What a pleasure to read, as the first line in the Preface, the phrase, “This monograph is an outgrowth of lectures given by the authors at Princeton University during the academic year 1949–1950 …” Just think of what this means: the material that is to follow is pretty close in spirit to what Niels Hendrik Abel meant when he said that in order to make true progress in mathematics, one needs “to study the masters, not their pupils.” Princeton, in the days right after World War II, was of course a geographical nexus for dramatic mathematical progress, with the same holding for theoretical physics “just down the street.” The only other places that come to mind sharing such a historical distinction are antebellum Göttingen and, at least as far as pure mathematics goes, Paris when Bourbaki descended on the scene some sixty years ago.
As far as the book under review is concerned the masters, or authors, are Menachem Schiffer and Donald Spencer. If I may recount a tangentially relevant anecdote regarding the latter: evidently he was for a little while uncomfortable with sheaves, then blowing in on the wind from France, what with the appearance of Serre’s unparalleled Faisceaux Algébriques Cohérents and, soon after that, the deluge that was Grothendieck’s revolution in commutative algebra and algebraic geometry. However, presently Spencer penetrated to the heart of the subject, saying enthusiastically that, yes, they have algebra “this way” (moving his hands up and down) and topology “that way” (moving his hands sideways). And then it was off to the races, of course.
The point is that Spencer, and in fact the present book, can be regarded as sitting at a juncture in mathematical practice, and for that reason if nothing more (but there is much more, of course), it is highly instructive to read and study Functionals of Finite Riemann Surfaces as a means to get a broader view of the subject than one would obtain from more modern treatments, to the exclusion of a more historical and holistic perspective. Along these lines, this book includes, for example, a closing chapter (of considerable length) on Kähler manifolds, a hot topic if ever there was one, where differential geometry meets physics (and a lot of other things). The book under review is something of a bridge between then and now, to put it colloquially.
Additionally, one of the major themes of Functionals of Finite Riemann Surfaces is the systematic application of methods from the calculus of variations to the low dimensional complex geometry at hand, which makes for any number of tantalizing suggestions and possibilities. This angle on the subject actually appears no earlier than the book’s seventh chapter, only after a very substantial foundation has been lain covering a good deal of general Riemann surface theory, couched in the context of differential geometry proper, and featuring particular themes such as harmonic differentials and eigenvalue problems (and therefore spectral theory) with an added setting of L2-spaces. This is followed by a lot of work on integral operators. And so it is that Schiffer and Spencer say at the start that their “main purpose … is the investigation of finite Riemann surfaces from the point of view of functional analysis, that is, the study of Abelian differentials of the surface in their dependence on the surface itself.”
A note of explanation of their nomenclature: by definition “a finite Riemann surface is a Riemann surface of finite genus with a finite number of nondegenerate boundary components” (as per Lars Ahlfors’ Bull. AMS review, pp. 581–584 in the volume of November, 1955). Speaking of masters, here is Ahlfors’ next comment, pithily characterizing the methodology Schiffer and Spencer employ: “Such a surface has a double [cf. p. 29ff., of the book under review], obtained by reflection across the boundary, and one of the main features of the book is the systematic use of this symmetrization process.” Nothing if not tantalizing.
Finally, as I already hinted above, Schiffer and Spencer are careful to build a strong foundation for what they do, and this is part and parcel of the book’s opening three chapters, taking the reader form the exterior calculus on Riemann surfaces, and a consideration of integration in that context, to period relations, Riemann-Roch, and conformality. This sets the stage for the deep and elegant material in the rest of the monograph. Indeed, brimming with exciting material that has aged well, Functionals of Finite Riemann Surfaces is truly a gem of a book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.