Deformations of complete minimal surfaces
HTML articles powered by AMS MathViewer
- by Harold Rosenberg PDF
- Trans. Amer. Math. Soc. 295 (1986), 475-489 Request permission
Abstract:
A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and ${R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.References
-
F. Gacksatter, Über Abelshe Minimalflächen, Math. Nachr. 74 (1976), 157-165.
B. Lawson, Lectures on minimal submanifolds, vol. 1, Math. Lecture Series 9, Publish or Perish, Berkeley, Calif., 1980.
- William H. Meeks III, A survey of the geometric results in the classical theory of minimal surfaces, Bol. Soc. Brasil. Mat. 12 (1981), no. 1, 29–86. MR 671473, DOI 10.1007/BF02588319
- Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278
- Max Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. (2) 63 (1956), 77–90. MR 74695, DOI 10.2307/1969991
- Frederico Xavier, The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere, Ann. of Math. (2) 113 (1981), no. 1, 211–214. MR 604048, DOI 10.2307/1971139
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 295 (1986), 475-489
- MSC: Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9947-1986-0833692-0
- MathSciNet review: 833692