Descripción
First edition, the rare offprint issue, of Riemann's important paper on minimal surfaces (i.e. surfaces of least area for a given boundary). "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, Bernhard Riemann 1826-1866 (1999), p. 142). "What [Riemann's] 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 (1826-66) 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 ('Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse', 1851). 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 1851 dissertation "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]" (Bottazzini & Gray, Hidden Harmony - Geometric Fantasies. The Rise of Complex Function Theory (2013), p. 540). "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" (ibid., p. 542). This problem was not solved in full generality until 1930. The simplest non-trivial case, that of a space quadrilateral, and with it the first non-trivial solution of Plateau's problem, is usually credited to Hermann Amandus Schwarz (1843-1921), but Riemann anticipated him in the present work: Schwarz's work was carried out in 1867, and won him a prize of the Berlin Academy, but it was not published until four years later. 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. 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. 4to, pp. 52. Stitched, with green paper spine strip, as issued, uncut and mostly unopened (first and last few leaves with light browning, soiling and faint damp staining, corners slightly worn). N° de ref. del artículo ABE-1541881128444
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