×

About the cover: the work of Jesse Douglas on minimal surfaces. (English) Zbl 1138.01011

Jesse Douglas was awarded one of the first Fields medals for his work on the Plateau problem of the existence of a least-area surface spanning a given contour. In this nicely written note, the authors analyze, both from historical and mathematical points of view, major contributions to this challenging problem concluding that ”Radó and Douglas share equal credit for solving the Plateau problem for disc-like surfaces spanning a single contour which bounds at least one disc-like surface of finite area.” However, ”Douglas’s contributions to the Plateau problem are more major, broader and deeper than those of Radó. Douglas’s ideas, as developed later by Courant (who brought the Dirichlet integral back to the forefront), have remained important in the theory of minimal surfaces up to the present.”

MSC:

01A60 History of mathematics in the 20th century
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Biographic References:

Douglas, Jesse
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263 – 321. · Zbl 0001.14102
[2] Douglas, J., 1931b, The Problem of Plateau for Two Contours, Journal of Mathematics and Physics 10, 315-359. · Zbl 0004.15403
[3] Jesse Douglas, Minimal surfaces of higher topological structure, Ann. of Math. (2) 40 (1939), no. 1, 205 – 298. · Zbl 0020.37402 · doi:10.2307/1968552
[4] (1927, 143-4, 32) Reduction of the problem of Plateau to an integral equation.
[5] (1927, 259, 5) Reduction to integral equations of the problem of Plateau for the case of two contours. · JFM 53.0385.09
[6] (1928, 405, 5) Reduction of the problem of Plateau to the minimization of a certain functional. \begin{extrabibtext} Other authors \end{extrabibtext}
[7] R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math. 72 (1940), 51 – 98. · JFM 66.0485.02 · doi:10.1007/BF02546328
[8] René Garnier, Le problème de Plateau, Ann. Sci. École Norm. Sup. (3) 45 (1928), 53 – 144 (French). · JFM 54.0748.04
[9] Gray, J.J., and Micallef, M.J., Minimal surfaces and the work of Jesse Douglas, book, in preparation.
[10] Stefan Hildebrandt and Heiko von der Mosel, On two-dimensional parametric variational problems, Calc. Var. Partial Differential Equations 9 (1999), no. 3, 249 – 267. · Zbl 0934.49022 · doi:10.1007/s005260050140
[11] Jürgen Jost, Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reine Angew. Math. 359 (1985), 37 – 54. · Zbl 0568.49025 · doi:10.1515/crll.1985.359.37
[12] Koebe, P., 1917, Abhandlungen zur Theorie der konformen Abbildung, III Der allgemeine Fundamentalsatz der konformen Abbildung nebst einer Anwendung auf die konforme Abbildung der Oberfläche einer körperlichen Ecke, Journal für die reine und angewandte Mathematik 147, 67-104. · JFM 46.0545.01
[13] Paul Koebe, Allgemeine Theorie der Riemannschen Mannigfaltigkeiten, Acta Math. 50 (1927), no. 1, 27 – 157 (German). Konforme Abbildung und Uniformisierung. · JFM 53.0320.01 · doi:10.1007/BF02421322
[14] Frank Morgan, On the singular structure of two-dimensional area minimizing surfaces in \?\(^{n}\), Math. Ann. 261 (1982), no. 1, 101 – 110. · Zbl 0549.49029 · doi:10.1007/BF01456413
[15] Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807 – 851. · Zbl 0033.39601 · doi:10.2307/1969401
[16] Radó, T., 1930a, Some remarks on the problem of Plateau, Proceedings of the National Academy of Sciences 16, 242-248. · JFM 56.0437.01
[17] Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457 – 469. · JFM 56.0437.02 · doi:10.2307/1968237
[18] Tibor Radó, The problem of the least area and the problem of Plateau, Math. Z. 32 (1930), no. 1, 763 – 796. · JFM 56.0436.01 · doi:10.1007/BF01194665
[19] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1 – 24. · Zbl 0462.58014 · doi:10.2307/1971131
[20] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127 – 142. · Zbl 0431.53051 · doi:10.2307/1971247
[21] Friedrich Tomi and Anthony J. Tromba, Existence theorems for minimal surfaces of nonzero genus spanning a contour, Mem. Amer. Math. Soc. 71 (1988), no. 382, iv+83. · Zbl 0638.58004 · doi:10.1090/memo/0382
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.