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The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

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Abstract

Let S ⊂ ℝn be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x1 (M1) ∪ … ∪ xr (Mr) where each Mj is connected, xj: Mj → Vj is a classical minimal immersion into an affine subspace Vj of ℝn, and the subspaces V1,…, Vr are pairwise orthogonal. Here we prove that if Mj is orientable, then xj (Mj) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.

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References

  1. Farkas, H.M. and Kra, I.Riemann Surfaces, Grad. Texts. in Math.,71, Springer-Verlag, Berlin.

  2. Hardt, R. and Lin, F.-H. Complete orientable surfaces in ℝ4 that minimize area amongst possibly nonorientable surfaces, preprint.

  3. Hoffman, D.A. and Osserman, R. The geometry of the generalized Gauss map,Memoirs A.M.S.,28, (1980).

  4. Micallef, M.J. Stable minimal surfaces in Euclidean space,J. Diff. Geom.,19, 64–76, (1984).

    MathSciNet  Google Scholar 

  5. Morgan, F. Examples of unoriented area-minimizing surfaces,Trans. A.M.S.,283, 225–237, (1984).

    Article  MATH  Google Scholar 

  6. Morgan, F. On the singular structure of two-dimensional area-minimizing surfaces in ℝn,Math. Ann.,261, 101–110, (1982).

    Article  MathSciNet  MATH  Google Scholar 

  7. Morgan, F. Harnack-type mass bounds and Bernstein theorems for area-minimizing flat chains modulov, Comm. P.D.E.,11, 1257–1283, (1986).

    MATH  Google Scholar 

  8. Morgan, F.Beginner’s Guide to Geometric Measure Theory, Academic Press, San Diego, 1987.

    Google Scholar 

  9. Morgan, F. Calibrations modulov, Adv. Math.,64, 32–50, (1987).

    Article  MATH  Google Scholar 

  10. Ross, M. The end behavior of complete 2-dimensional area-minimizing mod 2 surfaces in ℝn,Duke Math. J.,63, 623–632, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  11. Ross, M. Complete minimal spheres and projective planes in ℝn with simple ends,Math. Z.,201, 375–380, (1989).

    Article  MathSciNet  MATH  Google Scholar 

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  1. Antartic CRL, Box 252-C, 7001, Hobart, Tas, Australia

    Marty Ross

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  1. Marty Ross
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Correspondence to Marty Ross.

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Ross, M. The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn . J Geom Anal 8, 313–317 (1998). https://doi.org/10.1007/BF02921644

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  • DOI: https://doi.org/10.1007/BF02921644

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