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The Mathematics of Lars Valerian Ahlfors Frederick Gehring; Irwin Kra; Steven G

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The Mathematics of Lars Valerian Ahlfors Frederick Gehring; Irwin Kra; Steven G The Mathematics of Lars Valerian Ahlfors Frederick Gehring; Irwin Kra; Steven G. Krantz, Editor; and Robert Osserman Lars Valerian Ahlfors died October 11, 1996, and function theory to bring to bear in studying the a memorial article appears elsewhere in this issue. geometry of the surface. The most notable suc- This feature article contains three essays about dif- cesses of this approach have been in the study of ferent aspects of his mathematics. Ahlfors’s con- minimal surfaces, as exemplified in the contribu- tributions were so substantial and so diverse that tions to that subject made by some of the leading it would not be possible to do all of them justice in function theorists of the nineteenth century: Rie- an article of this scope (the reference [36] serves as mann, Weierstrass, and Schwarz. a detailed map of Ahlfors’s contributions to the In the other direction, given a Riemann surface, subject). These essays give the flavor of some of the one can consider those metrics on the surface that ideas that Ahlfors studied. induce the given conformal structure. By the Koebe —Steven G. Krantz, Editor uniformization theorem, such metrics always exist. In fact, for “classical Riemann surfaces” of the sort originally considered by Riemann, which are Conformal branched covering surfaces of the plane, there is the natural euclidean metric obtained by pulling back the standard metric on the plane under the Geometry projection map. One can also consider the Rie- Robert Osserman mann surface to lie over the Riemann sphere model of the extended complex plane and to lift the There are two directions in which one can pursue spherical metric to the surface. Both of those met- the relations between Riemann surfaces and Rie- rics prove very useful for obtaining information mannian manifolds. First, if a two-dimensional about the complex structure of the surface. Riemannian manifold is given, then not only lengths For simply connected Riemann surfaces the but also angles are well defined, so that it inher- Koebe uniformization theorem tells us that they its a conformal structure. Furthermore, there al- are all conformally equivalent to the sphere, the ways exist local isothermal coordinates, which are plane, or the unit disk. Since the first case is dis- local conformal maps from the plane into the sur- tinguished from the other two by the topological face. (That was proved by Gauss for real analytic property of compactness, the interesting question surfaces and early in the twentieth century for the concerning complex structure is deciding in the general case by Korn and Lichtenstein.) The set of noncompact case whether a given surface is con- all such local maps forms a complex structure for formally the plane or the disk, which became the manifold, which can then be thought of as a known as the parabolic and hyperbolic cases, re- Riemann surface. One then has all of complex spectively. In 1932 Andreas Speiser formulated the “problem of type”, which was to find criteria Steven G. Krantz is professor of mathematics at Wash- that could be applied to various classes of Riemann ington University in St. Louis. His e-mail address is surfaces to decide whether a given one was para- [email protected]. bolic or hyperbolic. That problem and variants of Robert Osserman is professor of mathematics emeritus at it became a central focus of Ahlfors’s work for sev- Stanford University and special projects director at MSRI. eral decades. He started by obtaining conditions FEBRUARY 1998 NOTICES OF THE AMS 233 for a branched prising: that Nevanlinna could develop his entire surface to be of theory without the geometric picture to go with it parabolic type in or that Ahlfors could condense the whole theory terms of the into fourteen pages. number of Not satisfied that he had yet got to the heart of branch points Nevanlinna theory from a geometric point of view, within a given Ahlfors went on to present two further versions distance of a of the theory. The first, also from 1935, was one fixed point on of his masterpieces: the theory of covering surfaces. the surface, first The guiding intuition of the paper is this: If a mero- using the euclid- morphic function is given, then the fundamental ean metric and quantities studied in Nevanlinna theory, such as later realizing the counting function, determined by the number that a much bet- of points inside a disk of given radius where the ter result could function assumes a given value, and the Nevanlinna be obtained characteristic function, measuring the growth of from the spher- the function, can be reinterpreted as properties of ical metric. But the Riemann surface of the image of the function, perhaps his viewed as a covering surface of the Riemann sphere. main insight The counting function, for example, just tells how was that one many points in the part of the image surface given could give a nec- by the image of the disk lie over the given point essary and suf- on the sphere. The exhaustion of the plane by Lars Valerian Ahlfors ficient condition disks of increasing radii is replaced by an ex- by looking at the haustion of the image surface. Ahlfors succeeds in totality of all showing that by using a combination of metric conformal metrics on the surface. and topological arguments (the metric being that The problem of type may be viewed as a spe- of the sphere and its lift to the covering surface), cial case of the general problem of finding con- one can not only recover basically all of standard formal invariants. There one has some class of Nevanlinna theory but that—quite astonishingly— topologically defined objects, such as a simply or the essential parts of the theory all extend to a far double-connected domain or a simply connected wider class of functions than the very rigid spe- domain with boundary and four distinguished cial case of meromorphic functions, namely, to points on the boundary, and one seeks to define functions that Ahlfors calls “quasiconformal”; in quantities that determine when two topologically this theory the smoothness requirements may be equivalent configurations are conformally equiv- almost entirely dropped, and, asymptotically, im- alent. One example is the “extremal length” of a ages of small circles—rather than having to be cir- family of curves in a domain, which is defined by cles—can be arbitrary ellipses as long as the ratio a minimax expression in terms of all conformal of the radii remains uniformly bounded. metrics on the domain and is thereby automatically Of his three geometric versions of Nevanlinna a conformal invariant. Ahlfors and Beurling, first theory, Ahlfors has described the one on covering independently and then jointly, developed the idea surfaces as a “much more radical departure from into a very useful tool that has since found many Nevanlinna’s own methods” and as “the most orig- further applications. inal of the three papers,” which is certainly the case. From the first, Ahlfors viewed classical results (According to Carathéodory that paper was singled like Picard’s theorem and Bloch’s theorem as spe- out in the decision of the selection committee to cial cases of the problem of type, in which condi- award the Fields Medal to Ahlfors.) Nevertheless, tions such as the projection of a Riemann surface the last of the three, published two years later in omitting a certain number of points would imply 1937, was destined to be probably at least as in- that the surface was hyperbolic and hence could fluential. Here the goal was to apply the methods not be the image of a function defined in the whole of differential geometry to the study of covering plane. He felt that Nevanlinna theory should also surfaces. The paper is basically a symphony on the be fit into that framework. Finally, in 1935 he pro- theme of Gauss-Bonnet. The explicit relation be- duced one of his most important papers, in which tween topology and total curvature of a surface, he used the idea of specially constructed confor- now called the “Gauss-Bonnet theorem”, had not mal metrics (or “mass distributions” in the termi- been around all that long at the time, perhaps first nology of that paper) to give his own geometric ver- appearing in Blaschke’s 1921 Vorlesungen über sion of Nevanlinna theory. When he received his Differentialgeometrie [15]. Fields Medal the following year, Carathéodory re- It occurred to Ahlfors that if one hoped to de- marked that it was hard to say which was more sur- velop a higher-dimensional version of Nevanlinna 234 NOTICES OF THE AMS VOLUME 45, NUMBER 2 theory, it might be useful to have a higher-di- thereby get mensional Gauss-Bonnet formula, a fact that he the size of mentioned to André Weil in 1939, as Weil recounts the largest in his collected works. (A letter from Weil says, “I circular disk learnt from Ahlfors, in 1939, all the little I ever in the image, knew about Gauss-Bonnet (in dimension 2).”) When which is just Weil spent the year 1941–42 at Haverford, where the circum- Allendoerfer was teaching, he heard of Allen- scribed circle doerfer’s proof of the higher-dimensional Gauss- of one of the Bonnet theorem; and remembering Ahlfors’s sug- equilateral gestion, he worked with Allendoerfer on their joint triangles paper, proving the generalized Gauss-Bonnet the- whose ver- orem for a general class of manifolds that need not tices are the be embedded. That in turn led to Chern’s famous branch intrinsic proof of the general Gauss-Bonnet theo- points of the rem.
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