Lars Ahlfors

How to use this course

Lars Ahlfors worked in complex analysis for nearly seventy years. His career falls into two periods. Before the Second World War he solved Denjoy’s conjecture, recast Rolf Nevanlinna’s value distribution theory as the geometry of covering surfaces, and generalized Schwarz’s lemma to curved metrics. After the war, at Harvard, he made quasiconformal mappings a central subject, gave the first complete proofs of Oswald Teichmüller’s theorems, and opened the modern study of Kleinian groups. His textbook Complex Analysis has taught the subject to generations of students.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work in order: entire functions (1929), covering surfaces (1935), curvature (1938), conformal invariants (1947–1950), quasiconformal maps (1953–1963) and Kleinian groups (1964 onward).

Timeline

Helsingfors, 1907–1928

Lars Valerian Ahlfors was born on 18 April 1907 in Helsingfors, the Swedish name of Helsinki, then capital of a grand duchy of the Russian Empire. His mother died giving birth to him. He spent his first years with two aunts in the Åland Islands before returning to his father, Axel Ahlfors, a professor of mechanical engineering. The family spoke Swedish, and Lars attended a private Swedish-language school. He later became nearly fluent in Finnish, German and English.

Finland declared independence in 1917 and fell into civil war in 1918. Food was short, and his father was briefly imprisoned by the Red Guard. The boy taught himself mathematics from his sister’s homework and from his father’s engineering books, which included calculus. His school did not teach it.

He entered the University of Helsinki in 1924. Finland had a remarkable school of complex function theory founded by Ernst Lindelöf, and Ahlfors studied with Lindelöf and with Rolf Nevanlinna, who had recently created the theory of value distribution of meromorphic functions. Ahlfors completed his master’s examinations in the spring of 1928.

Zürich and Denjoy’s conjecture, 1928–1930

In the autumn of 1928 Nevanlinna was invited to the ETH in Zürich to replace Hermann Weyl, who was away for a year. Ahlfors went with him. He later described the move as going from the periphery to the centre of Europe. In Zürich he attended Nevanlinna’s lectures and decided to become a research mathematician.

Nevanlinna lectured on a problem posed by Arnaud Denjoy in 1907: an entire function of finite order \(\rho\) should have at most \(2\rho\) distinct finite asymptotic values. Torsten Carleman had proved a weaker bound. Ahlfors found a proof of the full conjecture using conformal mapping of strip-like domains, a method whose key estimate is now called the Ahlfors distortion theorem (Lesson 04). He published it in 1929, at 22, and was immediately known to every function theorist in Europe.

He spent 1929–30 as a lecturer at Åbo Akademi, the Swedish-language university in Turku, and defended his doctoral thesis at Helsinki in the spring of 1930, with Lindelöf and Nevanlinna as his teachers.

Paris, Turku, Harvard, Oslo, 1930–1938

From 1930 to 1932 Ahlfors travelled on the Continent, with a long stay in Paris on a Rockefeller Foundation fellowship. In 1933 he became an adjunct professor at the University of Helsinki and married Erna Lehnert, an Austrian. He later called the marriage the happiest event of his life.

In 1935 he published Zur Theorie der Überlagerungsflächen in Acta Mathematica. It rebuilt Nevanlinna theory as a theory of covering surfaces of the sphere, using only metric and topological arguments. The paper also used maps with bounded distortion, under the name quasiconformal mappings, which Herbert Grötzsch had introduced in 1928.

That year, on the recommendation of Constantin Carathéodory, Harvard offered him a three-year appointment. At the International Congress in Oslo in July 1936, the first two Fields Medals were awarded, to Ahlfors and to Jesse Douglas of MIT. Ahlfors learned of the prize only hours before the ceremony. The committee cited above all his work on covering surfaces and called him one of the most brilliant representatives of Lindelöf’s Finnish school.

In 1938 he accepted a chair at Helsinki, from loyalty to his teachers and his country. In the same year he published a short paper extending Schwarz’s lemma to metrics of negative curvature. It became one of the most cited results in geometric function theory (Lesson 06).

Asymptotic values and the distortion theorem

For the general reader. An entire function, such as \(e^z\), is defined on the whole complex plane. Along some paths to infinity it may approach a fixed value. Denjoy guessed that slowly growing functions have few such limiting values. Ahlfors proved it by measuring how much a narrow, winding channel must be stretched when it is straightened out into a strip.

Let \(f\) be entire with maximum modulus \(M(r)=\max_{|z|=r}|f(z)|\). Its order is \(\rho=\limsup_{r\to\infty}\frac{\log\log M(r)}{\log r}\). A value \(a\in\mathbb C\) is an asymptotic value if \(f(\gamma(t))\to a\) along some path \(\gamma(t)\to\infty\).

The distortion theorem

Let \(\Omega\) be a domain lying in a strip around the positive real axis, and suppose each vertical line \(\operatorname{Re}z=x\) meets \(\Omega\) in a cross-cut of length \(\theta(x)\). Let \(F\) map \(\Omega\) conformally onto the strip \(\{0<\operatorname{Im}w<1\}\), with the two ends corresponding.

For a straight strip of width \(\theta\), the map is \(w=z/\theta\) and the advance is exactly \(\int dx/\theta\). The theorem says a winding channel is never conformally shorter than its width profile suggests, up to a fixed error. Its modern proof uses extremal length (Lesson 07).

Proof idea for Denjoy

Suppose \(f\) has \(n\) distinct finite asymptotic values. Between the asymptotic paths lie \(n\) disjoint unbounded domains \(D_k\) on which \(\log|f|\) is large and tends to infinity. In logarithmic coordinates \(\zeta=\log z\), each \(D_k\) becomes a channel whose cross-cut at \(\operatorname{Re}\zeta=\log r\) has length \(\theta_k(r)\), with \(\sum_k\theta_k(r)\le2\pi\). Harmonic-measure estimates plus the distortion theorem give \(\log\log M(r)\ge\pi\int^{r}\frac{dt}{t\,\theta_k(t)}-C\) for each \(k\). By Cauchy–Schwarz, \(\sum_k1/\theta_k\ge n^2/\sum_k\theta_k\ge n^2/(2\pi)\), so for some \(k\), along a sequence of radii, the integral is at least \(\frac{n}{2\pi}\log r\) up to constants. Hence \(\log\log M(r)\ge\frac n2\log r-C'\), and \(\rho\ge n/2\).

Covering surfaces and the five islands

For the general reader. Picard’s theorem says a non-constant entire function takes every complex value with at most one exception. Nevanlinna turned this into precise counting formulas. Ahlfors found that the heart of Nevanlinna’s theory is a fact about how a surface can be spread over a sphere, and that it is close to the Euler characteristic of topology. This is the work that won the Fields Medal.

Let \(f\) be meromorphic on a domain containing the closed disc \(|z|\le r\), regarded as a map to the Riemann sphere \(\hat{\mathbb C}\) with its spherical metric of total area \(4\pi\). Define

\(S(r)\) is the spherical area of the image counted with multiplicity, divided by the area of the sphere: the mean sheet number. \(L(r)\) is the spherical length of the image of the boundary circle. Let \(D_1,\dots,D_q\) be Jordan domains on the sphere with pairwise disjoint closures. An island over \(D_\nu\) is a component of \(f^{-1}(D_\nu)\) whose closure lies in \(|z|<r\); it is simple if \(f\) maps it one-to-one onto \(D_\nu\).

The number \(q-2\) is minus the Euler characteristic of the sphere with \(q\) holes, and the proof is a careful comparison of Euler characteristics, with \(hL\) bounding the error from pieces that touch the boundary. Ahlfors also proved the length–area principle: if \(f\) is meromorphic in the plane and non-constant, then \(L(r)=o(S(r))\) as \(r\to\infty\) outside a set of \(r\) of finite logarithmic measure. In this sense the boundary is negligible.

Proof sketch from the main theorem. Suppose no island over any \(D_\nu\) is simple, so every simply connected island covers \(D_\nu\) at least twice. The islands over \(D_\nu\) cover it at most \(S(r)+O(L(r))\) times in total, so \(n(r,D_\nu)\le\tfrac12 S(r)+O(L(r))\). Then \(3S\le\tfrac52S+O(L)\), that is, \(\tfrac12S(r)=O(L(r))\), contradicting \(L=o(S)\).

Ahlfors’ lemma and Bloch’s constant

For the general reader. Schwarz’s lemma says that a holomorphic map of the unit disc into itself cannot increase hyperbolic distances. In 1938 Ahlfors saw that this is really a statement about curvature: any metric on the disc that is at least as negatively curved as the hyperbolic metric must be smaller than it. This idea later grew into Kobayashi hyperbolicity and much of complex geometry.

A conformal metric \(\lambda(z)|dz|\) with \(\lambda>0\) of class \(C^2\) has Gaussian curvature

The Poincaré metric on the unit disc \(\mathbb D\) is \(\lambda_{\mathbb D}=\frac{2}{1-|z|^2}\), with \(K\equiv-1\).

Proof

Fix \(0<R<1\) and compare \(\lambda\) on the disc \(|z|<R\) with \(\mu_R(z)=\frac{2R}{R^2-|z|^2}\), the Poincaré metric of that disc, which also has curvature \(-1\) and tends to infinity at \(|z|=R\). The function \(u=\log(\lambda/\mu_R)\) tends to \(-\infty\) at the boundary, so if it is positive somewhere it has an interior maximum at a point with \(\lambda>0\). There \(\Delta u\le0\). But

using \(\Delta\log\lambda=-K_\lambda\lambda^2\ge\lambda^2\) and \(\Delta\log\mu_R=\mu_R^2\). Contradiction. So \(\lambda\le\mu_R\), and letting \(R\to1\) gives the result.

If \(f:\mathbb D\to\mathbb D\) is holomorphic, the pull-back \(\lambda=\lambda_{\mathbb D}(f(z))|f'(z)|\) has curvature \(-1\) where \(f'\ne0\). The lemma gives the Schwarz–Pick inequality \(\frac{|f'(z)|}{1-|f(z)|^2}\le\frac{1}{1-|z|^2}\).

Bloch’s constant

André Bloch proved that there is an absolute constant \(B>0\) such that every holomorphic \(f\) on \(\mathbb D\) with \(f'(0)=1\) maps some subdomain one-to-one onto a disc of radius \(B\). Applying his lemma to a cleverly built metric of curvature \(\le-1\), Ahlfors showed \(B\ge\sqrt3/4\approx0.433\). With Helmut Grunsky he had earlier shown \(B\le0.472\) and conjectured that this upper bound is exact. Despite many efforts the lower bound has been improved only in the fourth decimal place.

Extremal length and analytic capacity

For the general reader. How “wide” is a family of curves? Ahlfors and the Swedish analyst Arne Beurling answered with a number, the extremal length, that does not change under conformal maps. It became the universal tool for estimating conformal maps. In the same period Ahlfors asked which compact sets are too small to support a bounded analytic function, a question that was settled only in 2003.

Let \(\Gamma\) be a family of locally rectifiable curves in a domain \(\Omega\). For a Borel function \(\rho\ge0\) on \(\Omega\), set \(L(\Gamma,\rho)=\inf_{\gamma\in\Gamma}\int_\gamma\rho\,|dz|\) and \(A(\rho)=\iint_\Omega\rho^2\,dx\,dy\).

Invariance under conformal maps follows by transporting \(\rho\) to \(\rho\circ f^{-1}\,|(f^{-1})'|\), which preserves both lengths and area. The ideas were developed first independently by both men, then jointly in their 1950 paper in Acta Mathematica on conformal invariants and function-theoretic null-sets.

Analytic capacity

For a compact set \(E\subset\mathbb C\), let

where \(f'(\infty)=\lim_{z\to\infty}z\,f(z)\). A set is removable for bounded analytic functions exactly when \(\gamma(E)=0\). Ahlfors introduced \(\gamma\) in 1947 and proved that an extremal function exists and is unique; it is now called the Ahlfors function. Painlevé’s problem, to characterize removable sets geometrically, was solved through work of Guy David (1998) and Xavier Tolsa (2003), who showed that \(\gamma\) is comparable to a capacity defined by Menger curvature.

Quasiconformal maps and the Beltrami equation

For the general reader. Teichmüller had shown, in papers few people read, that the best way to deform one Riemann surface into another stretches uniformly in one direction. After the war Ahlfors gave the theory solid foundations. With Lipman Bers he then proved that one can prescribe the infinitesimal stretching at every point, almost arbitrarily, and still find a map realizing it. This theorem underlies Teichmüller theory and complex dynamics.

A quadrilateral \(Q\) is a Jordan domain with four marked boundary points; its modulus \(\operatorname{mod}Q\) is the extremal length of the curves joining one pair of opposite sides, which equals \(a/b\) when \(Q\) is mapped conformally to an \(a\times b\) rectangle with vertices corresponding.

This definition needs no differentiability. Ahlfors proved that such maps are differentiable almost everywhere and satisfy the Beltrami equation \(f_{\bar z}=\mu f_z\) with \(\|\mu\|_\infty\le\frac{K-1}{K+1}\). In the same 1953–54 paper he gave a complete proof of Teichmüller’s theorem on extremal maps between closed Riemann surfaces, using the length–area method in the metric of a quadratic differential.

Existence for measurable \(\mu\) goes back to Charles Morrey (1938). The Ahlfors–Bers contribution was a new proof through singular integrals, with the Beurling transform \(T\) whose \(L^p\) norms tend to 1 as \(p\to2\), and the holomorphic dependence on parameters, which is what Teichmüller theory and holomorphic dynamics need. Adrien Douady remarked at the 1986 Congress that before mating two polynomials one must first “Ahlfors–Bers” the structure.

Quasicircles and universal Teichmüller space

For the general reader. Which closed curves are images of a circle under a quasiconformal map of the plane? In 1963 Ahlfors gave a strikingly simple answer: a curve qualifies exactly when it does not double back on itself too sharply, a condition on any three of its points. The same paper showed that the space of all such shapes is an open set in a space of holomorphic functions, the universal Teichmüller space of Bers.

A quasicircle is the image of the unit circle under a quasiconformal homeomorphism of \(\hat{\mathbb C}\).

The proof builds a quasiconformal reflection across \(C\): an orientation-reversing quasiconformal map of the sphere that fixes \(C\) pointwise and swaps its two sides. Its existence is equivalent to \(C\) being a quasicircle.

Boundary values

In 1956 Beurling and Ahlfors characterized the boundary values of quasiconformal self-maps of the upper half-plane: an increasing homeomorphism \(h\) of \(\mathbb R\) extends to a quasiconformal map of \(\mathbb H\) if and only if it is quasisymmetric, meaning

Their extension formula is explicit. They also showed that such \(h\) can be singular with respect to Lebesgue measure. The space of quasisymmetric maps modulo Möbius maps is the universal Teichmüller space, which contains the Teichmüller space of every Riemann surface. Ahlfors’ 1963 paper showed that its image under the Schwarzian derivative is open in the Banach space of bounded holomorphic quadratic differentials on the half-plane.

Kleinian groups: finiteness and measure

For the general reader. A Kleinian group is a discrete group of Möbius transformations of the sphere. Where the group acts nicely, the quotient is a collection of Riemann surfaces. In 1964 Ahlfors proved that a group with finitely many generators produces only finitely many surfaces, each of finite type. He also posed a problem about the leftover “limit set” that took forty years to solve.

Let \(\Gamma\subset\mathrm{PSL}_2(\mathbb C)\) be a discrete group acting on \(\hat{\mathbb C}\). Its region of discontinuity \(\Omega(\Gamma)\) is the set of points with a neighbourhood \(U\) such that \(\gamma U\cap U\ne\varnothing\) for only finitely many \(\gamma\). The complement \(\Lambda(\Gamma)\) is the limit set. The group is Kleinian if \(\Omega\ne\varnothing\).

Ahlfors proved this with Eichler cohomology and quasiconformal deformations: each component of the quotient contributes independently to a space of deformations of \(\Gamma\), which is finite-dimensional because \(\Gamma\) is finitely generated. A gap concerning thrice-punctured spheres was closed by later work of Bers, Leon Greenberg and Ahlfors. In 1967 Bers proved a quantitative version: if \(\Gamma\) is generated by \(N\ge2\) elements, the total hyperbolic area of \(\Omega/\Gamma\) is at most \(4\pi(N-1)\).

Ahlfors proved it for geometrically finite groups. Dennis Sullivan proved around 1980 that the limit set supports no invariant line field, the analytic statement Ahlfors’ question was really about. The full conjecture followed from the tameness theorem, proved in 2004 by Ian Agol and independently by Danny Calegari and David Gabai, combined with earlier work of Richard Canary. William Thurston’s geometrization of 3-manifolds had in the meantime shown that Kleinian groups are the same subject as hyperbolic 3-manifolds.

War, Zürich, Harvard, 1938–1977

Ahlfors returned to Helsinki in 1938. The Winter War with the Soviet Union began in November 1939, and the Continuation War followed in 1941, with Finland fighting alongside Germany. Helsinki was bombed, and the university nearly emptied because the male students were at the front. Ahlfors was exempt from military service because of an earlier medical condition. He sent his wife and children to safety in Sweden and kept teaching. He later recalled writing one of his best papers in an air-raid shelter.

Finland signed an armistice with the Soviet Union in September 1944. In the spring of 1945 Ahlfors took up a professorship in Zürich, reaching it after a difficult journey across a disrupted continent; MacTutor reports that he pawned his Fields Medal to help pay for the trip out of Finland. He was not happy in Zürich, and in 1946 he accepted Harvard’s offer to return. He stayed for the rest of his career, chaired the department, became William Caspar Graustein Professor in 1964 and retired in 1977.

At Harvard his research entered its second great period. Lipman Bers, at Syracuse, Princeton and then Columbia, became his closest mathematical partner. Their joint seminar culture, and the regular conferences on Riemann surfaces and Kleinian groups that began in the 1960s, created the school from which Teichmüller theory, Kleinian groups and holomorphic dynamics grew. He also read and championed Teichmüller’s work and, with Frederick Gehring, edited Teichmüller’s collected papers in 1982.

Teacher, author, legacy

Ahlfors supervised 24 doctoral students and many postdoctoral visitors. They include Paul Garabedian, James Jenkins, George Springer, Halsey Royden, Robert Osserman, Clifford Earle, Albert Marden, Henry Pollak and Dale Husemoller. Royden proved that the Teichmüller metric equals the Kobayashi metric; Earle and Marden became leaders of Teichmüller and Kleinian group theory.

Books

• Complex Analysis (1953; 3rd ed. 1979), the classic graduate text, admired for its economy and precision.
• Riemann Surfaces, with Leo Sario (1960).
• Lectures on Quasiconformal Mappings (1966), the short book from which most of the field learned the subject.
• Conformal Invariants: Topics in Geometric Function Theory (1973).
• Möbius Transformations in Several Dimensions (1981).

Honours

Ahlfors received the Fields Medal (1936), the Wihuri International Prize (1968), the Wolf Prize (1981, shared with Oscar Zariski) and the Steele Prize of the American Mathematical Society (1982). He was a member of the US National Academy of Sciences and of several European academies, and honorary president of the 1986 International Congress in Berkeley. The Wolf Prize jury said that every complex analyst is in some sense his student. His Collected Papers appeared in two volumes in 1982.

He remained mathematically active long after retirement. He died on 11 October 1996 in Pittsfield, Massachusetts, at the age of 89, survived by his wife Erna and their three daughters.

Influence

• Value distribution: the geometric theory of covering surfaces and the five islands theorem, now used in complex dynamics and in the study of normal families.
• Geometry of curvature: Ahlfors’ lemma is the prototype of the Schwarz lemmas of Shiing-Shen Chern, Shing-Tung Yau and Shoshichi Kobayashi.
• Quasiconformal maps: the Ahlfors–Bers theorem drives Teichmüller theory and holomorphic dynamics, from Douady and Hubbard to Curtis McMullen.
• Kleinian groups: the finiteness theorem and the measure conjecture shaped the field that Thurston connected to 3-manifold topology.

Connections

People linked to Lars Ahlfors across this collection. A name without a link has no page here yet. See Lars Ahlfors in the network of all 337 lives →

Teachers and advisors

• Ernst Lindelöf 1924-1930
• Rolf Nevanlinna 1924-1930

Students

• Albert Marden
• Clifford Earle 1962
• George Springer
• Halsey Royden
• James A. Jenkins
• Paul Garabedian
• Robert Osserman

Influenced by

• Oswald Teichmüller
• Arnaud Denjoy

Influenced

• Curtis McMullen
• Dennis Sullivan
• Howard Masur
• William Thurston

Colleagues and collaborators

• Oscar Zariski 1981
• Arne Beurling 1946-1956
• Constantin Carathéodory
• Frederick Gehring
• Lipman Bers 1950s-1970s

Also mentioned in this life

• Stefan Banach
• Eugenio Beltrami
• Augustin-Louis Cauchy
• Leonhard Euler
• Carl Friedrich Gauss
• Henri Lebesgue
• Joseph Liouville
• August Ferdinand Möbius
• Henri Poincaré
• Bernhard Riemann
• Karl Weierstrass
• Hermann Weyl

Fields

• complex analysis 13 lives
• riemann surfaces 3 lives
• kleinian groups 4 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Lars Valerian Ahlfors, MacTutor History of Mathematics.
2. F. W. Gehring, Lars Valerian Ahlfors 1907–1996, Biographical Memoirs of the National Academy of Sciences 87 (2005).
3. Lars Ahlfors, Wikipedia.
4. Mathematics Genealogy Project (entry for C. J. Earle, Ahlfors’ student).
5. L. V. Ahlfors, Collected Papers, 2 vols., Birkhäuser, 1982 (with his own commentaries).

Mathematics

1. L. V. Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935).
2. L. V. Ahlfors, An extension of Schwarz’s lemma, Trans. AMS 43 (1938).
3. L. V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947).
4. L. V. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950).
5. L. V. Ahlfors, On quasiconformal mappings, J. Analyse Math. 3 (1953–54).
6. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956).
7. L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. 72 (1960).
8. L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963).
9. L. V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964).
10. L. V. Ahlfors, Lectures on Quasiconformal Mappings, 2nd ed., AMS, 2006.
11. L. V. Ahlfors, Conformal Invariants, McGraw-Hill, 1973; reprinted AMS Chelsea, 2010.
12. W. K. Hayman, Meromorphic Functions, Oxford, 1964 (Ahlfors’ theory of covering surfaces, chapter 5).
13. J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge, 2005 (extremal length and the distortion theorem).