Abstract
Three months before his death in 1866, Riemann left a set of notes to K. Hattendorff, a disciple of his, on minimal surfaces with boundary. Afterwards, Hattendorff supplied the text to the notes mostly consisting of computations, which became the two papers on the subject: “On the surface of least area with a given boundary” and “Examples of surfaces of least area with a given boundary.” We will go over the expositions and provide an overview from the modern viewpoint, make some comments on Riemann-Hattendorff’s text, and compare the work with that of Weierstrass on the same subject.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Ahlfors, Complex Analysis, International Series in Pure & Applied Mathematics (McGraw-Hill, 1979)
R. Courant, Dirichlet’s Principle, Conformal Mapping and Minimal Surafaces, Pure and applied mathematics, 3, (Interscience, 1950)
B. Daniel, Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space. J. Differ. Geom. 72, 467–508 (2006)
C.F. Gauss, General Investigations of Curved Surfaces (1825) and (1827). Princeton University Library, also available as General Investigations of Curved Surfaces (Dover, 2005)
H. Lewy, Introduction in Collected Works of Bernhard Riemann (Dover, 1953)
W.H. Meeks III, J. Pérez, The Riemann minimal examples, in The Legacy of Bernhard Riemann After One Hundred and Fifty Years ALM vol. 35, pp. 417–457
W.H. Meeks III, J. Pérez, A. Ros, Properly embedded minimal planar domains. Ann. Math. 181, 473–546 (2015)
B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, dissertation, Göttingen (1851)
B. Riemann, Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung (On the surface of least area with a given boundary) Abh. Königl, d. Wiss. Göttingen Mathem. Cl. 13, 3–52 (1867)
B. Riemann, Beispiele von Flächen kleinsten Inhalts bei gegebener Begrenzung (Examples of surfaces of least area with a given boundary), Chap. XXVII in Gesammelte mathematishe Werke 2nd edn. (B.G. Teubner, Leibzig, 1892)
J.C.C. Nitsche, Lectures on Minimal Surfaces: Volume 1, Introduction, Fundamentals, Geometry and Basic Boundary Value Problems (Cambridge University Press, 2011)
R. Osserman, A Survey of Minimal Surfaces (Dover, 1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Yamada, S. (2017). Riemann’s Work on Minimal Surfaces. In: Ji, L., Papadopoulos, A., Yamada, S. (eds) From Riemann to Differential Geometry and Relativity. Springer, Cham. https://doi.org/10.1007/978-3-319-60039-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-60039-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-60038-3
Online ISBN: 978-3-319-60039-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)