Elliptic functions and nonexistence of complete minimal surfaces of certain type
HTML articles powered by AMS MathViewer
- by Chi Cheng Chen
- Proc. Amer. Math. Soc. 79 (1980), 289-293
- DOI: https://doi.org/10.1090/S0002-9939-1980-0565356-5
- PDF | Request permission
Abstract:
It is proved that any complete minimal surfaces in ${{\mathbf {R}}^n}(n \geqslant 3)$ with total curvature $- 4\pi$ is conformally equivalent to the complex plane or the punctured plane, just like the case in ${{\mathbf {R}}^3}$.References
- E. Calabi, Quelques applications l’analyse complexe aux surface d’aire minima, Topics in Complex Manifolds, Les Presses de L’Université de Montréal, Montréal, 1968.
- Chi Cheng Chen, Complete minimal surfaces with total curvature $-2\pi$, Bol. Soc. Brasil. Mat. 10 (1979), no. 1, 71–76. MR 552036, DOI 10.1007/BF02588341
- Chi Cheng Chen, A characterization of the catenoid, An. Acad. Brasil. Ciênc. 51 (1979), no. 1, 1–3. MR 533576
- Shiing-shen Chern and Robert Osserman, Complete minimal surfaces in euclidean $n$-space, J. Analyse Math. 19 (1967), 15–34. MR 226514, DOI 10.1007/BF02788707
- Fritz Gackstatter, Über die Dimension einer Minimalfläche und zur Ungleichung von St. Cohn-Vossen, Arch. Rational Mech. Anal. 61 (1976), no. 2, 141–152 (German). MR 420447, DOI 10.1007/BF00249702
- Tilla Klotz and Leo Sario, Existence of complete minimal surfaces of arbitrary connectivity and genus, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 42–44. MR 178408, DOI 10.1073/pnas.54.1.42
- Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
- Robert Osserman, Global properties of minimal surfaces in $E^{3}$ and $E^{n}$, Ann. of Math. (2) 80 (1964), 340–364. MR 179701, DOI 10.2307/1970396
Bibliographic Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 289-293
- MSC: Primary 53A10
- DOI: https://doi.org/10.1090/S0002-9939-1980-0565356-5
- MathSciNet review: 565356