On uniqueness of Riemann’s examples

Fang, Yi; Wei, Fusheng

doi:10.1090/S0002-9939-98-04441-4

On uniqueness of Riemann’s examples
HTML articles powered by AMS MathViewer

by Yi Fang and Fusheng Wei PDF

Proc. Amer. Math. Soc. 126 (1998), 1531-1539 Request permission

Abstract:

We prove that a properly embedded minimal annulus with one flat end, bounded in a slab by lines or circles, is a part of a Riemann’s example.

References

J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in $R^{3}$, Amer. J. Math. 98 (1976), no. 2, 515–528. MR 413172, DOI 10.2307/2373899
Michael Callahan, David Hoffman, and William H. Meeks III, The structure of singly-periodic minimal surfaces, Invent. Math. 99 (1990), no. 3, 455–481. MR 1032877, DOI 10.1007/BF01234428
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, and Ortwin Wohlrab, Minimal surfaces. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 295, Springer-Verlag, Berlin, 1992. Boundary value problems. MR 1215267
Yi Fang, On minimal annuli in a slab, Comment. Math. Helv. 69 (1994), no. 3, 417–430. MR 1289335, DOI 10.1007/BF02564495
Yi Fang, Total curvature of branched minimal surfaces, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1895–1898. MR 1322922, DOI 10.1090/S0002-9939-96-03296-0
David Hoffman, Hermann Karcher, and Harold Rosenberg, Embedded minimal annuli in $\textbf {R}^3$ bounded by a pair of straight lines, Comment. Math. Helv. 66 (1991), no. 4, 599–617. MR 1129800, DOI 10.1007/BF02566668
David Hoffman and William H. Meeks III, The asymptotic behavior of properly embedded minimal surfaces of finite topology, J. Amer. Math. Soc. 2 (1989), no. 4, 667–682. MR 1002088, DOI 10.1090/S0894-0347-1989-1002088-X
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
F. López, M. Ritoré and F. Wei. A characterization of Riemann’s minimal surfaces. to appear in Journal of Differential Geometry.
William H. Meeks III, The geometry, topology, and existence of periodic minimal surfaces, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 333–374. MR 1216594, DOI 10.1090/pspum/054.1/1216594
Johannes C. C. Nitsche, Lectures on minimal surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems; Translated from the German by Jerry M. Feinberg; With a German foreword. MR 1015936
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Joaquín Pérez and Antonio Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces, Math. Ann. 295 (1993), no. 3, 513–525. MR 1204835, DOI 10.1007/BF01444900
Pascal Romon, A rigidity theorem for Riemann’s minimal surfaces, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 2, 485–502 (English, with English and French summaries). MR 1220280
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Éric Toubiana, On the minimal surfaces of Riemann, Comment. Math. Helv. 67 (1992), no. 4, 546–570. MR 1185808, DOI 10.1007/BF02566518