Abstract
Minimal surfaces in a Riemannian manifold \(M^n\) are surfaces which are stationary for area: the first variation of area vanishes. In this paper, we treat two topics on branch points of minimal surfaces. In the first, we show that a minimal surface \(f:\mathbb RP^2\rightarrow M^3\) which has the smallest area, among those mappings from the projective plane which are not homotopic to a constant mapping, is an immersion. That is, \(f\) is free of branch points, including especially false branch points. As a major step toward treating minimal surfaces of the type of the projective plane, we extend the fundamental theorem of branched immersions to the nonorientable case. In the second topic, we resolve, in the negative, a question on the directions of curves of self-intersection at a true branch point, which was posed by Courant (Dirichlet’s principle, conformal mapping and minimal surfaces. Wiley, New York, 1950).
Similar content being viewed by others
References
Alt, W.H.: Verzweigungspunkte von H-Flächen I. Math. Z. 127, 333–362 (1972)
Alt, W.H.: Verzweigungspunkte von H-Flächen II. Math. Annalen 201, 33–56 (1973)
Bray, H., Brendle, S., Eichmair, M., Neves, A.: Area-minimizing projective planes in 3-manifolds. Commun. Pure Appl. Math. 63, 1237–1247 (2010)
Courant, R.: On a generalized form of Plateau’s problem. Trans. Am. Math. Soc. 50, 40–47 (1941)
Courant, R.: Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces. Wiley, New York (1950)
Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33, 263–321 (1931)
Douglas, J.: One-sided minimal surfaces with a given boundary. Trans. Am. Math. Soc. 34, 731–756 (1932)
Douglas, J.: Minimal surfaces of higher topological structure. Ann. Math. 40, 205–298 (1939)
Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97, 275–305 (1973)
Gulliver, R.: Branched immersions of surfaces and reduction of topological type, I. Math. Z. 145, 267–288 (1975)
Gulliver, R.: Branched immersions of surfaces and reduction of topological type, II. Math. Ann. 230, 25–48 (1977)
Gulliver, R., Osserman, R., Royden, H.: A theory of branched immersions of surfaces. Am. J. Math 95, 750–812 (1973)
Hartman, P., Wintner, A.: On the local behavior of solutions of non-parabolic partial differential equations. Am. J. Math. 75, 449–476 (1953)
Osserman, R.: A Survey of Minimal Surfaces. Van Nostrand Reinhold, New York (1969)
Osserman, R.: A proof of the regularity everywhere of the classical solution to Plateau’s problem. Ann. Math. 91, 550–569 (1970)
Radó, T.: The problem of least area and the problem of Plateau. Math. Z. 32, 763–796 (1930)
Tromba, A.: A Theory of Branched Minimal Surfaces. Springer, New York (2012)
Zaslevsky, T.: The largest parity demigenus of a simplicial graph. J. Comb. Theory 70, 325–345 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Robert Osserman.
Rights and permissions
About this article
Cite this article
Gulliver, R. Branch points of area-minimizing projective planes and a question of Courant. Math. Ann. 362, 389–400 (2015). https://doi.org/10.1007/s00208-014-1121-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-014-1121-8