References
S. Berstein,Sur les surfaces définies au moyen de leur courbure moyenne ou totale, Ann. Ec. Norm. Sup.27 (1910), 233–256.
W. Blaschke,Sulla geometria differenziale delle superficie S 2 nello spazio euclideo S 4, Ann. Mat. Pura Appl. (4)28 (1949), 205–209.
C. C. Chen, On the image of the generalized Gauss map of a complete minimal surface inR 4, Pacific J. Math. (to appear).
S.-S. Chern Minimal surfaces in an euclidean space of N dimensions, Differential and Combinatorial Topology, Princeton University Press (1965), 187–198.
S.-S. Chern,On the curvature of a piece of hypersurface in Euclidean space, Abh. Math. Sem. Hamburg29 (1965), 77–91.
S.-S. Chern andR. Osserman,Complete minimal surfaces in Euclidean n-space, J. Analyse Math.19 (1967), 15–34.
S.-S. Chern andE. Spanier,A theorem on orientable surfaces in four-dimensional space, Comment. Math. Helv.25 (1951), 1–5.
J. Eells andL. Lemaire,A report on harmonic maps, Bull. London Math. Soc.10 (1978), 1–68.
D. Fischer-Colbrie andR. Schoen,The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Applied Math.33 (1980), 199–211.
Erhard Heinz, Über Flächen mit eineindeutiger Projection auf einer Ebene, deren Krümmungen durch Ungleichugen eingeschränkt sind, Math. Ann.192 (1955), 451–454.
S. Hildebrandt, J. Jost andK.-O. Widman,Harmonic mappings and minimal submanifolds, Invent. Math.62 (1980), 269–298.
D. A. Hoffman,Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry8 (1973), 161–176.
D. A. Hoffman andR. Osserman,The geometry of the generalized Gauss map, Amer. Math. Soc. Memoir No. 236, 1980.
D. A. Hoffman andR. Osserman,The Gauss map of surfaces in R n (preprint).
K. Kenmotsu,Wierstrass formula for surfaces of prescribed mean curvature, Math. Ann.245 (1979), 89–99.
R. Osserman,Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math.12 (1959), 229–232.
R. Osserman,Global properties of minimal surfaces in E 3 and E n, Ann. of Math.80 (1964), 340–364.
E. A. Ruh,Asymptotic behavior of non-parametric minimal hypersurfaces, J. Differential Geometry4 (1970), 509–513.
E. A. Ruh andJ. Vilms,The tension field of the Gauss map, Trans. Amer. Math. Soc.149 (1970), 569–573.
R. Schoen andS.-T. Yau,On univalent harmonic maps between surfaces, Invent. Math.44 (1978), 265–278.
J. L. Weiner,The Gauss map for surfaces in 4-space (preprint).
F. Xavier,The Gauss map of a complete non-flat minimal surface cannot omit 7 points on the sphere, Ann. of Math.13 (1981), 211–214.
S.-T. Yau,Submanifolds with constant mean curvature, II, Amer. J. Math.97 (1975), 76–100.
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This work was supported in part by NSF grants at the University of Massachusetts, Amherst; Stanford University, Stanford; University of California, Berkeley.
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Hoffman, D.A., Osserman, R. & Schoen, R. On the Gauss map of complete surfaces of constant mean curvature in R3 and R4 . Commentarii Mathematici Helvetici 57, 519–531 (1982). https://doi.org/10.1007/BF02565874
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DOI: https://doi.org/10.1007/BF02565874