Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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NON-COMPACT DOUGLAS-PLATEAU PROBLEM

  • JIN, SUN SOOK (College of Electronics and Information Kyung Hee University)
  • Published : 2005.09.01
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Abstract

In this article, we prove the existence of two embedded minimal annuli in a slab which are all bounded by a Jordan convex curve and a straight line.

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References

  1. M. Anderson, Curvature estimates for minimal surfaces in 3-manifolds, Ann. Sci. Ecole. Norm. Sup. (4) 18 (1985), 89-105 https://doi.org/10.24033/asens.1485
  2. T. H. Colding and W. P. Minicozzi, Minimal surfaces, Courant Lect. Notes Math. 4 (1999)
  3. J. Douglas, The problem of Plateau for two contours, J. Math. Phys. Masschusetts Inst. of Tech. 10 (1930/31), 315-359
  4. Yi Fang and Jenn-Fang Hwang, Curvature estimates for minimal annuli and non- compact Douglas-Plateau problem, Comm. Anal. Geom. 8 (2000), no. 4, 871-904 https://doi.org/10.4310/CAG.2000.v8.n4.a6
  5. D. Hoffman and W. Meeks, A variational approach to the existence of complete embedded minimal surfaces, Duke. Math. J. 57 (1988), 877-893 https://doi.org/10.1215/S0012-7094-88-05739-0
  6. D. Hoffman and W. Meeks, The strong halfspace theorem for minimal surface, Invent. Math. 101 (1990), 373-377 https://doi.org/10.1007/BF01231506
  7. H. Jenkins and J. Serrin, Variational problems of minimal surface type, 2, Bound-ary value problems for the minimal surface equation, Arch. Ration. Mech. Anal.21 (1966), 321-342
  8. N. Korevaar, R. Kusner, and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465-503 https://doi.org/10.4310/jdg/1214443598
  9. F. Lopez, The classification of complete minimal surfaces with total curvature greater than $-12{\pi}$, Trans. Amer. Math. Soc. 334 (1992), 49-74 https://doi.org/10.2307/2153972
  10. W. Meeks and B. White, Minimal surfaces bounded by convex curves in parallel planes, Comm. Math. Helv. 66 (1991), 263-278 https://doi.org/10.1007/BF02566647
  11. J. Perez and A. Ros, Properly embedded minimal annuli bounded by a convex curve, J. Inst. Math. Jussieu 1(2) (2002), 293-305 https://doi.org/10.1017/S1474-748002000075
  12. J. Perez and A. Ros, Properly embedded minimal surfaces with finite total curvature, in the global theory of minimal surfaces in flat spaces, Lecture Notes in Math. 1775 (2002), 15-66
  13. B. Riemann, Uber die Flache vom kleinsten Inhalt bei gegebener Begrebzung, Abh. Konigl. d. Wiss. Gottingen, Mathem. Cl. 13 (1867), 3-52
  14. M. Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. 63 (1956), 77-90 https://doi.org/10.2307/1969991
  15. B. White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987), 243-256 https://doi.org/10.1007/BF01388908