Biographical Memoirs Volume 87 (2005) / Chapter Skim
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Lars Valerian Ahlfors
Lars Valerian Ahlfors
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Biographical Memoirs VOLUME 87
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For historical reasons the inhabitants of Finland are divided into those who have Finnish or Swedish as their mother tongue. The Ahlfors family was Swedish speaking, so Lars attended a private school where all classes were taught in Swedish.
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In 1924 Lars Ahlfors entered the University of Helsinki, where his teachers were two internationally known mathematicians, Ernst Lindelöf and Rolf Nevanlinna. At that time the university was still run on the system of one professor for each subject.
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Half a century later, in the preface to his Collected Papers, Lars wrote, "This was the happiest and most important event in my life." In 1935 Lars was first offered and then accepted a threeyear appointment at Harvard University as a visiting lecturer. One year later, at the quadriennal International Congress of Mathematicians in Oslo, he was awarded a Fields Medal, the equivalent for mathematicians of a Nobel Prize.
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The first two Fields Medals were then awarded to Lars Ahlfors and Jesse Douglas of the United States in 1936. That he was to be awarded a Fields Medal came as a complete surprise for Lars.
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The set of all such local maps forms a complex structure for the manifolds, which can then be thought of as a Riemann surface. One then has all of complex function theory to bring to bear in studying the geometry of the surface.
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In fact, for "classical Riemann surfaces" of the sort originally considered by Riemann, which are branched covering surfaces of the plane, there is the natural Euclidean metric obtained by pulling back the standard metric on the plane under the projection map. One can also consider the Riemann surface to lie over the Riemann sphere and to lift the spherical metric to the surface.
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to give his own geometric version of Nevanlinna theory. When he received his Fields Medal the following year, Carathéodory remarked that it was hard to say which was more surprising: that Nevanlinna could develop his entire theory without the geometric picture to go with it or that Ahlfors could condense the whole theory into 14 pages.
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, one can not only recover basically all of standard Nevanlinna theory but that -- quite astonishingly -- the essential parts of the theory all extend to a far wider class of functions than the very rigid special case of meromorphic functions, namely, to functions that Ahlfors calls "quasiconformal"; in this theory the smoothness requirements may be almost entirely dropped, and asymptotically, images of small circles -- rather than having to be circles -- can be arbitrary ellipses as long as the ratio of the radii remains uniformly bounded. Of his three geometric versions of Nevanlinna theory, Ahlfors has described the one on covering surfaces as a "much more radical departure from Nevanlinna's own methods" and as "the most original of the three papers," which is certainly the case.
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As for Ahlfors's idea of adapting the method to obtain a higher-dimensional Nevanlinna theory, that had to wait until the paper by Bott and Chern in 1963. The year following his Gauss-Bonnet Nevanlinna theory paper, Ahlfors published a deceptively short and unassuming paper called "An Extension of Schwarz's Lemma" (1938)
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. The conjectured extremal function maps the unit disk onto a Riemann surface with simple branch points in every sheet over the lattice formed by the vertices obtained by repeated reflection over the sides of an equilateral triangle, where the center of the unit disk maps onto the center of one of the triangles.
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It demonstrates perhaps more strikingly than anywhere else the power that Ahlfors was able to derive from his unique skill in melding the complex analysis of Riemann surfaces with the metric approach of Riemannian geometry. KLEINIAN GROUPS BY IRWIN KRA Of the many significant contributions of Lars Ahlfors to the modern theory of Kleinian groups, I will discuss only two, which are closely related: the Ahlfors finiteness theory and the use of Eichler cohomology as a tool for proving this and related results.
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The study of subgroups of PSL(2,C) was successful because of its connection to classical function theory and to 2-dimensional topology and geometry, about which a lot was known, including the uniformization theorem classifying all simply connected Riemann surfaces.
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Bers constructs Eichler cohomology classes from analytic potentials by integrating cusp forms sufficiently many times, using methods developed by Eichler (1957) for number theory.
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. Since the minimal area of a hyperbolic orbifold is p/21, Bers's area theorem gives an upper bound on the number of connected Riemann surfaces represented by a nonelementary Kleinian groups as 84(N 1)
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to establish the measure zero conjecture led Sullivan to prove a finiteness theorem on the number of maximal conjugacy classes of purely parabolic subgroups of a Kleinian group. The measure zero problem not only opened up a new industry in the Kleinian groups "industrial park," it also revived the connection with 3-dimensional topology following the fundamental work of Marden and Thurston.
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Teichmüller's theorem concerned the nature of the quasiconformal mappings between two Riemann surfaces S and S with minimum maximal dilatation. This and the fact that any useful theory that generalizes conformal mappings should have compactness and reflection properties led Ahlfors to formulate a geometric definition that was free of all a priori smoothness hypotheses.
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His inspired idea to drop all analytic hypotheses eventually led to striking applications of these mappings in other parts of complex analysis such as discontinuous groups, classical function theory, complex iteration, as well as in other fields of mathematics, including harmonic analysis, partial differential equations, differential geometry, and topology.
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Mathematicians asked whether this conclusion holds when f is K quasiconformal. By composing f with a pair of conformal mappings, one can reduce the problem to the case where D = D = H, where H is the upper half plane and f()
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The main result of this article, joint with Lipman Bers, states that for any function m that is measurable with |m| k < 1 a.e. in D, there exists a K quasiconformal mapping f of D that has m as its complex dilatation.
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This theorem may already have become a verb in complex dynamics. For at a plenary lecture at the International Congress of Mathematicians in 1986 a distinguished French mathematician Adrien Douady was heard to explain that "before mating two polynomials, one must first Ahlfors-Bers the structure." QUASICONFORMAL REFLECTIONS (1963)
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. It also led to another surprising connection between quasiconformal mappings and classical function theory, namely, that a simply connected domain D is a quasidisk if and only if each function f, analytic with small Schwarzian derivative Sf in D, is injective.
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In discussing Lars's work, Carathédory said that this article opened a completely new chapter in analysis, one that could be called "metric topology." In a commentary on this article Lars wrote, "Little did I know at the time what an important role quasiconformal mappings would come to play in my own work." Lars's geometric approach to quasiconformal mappings stimulated their development in higher dimensional Euclidean space and recently in general spaces, such as the Heisenberg and Carnot groups. The fact that this class is so natural and flexible has led to striking applications to Kleinian groups, classical function theory, complex dynamics, and to other parts of mathematics, including harmonic analysis, differential geometry, elasticity, and topology.
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In describing the mathematical achievements of the prizewinner, the jury ended its report with the following final comment: "Everyone working today in complex analysis is in some sense a student of Ahlfors." At each International Congress of Mathematicians it is customary for the host country to nominate its most prestigious mathematician as honorary president of the congress. This congress was held in the United States in 1986, 50 years after Lars had been awarded the Fields Medal in Oslo.
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1980. On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics.
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The boundary correspondence under quasiconformal mappings.
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15:257-263. 1969 The structure of a finitely generated Kleinian group.
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