# Ahlfors, Lars

# AHLFORS, LARS

(*b.*Helsigfors, Finland, 18 April 1907; *d*. Pittsfield, Massachusetts, 11 October 1996),

*mathematics, complex function theory, Riemann surface theory, conformal geometry, quasiconformal mappings, theory of Kleinian groups*.

Ahlfors was one of the leaders in the field of complex function theory for more than fifty years. Among his earliest successes was a geometric reformulation and deepening of Rolf Nevanlinna’s profound study of the values of complex functions. Later he was instrumental in the revival of interest in the topic of Kleinian groups, and he enriched function theory with the techniques in the theory of quasiconformal mappings. He was also an influential teacher and textbook writer.

**Life and Career.** Ahlfors’s mother, Sievä Helander, died giving birth to him, and his father, Axel Ahlfors, an engineer, sent him to two aunts for his early upbringing. He soon showed an aptitude for mathematics, doing his older sisters’ homework for them and quickly outstripping his teachers at the local school. He also read mathematics books on clandestine trips to his father’s library, which is how he came to learn the calculus. When Lars went to Helsinki University in 1924, he studied mathematics under Ernst Lindelöf and Rolf Nevanlinna, two of the major figures in the field of complex function theory, a branch of mathematics that was particularly cherished in Finland at the time. At age twenty-one, Ahlfors followed Nevanlinna to the Eidgenösische Technische Hochschule (ETH, also known as the Zürich Polytechnic School) in Zürich, Switzerland, where he began to do original work on his own, and he was awarded his PhD in 1930 from Helsinki University for the work he did in Zürich. In 1933 he returned from Zürich to Helsinki and married Erna Lehnert; the marriage proved to be a happy one and ended only with his death.

In 1935 Ahlfors took up a three-year teaching position at Harvard University in Cambridge, Massachusetts, and in 1936 he was awarded one of the first two Fields Medals at the International Congress of Mathematicians in Oslo, Norway, for his work on complex function theory. But he became homesick and returned to Helsinki in 1938, where he spent most of the World War II as a professor at Helsinki University before being appointed to a professorship at the University of Zürich in 1944. He had to pawn his Fields Medal to get enough money to make the trip; later, some Swiss mathematicians helped him retrieve it.

In 1946 Ahlfors returned to Harvard, becoming the William Caspar Graustein Professor in 1964, a position he held until his retirement in 1977. The first edition of his book *Complex Analysis* came out in 1953; it ran to three editions and was for decades the incontestable standard introduction to the subject. Students attending his lectures, delivered in a thundering basso and in a strong Scandinavian accent, recalled them as stunningly beautiful. In 1981 he was awarded a prestigious Wolf Prize in mathematics “for seminal discoveries and the creation of powerful new methods in geometric function theory.”

**Complex Function Theory.** A striking result of Émile Picard’s in 1880 shows that a meromorphic function that fails to take three values in the extended complex plane is in fact constant. Much energy was expended over the next forty years in finding elementary proofs of this result and exploring its implications, and the definitive account was given only in 1925 by Rolf Nevanlinna. The theory, called value distribution theory, is both profound and technically very difficult. Ahlfors earned his Fields Medal in 1936 in part because of the way his use of geometric ideas greatly simplified it. In presenting him with the medal, Constantin Carathéodory said it was hard to say if it was more surprising that Nevanlinna could develop his theory without the aid of geometry or that Ahlfors could condense the whole theory to a mere fourteen pages.

In his 1935 account, “Zur Theorie der Überlagerungsflächen,” Ahlfors introduced geometry by regarding the image of a meromorphic function as a covering surface of the Riemann sphere. He then interpreted the fundamental concepts of Nevanlinna theory as properties of the covering surface. For example, Nevanlinna’s counting function says how many times the meromorphic function on a disc of a given radius takes a given value. Geometrically, this says how many points on the image surface lie over a given point of the Riemann sphere. Ahlfors then derived the main results in what became a characteristic method of his: exploiting a combination of metrical and topological arguments. But he did more, because his formulation of the theory extended it to a much wider class of functions than the meromorphic ones, namely the quasiconformal mappings. A further reworking of the Nevanlinna theory made greater use of the methods of differential geometry in the hope that a generalization to higher dimensions could be found. But that had to wait almost thirty years, with the work of Raoul Bott and Shiing-Shen Chern in 1965 and then the work of Phillip Griffiths.

**Quasiconformal Mappings.** Quasiconformal mappings were introduced by Herbert Grötzsch and Oswald Teichmüller in the mid-1930s and applied very successfully by Teichmüller to the study of the moduli space of a Riemann surface, the space of all complex structures on the surface. Teichmüller, a committed Nazi, died on the Russian front in World War II, and after the war Ahlfors and his close colleague Lipman Bers, at Columbia University in New York, decided to rescue the theory from the shadow cast upon it by its creator. As Bers, quoting Plutarch, noted in 1960 in his paper “Quasiconformal Mappings and Teichmüller’s Theorem,” “it does not of necessity follow that, if the work delights you with its grace, the one who wrought it is worthy of your esteem”(p. 90).

Informally, a map between two Riemann surfaces is quasiconformal if it maps infinitesimal circles to infinitesimal ellipses (a conformal map will map infinitesimal circles to infinitesimal circles); and if the eccentricity of the resulting ellipses is bounded, the quasiconformal map is said to be of bounded distortion. Teichmüller showed that among quasiconformal mappings from one Riemann surface to another of the same genus, there is a unique one with a maximal degree of distortion, modulo a certain equivalence relation that involves a homotopy consideration. Furthermore, these equivalence classes form a space *T _{p}* that describes all the complex structures on a Riemann surface of the given genus. The description is, however, redundant, and a certain group intervenes that acts on the space

*T*. This group is called the mapping class group, the corresponding quotient space is the space

_{p}*M*of complex structures on the Riemann surface, and Teichmüller’s space is the universal covering space of

_{p}*M*. Teichmüller also gave another description of his space in terms of quadratic differentials on the Riemann surface, and this connected it with a rich seam of technical mathematics.

_{p}One of Ahlfors’s first contributions to Teichmüller theory after the war was to simplify and generalize the definition of a quasiconformal mapping. He then gave clear new proofs of Teichmüller’s main results. In this way he made the theory accessible for the first time to a whole generation of mathematicians, and the theory blossomed. It was extended to mappings between spaces of higher dimension, and in many areas of mathematics, quasicon-formal mappings proved the right generalization of the much more restricted class of conformal mappings. For example, a classical theorem of Marcel and Frigyes Riesz states that a map between Jordan domains induces a map between their boundaries that is absolutely continuous with respect to a linear measure if the boundaries are rectifiable and the map is conformal. In 1956 Ahlfors and Arne Beurling showed in the joint paper “The Boundary Correspondence under Quasiconformal Mappings” that if the map is only quasiconformal, the induced map in the boundaries can be completely singular.

Ahlfors also found results concerning the existence of quasiconformal mappings with prescribed distortion, and these have had profound implications elsewhere, for example, in Dennis Sullivan’s solution of the Fatou-Julia problem on wandering domains in complex iteration theory. Iteration theory is also a source of geometric objects called quasi-circles, the image of a circle under a quasiconformal map of the extended complex plane to itself. These can be very complicated objects; they may, for example, have any Hausdorff measure *m* in the range 0 <*m* < 2. Even so, Ahlfors was able to give a very simple characterization of quasi-circles that has become fundamental in the study of these objects.

**Kleinian Groups.** Ahlfors, Bers, and their students also did major work in the 1950s and 1960s on Kleinian groups. Fuchsian and Kleinian groups had been introduced by Henri Poincaré in the 1880s as discrete groups of transformations of, respectively, non-Euclidean two-dimensional and non-Euclidean three-dimensional space. Neither theory is simple, but it is fair to say that the Kleinian theory more or less languished from the point when Poincaré abandoned it until Ahlfors came along. He took a different approach to Poincaré and revived the subject. Bers had shown in 1965 that Fuchsian groups obey a finiteness condition, which says (to oversimplify) that the quotient of the upper half plane by a finitely generated Fuchsian group is an orbifold of finite type. More accurately, the set of nonsingular points in the quotient has that property. Indeed, as Bers knew, this was an old result; the significance of his work was that it derived the result by modern methods, specifically, Eichler cohomology and the use of analytic potentials. This was the stepping-stone for Ahlfors to come up with the generalization to Kleinian groups, but this was no trivial advance. Ahlfors had to replace the analytic potentials with the much larger class of smooth potentials and so to rebuild Eichler cohomology on new foundations.

Ahlfors, Bers, and their students also went back to the original suggestion of Poincaré that the behavior of a Kleinian group can be studied by what it does on the boundary of non-Euclidean space. The action of the Kleinian group shunts various points out to the boundary, and Ahlfors conjectured that this limit set would necessarily have two-dimensional Lebesgue measure zero. This result is still unproved, but it reopened the connection to three-dimensional topology that Poincaré had stressed, and in the 1970s this approach was developed extensively by William Thurston in a series of papers that have shaped the field ever since.

A number of Ahlfors’s other ideas deserve mention.

The uniformization theorem of Poincaré and Paul Koebe, dating from 1907, says that every Riemann surface is covered either by the sphere, the Euclidean plane, or the non-Euclidean plane. The last two are topologically indistinguishable, so the question of determining whether the Riemann surface of a given complex function is covered by the Euclidean or the non-Euclidean plane is delicate.

The first case is called, for reasons that go back to the work of Felix Klein in the 1880s, the parabolic case, the second the hyperbolic case, and distinguishing between the two is called the type problem. Very early in his career, Ahlfors was able to give a sufficient criterion for a function to be of parabolic type. The criterion makes fundamental use of the different ways the circumference of a circle depends on the radius in the two cases.

**Bloch’s Constant.** In 1936 and 1937, Ahlfors and Helmut Grunsky gave a striking estimate of what is called Bloch’s constant. Given an analytic map defined on the unit disc and normalized so that its derivative at the origin has modulus 1, Bloch had proved that there is a subdomain of the disc that is mapped one-to-one and conformally onto a disc of a certain radius that is independent of the function studied. Bloch’s constant is the maximum such radius. Ahlfors found a lower bound of for this constant, which has scarcely been improved to this day.

More importantly, the way Ahlfors came to this result was a novel reinterpretation of Schwarz’s lemma, which says that an analytic function mapping the unit disc to itself and fixing the origin satisfies ǀf '(0)ǀ = 1 unless the map is a rotation, in which case ǀf '(0)ǀ = 1. In 1916 George Pick observed that this implies that any analytic map of the disc to itself decreases hyperbolic (non-Euclidean) lengths unless it is a hyperbolic isometry. Ahlfors reinterpreted this as a statement about two metrics in the disc and then found new results by varying the metric. This idea has since proved its worth in many areas on the complex function theory of one and several variables.

**Bloch’s Principle.** Bloch himself had proclaimed a heuristic principle of considerable depth, that any qualitative principle known to be exact in a certain domain remains correct in a new setting if one modifies its statement in a continuous way. Ahlfors triumphantly vindicated this principle in his work on value distribution theory. It is one of the main results of value distribution theory that given five distinct points in the complex plane, a_{1}, a_{2}, …, a_{5}, and a meromorphic function *f*, at least one of the equations *f* (*z*) = a _{i} has a unique solution. This is a considerable generalization of Picard’s theorem that a meromorphic function on the complex plane cannot omit three values without being constant. To make this statement amenable to Bloch’s principle, it can be reformulated it this way: given three disjoint domains and a nonconstant meromorphic function, the pre-image of at least one of these domains has at least one bounded component. This invites the claim that given five disjoint domains in the complex plane whose boundaries are Jordan curves and a nonconstant meromorphic function *f*, there is a domain *D* that is mapped homeomorphically onto exactly one of these domains. This is called, for obvious reasons, the Five Islands Theorem. It was conjectured by Bloch and proved by Ahlfors in 1932, using his method of mixing topological considerations with metric distortion theorems.

## BIBLIOGRAPHY

*Ahlfors’s papers, and a complete bibliography of his work, can be found in the two volumes of his* Collected Papers, *edited by Lars Valerian Ahlfors (Boston: Birkhäuser, 1982)*.

### WORKS BY AHLFORS

“Zur Theorie der Überlagerungsflächen.” *Acta Mathematica* 65 (1935): 157–194.

*Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable*. New York: McGraw-Hill, 1953.

With Arne Beurling. “The Boundary Correspondence under Quasiconformal Mappings.” *Acta Mathematica* 96 (1956): 125–142.

*Lectures on Quasiconformal Mappings*. Princeton, NJ: Van Nostrand, 1966.

### OTHER SOURCES

Bers, Lipman. “Quasiconformal Mappings and Teichmüller’s Theorem.” In his *Analytic Functions*, 89–119. Princeton, NJ: Princeton University Press, 1960.

Bott, Raoul, Clifford Earle, Dennis Hejhal, et al. “Lars Valerian Ahlfors.” *Notices of the American Mathematical Society* 45, no. 2 (1998): 248–255.

Eremenko, Alexandre. “Ahlfors’ Contribution to the Theory of Meromorphic Functions.” Available from http://www.math.purdue.edu

Gehring, Frederick, Irwin Kra, Steven G. Krantz, et al. “The Mathematics of Lars Valerian Ahlfors.” *Notices of the American Mathematical Society* 45, no. 2 (1998): 233–242.

Gleason, Andrew, George Mackey, and Raoul Bott. “Faculty of Arts and Sciences—Memorial Minute: Lars Valerian Ahlfors.”*Harvard University Gazette* 25 (January 2001): 16.

Lehto, Olli. “On the Life and Works of Lars Ahlfors.” *Mathematical Intelligencer*20, no. 3 (1998): 4–8.

*Jeremy Gray*

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**Ahlfors, Lars**