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Minimal surfaces bounded by convex curves in parallel planes

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References

  1. M. Anderson. Curvatuve estimates for minimal surfaces in 3-manifolds.Ann. Scient. Éc. Norm. Sup. 18, 89–105 (1985).

    Google Scholar 

  2. J. L. Barbosa andM. Do Carmo On the size of a stable minimal surface in ℝ3.American Journal of Mathematics, 19(8), 515–528 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Callahan, D. Hoffman andW. H. Meeks III Embedded minimal surfaces with an infinite number of ends.Inventiones Math., 96, 459–505 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Callahan, D. Hoffman, andW. H. Meeks III. The structure of singly-periodic minimal surface.Inventiones Math., 99, 455–481 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. I, Chavel,Eigenvalues in Riemannian Geometry. Academic Press, 1984.

  6. A. Enneper Die cyklischen flachen.Z. Math. und Phys., 14, 393–421 (1869).

    Google Scholar 

  7. S. Hildebrandt Boundary behavior of minimal surfaces.Archive Rational Mech. Anal., 35, 47–81 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Hoffman andW. H. Meeks III Minimal surfaces based on the catenoid. Amer. Math. Monthly, special Geometry Issue, 97(8), 702–730 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. B. Lawson, Jr.,Lectures on Minimal Submanifolds. Publish or Perish Press, Berkeley, 1971.

    Google Scholar 

  10. W. H. Meeks III.Lectures on Plateau's Problem. Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil, 1978.

    Google Scholar 

  11. W. H. Meeks III. Uniqueness theorems for minimal surfaces.Illinois Journal of Math., 25, 318–336 (1981).

    MathSciNet  MATH  Google Scholar 

  12. W. H. Meeks III andB. White. The space of minimal annuli bounded by an extremal pair of planar curves. Preprint.

  13. W. H. Meeks III andS. T. Yau The classical Plateau problem and the topology of three-dimensional manifolds.Topology, 21, 409–422 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  14. W. H. Meeks III andS. T. Yau The existence of embedded minimal surfaces and the problem of uniqueness.Math. Z., 179, 151–168 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Riemann.Ouevres Mathématiques de Riemann, Gauthiers-Villars, Paris, 1898.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. M. Shiffman On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes.Annals of Math., 63, 77–90 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Smale. An infinite dimensional version of Sard's theorem.Americal Journal of Math., 87, 861–866, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Struwe. A Morse theory for annulus-type minimal surfaces.Journal für die reine und angewandte Mathematik, 368, 1–27 (1986).

    MathSciNet  MATH  Google Scholar 

  20. F. Tomi andA. J. Tromba Extreme curves bound an embedded minimal surface of disk type.Math. Z. 158, 137–145 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  21. B. White Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals.Inventiones Math., 88(2), 243–256 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  22. B. White. New applications of mapping degrees to minimal surface theory.Journal of Differential Geometry, 29, 143–162 (1989).

    MathSciNet  MATH  Google Scholar 

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The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285.

Funded by National Science Foundation grants DMS-8553231 (PYI) and DMS-8703537.

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Meeks, W.H., White, B. Minimal surfaces bounded by convex curves in parallel planes. Comment. Math. Helv. 66, 263–278 (1991). https://doi.org/10.1007/BF02566647

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