Göttingen: Dieterich, 1867.
First edition, the very rare offprint issue, of Riemann’s important paper on minimal surfaces (i.e. surfaces of least area for a given boundary). “What his work lacks in quantity is more than compensated for by its superb quality. One of the most profound and imaginative mathematicians of all time …” (DSB). The research first published in this paper was carried out in 1860-61; this delay in publication cost Riemann the credit for several fundamental discoveries contained in the present work. Most importantly, Riemann was the first to understand the intimate relationship between minimal surfaces and complex analytic functions, later credited to Karl Weierstrass (1815-97). Indeed, the present paper is a natural outgrowth of Riemann’s landmark doctoral dissertation on complex function theory (1851). Riemann also anticipated Hermann Amandus Schwarz (1843-1921) in the construction of a new family of minimal surfaces which provided the first solution to a non-trivial case of Plateau’s problem, the problem of finding a minimal surface having a given curve as its boundary (this problem was not solved in full generality until 1930). “The problem involves geometry and physics, and its treatment uses real and complex analysis. In other words, problem and treatment involve almost all of Riemann’s work areas” (Laugwitz, p. 142). ABPC/RBH list only a single copy.
The French mathematician Ossian Bonnet showed in 1860 a close connection between minimal surfaces and conformal mappings (transformations of one surface to another that preserve angles). Such mappings had been a major theme in Riemann’s great work of 1851 on complex function theory (Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse), “but neither [Bonnet] nor Minding before him took the step of thinking in terms of complex analytic maps, which suggests that even in 1860 such a connection might not have been generally appreciated. Instead, the first to put the picture together were the two leading complex analysts of the day, Riemann and Weierstrass. They worked independently. Riemann’s account was entrusted by him to [Karl] Hattendorff for editing in April 1866, but apparently dates from 1860 to 1861. The original manuscript consists entirely of formulae, and Hattendorff supplied a text; the result was published in 1867 as [the present work]. Weierstrass’s account was first given in a lecture at the Berlin Academy in 1866” (Bottazzini & Gray, p. 540). Shortly after handing his manuscript to Hattendorff, Riemann fled Göttingen when the armies of Hanover and Prussia clashed there at the start of the Austro-Prussian War. He was in any case probably too ill to work on the manuscript himself, having contracted tuberculosis in 1862. He died in Italy on 20 July, 1866.
“The much harder problem is to find a minimal surface that spans a given contour; this problem is called the Plateau problem after the blind Belgian physicist Joseph Plateau who, in the 1840s and 1850s had drawn attention to the fact that a soap film spanning a wire frame will take up the shape of a minimal surface. The contours can have quite arbitrary shapes. The disparity between mathematicians’ ability to solve the Plateau problem and the behaviour of soap films was a potent challenge, but even Riemann found the task of finding the right surface for a given boundary curve daunting. He was able to give explicit solutions only for simple boundaries: two skew lines in space (when the surface is a piece of a helicoid); two intersecting lines and a third lying in a plane parallel to the first two, three skew lines (which led to a generalization of the Riemann P-function); the space quadrilateral (later studied by Schwarz), and two circles in two parallel planes” (ibid., p. 542). The most interesting case, that of a space quadrilateral, and with it the first non-trivial solution of Plateau’s problem, is usually credited to Schwarz (see, for example, Rassias, p. 153). Schwarz’s work was carried out in 1867, and won him a prize of the Berlin Academy, but his work was not published until four years later (Bestimmung einer speziellen Minimalfläche, Berlin: Dümmler, 1871), so in this case Riemann’s work was not only carried out earlier, but also published first.
In section 12 of the present paper, Riemann also anticipated Schwarz in the discovery of the ‘reflection principle’ for analytic functions, later also named for Schwarz. This principle allows one to extend a complex analytic function initially defined on one side of a curve, and with real values along the curve, to an analytic function everywhere. “Riemann was interested in calculating a certain expression he denoted du/dt. He commented: ‘In order to form the expression for du/dt, we must observe that dt is always real along the boundary, and du is either real or purely imaginary. Hence (du/dt)2 is real when t is real. This function can be continuously extended across the line of real values of t by the condition that, for conjugate values of the variable, the function will have conjugate values. Then (du/dt)2 is determined for the whole t-plane and turns out to be single-valued’” (Bottazzini & Gray, p. 542).
The theory of minimal surfaces underwent at least two revivals in the twentieth century. The first major milestone was the complete solution of the Plateau problem by Jesse Douglas and Tibor Radó. The subject revived again in the 1980s when Celso Costa disproved the conjecture that the plane, catenoid and helicoid are the only ‘complete embedded minimal surfaces of finite topology’. Costa’s work demonstrated the importance of computer graphics to visualize minimal surfaces and to suggest the existence of new ones. In the present century minimal surfaces are studied in higher dimensions, and have become relevant to mathematical physics (e.g. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. the Smith conjecture, the Poincare conjecture, and the Thurston geometrization conjecture).
Bottazzini & Gray, Hidden Harmony – Geometric Fantasies. The Rise of Complex Function Theory, 2013. Laugwitz, Bernhard Riemann 1826-1866, 1999; Rassias (ed.), The Problem of Plateau: A Tribute to Jesse Douglas and Tibor Radó, 1992.
Offprint from Abhandlung der königliche Gesellschaft der Wissenschaften zu Göttingen 13 (1867). 4to (260 x 210 mm), pp. 52. Stitched, with green paper spine strip (as issued). One gathering has become loose. Light vertical fold.