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Minimal surface system in Euclidean four-space

이호주
대한수학회 (The Korean Mathematical Society)|2023.01|대한수학회지 (Journal of the KMS)|제60권 제1호|pp. 71-90|DOI 10.4134/JKMS.j220095

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참고문헌

키워드

Gauss mapMinimal surface systemScherk surface
주제분류수학

초록

We construct generalized Cauchy-Riemann equations of the first order for a pair of two $\mathbb{R}$-valued functions to deform a minimal graph in ${\mathbb{R}}^{3}$ to the one parameter family of the two dimensional minimal graphs in ${\mathbb{R}}^{4}$. We construct the two parameter family of minimal graphs in ${\mathbb{R}}^{4}$, which include catenoids, helicoids, planes in ${\mathbb{R}}^{3}$, and complex logarithmic graphs in ${\mathbb{C}}^{2}$. We present higher codimensional generalizations of Scherk's periodic minimal surfaces.

Abstract

We construct generalized Cauchy-Riemann equations of the first order for a pair of two $\mathbb{R}$-valued functions to deform a minimal graph in ${\mathbb{R}}^{3}$ to the one parameter family of the two dimensional minimal graphs in ${\mathbb{R}}^{4}$. We construct the two parameter family of minimal graphs in ${\mathbb{R}}^{4}$, which include catenoids, helicoids, planes in ${\mathbb{R}}^{3}$, and complex logarithmic graphs in ${\mathbb{C}}^{2}$. We present higher codimensional generalizations of Scherk's periodic minimal surfaces.

참고문헌 (29)

  1. S. T. Becker-Kahn, "Transverse singularities of minimal two-valued graphs in arbitrary codimension", J. Differential Geom, 107(2), pp.241~325 (2017)[DOI]
  2. R. Beyerstedt, "Removable singularities of solutions to elliptic Monge-Ampere equations", Math. Z, 208(3), pp.363~373 (1991)[DOI]
  3. E. Calabi, "Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)", pp.223~230 (1970), Amer. Math. Soc.
  4. E. Catalan, "Memoire sur les surfaces dont les rayons de courbure, en chaque point, sont egaux et de signer contraires", C. R. Acad. Sci. Paris, 41, pp.1019~1023 (1855)
  5. C. C. Chen, "Degenerate minimal surfaces in R4", Bol. Soc. Brasil. Mat, 14(1), pp.1~16 (1983)[DOI]
  6. S. Chern, "Complete minimal surfaces in euclidean n-space", J. Analyse Math, 19, pp.15~34 (1967)[DOI]
  7. J. Eells, "Minimal graphs", Manuscripta Math, 28(1-3), pp.101~108 (1979)[DOI]
  8. L. Fu, "An analogue of Bernstein's theorem", Houston J. Math, 24(3), pp.415~419 (1998)
  9. R. Harvey, "Calibrated geometries", Acta Math, 148, pp.47~157 (1982)[DOI]
  10. D. A. Homan, "The geometry of the generalized Gauss map", Mem. Amer. Math. Soc, 28(236) (1980)[DOI]
  11. H. Jenkins, "Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation", Arch. Rational Mech. Anal, 21, pp.321~342 (1966)[DOI]
  12. J. Jost, "A Bernstein theorem for special Lagrangian graphs", Calc. Var. Partial Differential Equations, 15(3), pp.299~312 (2002)[DOI]
  13. H. Karcher, "Embedded minimal surfaces derived from Scherk's examples", Manuscripta Math, 62(1), pp.83~114 (1988)[DOI]
  14. J. L. Lagrange, "Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies", Miscellanea Taurinensia 2, 325(1), pp.173~199
  15. J. M. Landsberg, "Minimal submanifolds defined by first-order systems of PDE", J. Dif-ferential Geom, 36(2), pp.369~415 (1992)
  16. H. B. Lawson, Jr., "Some intrinsic characterizations of minimal surfaces", J. Analyse Math, 24, pp.151~161 (1971)[DOI]
  17. H. B. Lawson, Jr., "Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system", Acta Math, 139(1-2), pp.1~17 (1977)[DOI]
  18. H. Lee, "Minimal surface systems, maximal surface systems and special Lagrangian equations", Trans. Amer. Math. Soc, 365(7), pp.3775~3797 (2013)[DOI]
  19. 이호주, "Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$", 대한수학회지, 57(1), pp.1~19 (2020), 대한수학회[DOI]
  20. M. J. Micallef, "The structure of branch points in minimal surfaces and in pseudoholomorphic curves", Ann. of Math. (2), 141(1), pp.35~85 (1995)[DOI]
  21. J. C. C. Nitsche, "Lectures on minimal surfaces. Vol. 1" (1989), Cambridge University Press
  22. R. Osserman, "Minimal varieties", Bull. Amer. Math. Soc, 75, pp.1092~1120 (1969)[DOI]
  23. R. Osserman, "Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)", pp.283~291 (1970), Amer. Math. Soc.
  24. R. Osserman, "A Survey of Minimal Surfaces" (1986), Dover Publications, Inc.
  25. F. Pacard, "Higher-dimensional Scherk's hypersurfaces", J. Math. Pures Appl. (9), 81(3), pp.241~258 (2002)[DOI]
  26. T. Rado, "Ergebnisse der Math. Band 2" (1933), Springer
  27. H. F. Scherk, "Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen", J. Reine Angew. Math, 13, pp.185~208 (1835)[DOI]
  28. M. -P. Tsui, "A Bernstein type result for special Lagrangian submanifolds", Math. Res. Lett, 9(4), pp.529~535 (2002)[DOI]
  29. Y. Yuan, "A Bernstein problem for special Lagrangian equations", Invent. Math, 150(1), pp.117~125 (2002)[DOI]

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