190 (2013). https://doi.org/10.1007/s11511-013-0101-2

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  • DOI: https://doi.org/10.1007/s11511-013-0101-2

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