TY - JOUR
T1 - Branch points of area-minimizing projective planes and a question of Courant
AU - Gulliver, Robert
N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2015/6/1
Y1 - 2015/6/1
N2 - Minimal surfaces in a Riemannian manifold Mn 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:RP2→M3 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).
AB - Minimal surfaces in a Riemannian manifold Mn 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:RP2→M3 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).
KW - 49Q05
KW - 53A10
KW - 58E12
UR - http://www.scopus.com/inward/record.url?scp=84939940816&partnerID=8YFLogxK
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U2 - 10.1007/s00208-014-1121-8
DO - 10.1007/s00208-014-1121-8
M3 - Article
AN - SCOPUS:84939940816
SN - 0025-5831
VL - 362
SP - 389
EP - 400
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -