New Brief Proofs of the Gulliver–Osserman–Royden Theorem

  • Anthony Tromba
Part of the Springer Monographs in Mathematics book series (SMM)


We would like to present very much simplified proofs of versions of the Gulliver–Osserman–Royden (GOR) theorem (1973), in the case Γ is C 2,α smooth. In the first proof instead of employing a topological theory of ramified coverings used in (GOR), we introduce a new analytical method of root curves. The surprising aspect of this proof is that it connects the issue of the existence of analytical false interior branch points with boundary branch points. We should note that this fact was also observed by F. Tomi (to appear) who has found his own very brief proof of (GOR) in the case ΓC 2,α which we also include.


Minimal Surface Branch Point Tangent Plane Relative Minimum Equivalent Point 
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Gulliver, R., Osserman, R., Royden, H.L.

  1. 1.
    A theory of branched immersions of surfaces. Am. J. Math. 95, 750–812 (1973) CrossRefzbMATHMathSciNetGoogle Scholar

Tomi, F.

  1. 1.
    Tomi, F. False branch points revisited (to appear) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

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