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. As for Ahlfors’s idea of adapting the method image sur- to obtain a higher-dimensional Nevanlinna theory, face. The that had to wait until the paper by Bott and Chern size of that in 1965. circle turns The year following his Gauss-Bonnet Nevan- out to be a linna theory paper, there appeared a deceptively bit under Photograph of Lars Ahlfors by Bill Graustein, the short and unassuming paper called “An extension 1/2, or ap- nephew of Harvard mathematician William of Schwarz’s lemma” [8, v. 1, p. 350]. The main the- proximately Caspar Graustein. Ahlfors was at the time the orem and its proof take up less than a page. That .472, which William Caspar Graustein Professor of is followed by two brief statements of more gen- is therefore Mathematics. eral versions of the theorem and then four pages an upper of applications. Initially, it was the applications that bound for Bloch’s constant B. received the most attention and that Ahlfors was As one application of his generalized Schwarz most pleased with, since they were anything but lemma, Ahlfors proves Bloch’s theorem with a a straightforward consequence of the main theo- lower bound for B of √3/4 .433. The method is ≈ rem. That is particularly true of the second appli- worth describing, since it so typifies Ahlfors’s ap- cation, which gives a new proof of Bloch’s theorem proach to many problems in function theory. in a remarkably precise form. Bloch’s theorem First we give the statement of his generalized states that there is a uniform constant B such that Schwarz lemma. The original Schwarz lemma said every function analytic in the unit disk, and nor- in particular that for an analytic function mapping malized so that its derivative at the origin has the unit disk into the disk, with f (0) = 0, one has modulus 1, must map some subdomain of the unit f (0) < 1, unless f is a rotation, in which case | 0 | disk one-to-one conformally onto a disk of radius f 0(0) =1. Pick observed in 1916 that since every B. Said differently, the image Riemann surface one-to-one| | conformal map of the unit disk onto it- contains an unbranched disk of radius B. The self is an isometry of the hyperbolic metric of the largest such B is known as Bloch’s constant. In 1937 disk, one could drop the assumption that the ori- Ahlfors and Grunsky [8, v. 1, p. 279] published a gin maps into the origin and conclude that for any paper giving an upper bound for B that they con- analytic map of the unit disk into itself, the hy- jectured to be the exact value. The conjectured ex- perbolic length of the image of any curve is at tremal function maps the unit disk onto a Rie- most equal to the hyperbolic length of the origi- mann surface with simple branch points in every nal curve and, in fact, those lengths are strictly de- sheet over the lattice formed by the vertices ob- creased unless the map is one-to-one onto or an tained by repeated reflection over the sides of an isometry preserving all hyperbolic lengths. equilateral triangle, where the center of the unit Ahlfors’s great insight came from viewing the disk maps onto the center of one of the triangles. Schwarz-Pick lemma as a statement about two One obtains the map by taking three circles or- conformal metrics on the unit disk: the original hy- thogonal to the boundary of the unit disk that perbolic metric and the pullback of the hyperbolic form an equilateral triangle centered at the origin metric on the image. That led him to a truly far- with 30o angles and mapping the interior of that reaching generalization, in which he replaces the triangle onto the interior of a euclidean equilateral second conformal metric by any one whose cur- triangle. Under repeated reflections one gets a vature is bounded above by the (constant) nega- map of the entire unit disk onto the surface de- tive curvature of the original hyperbolic metric. The scribed. Ahlfors and Grunsky write down an explicit conclusion is that, again, the lengths of curves in expression for the function of that description the second metric are at most equal to the origi- with the right normalization at the origin and nal hyperbolic lengths. That is the main result of

FEBRUARY 1998 NOTICES OF THE AMS 235 his paper and is the content of the Ahlfors-Schwarz For the purposes of this note a Kleinian group lemma. The proof requires the metrics to be G will always be finitely generated, nonelementary, smooth, but in his generalized versions he shows and of the second kind. Thus it consists of Möbius that the conclusion continues to hold in particu- transformations (a subgroup of PSL (2, C)) and acts lar for various piecewise smooth metrics. In order discontinuously on a nonempty maximal open set to apply the result, Ahlfors constructs specific C , the region of discontinuity of G, ⊂ ∪ {∞} conformal metrics adapted to specific problems. whose complement in C , the limit set of ∪ {∞} As already mentioned, that is anything but straight- GΩ, is an uncountable perfect nowhere dense sub- forward and may take considerable ingenuity in set of the Riemann sphere.Λ each case. For Bloch’s theorem he constructs a In the early sixties not much was known about conformal metric based on the conjectured opti- Kleinian groups. Around the beginning of this cen- mal surface described above and uses it to obtain tury Poincaré suggested a program for studying dis- the lower bound of √3/4 for Bloch’s constant.1 crete subgroups of PSL(2, C); Poincaré’s program When his collected papers [8] were published in was based on the fact that PSL(2, R) acts on the 1982, Ahlfors commented that this particular paper upper half plane H2, a model for hyperbolic 2- “has more substance than I was aware of,” but he space. The quotient of H2 by a discrete subgroup also said: “Without applications my lemma would (a Fuchsian group) of PSL(2, R) is a 2-dimensional have been too lightweight for publication.” It is a orbifold (a Riemann surface with some “marked” lucky thing for posterity that he found applications points). By analogy PSL(2, C) acts on H3, hyperbolic that he considered up to his standard, since it 3-space, and the quotient of H3 by a torsion free would have been a major loss for us not to have discrete subgroup of PSL(2, ) is a 3-dimensional the published version of the Ahlfors-Schwarz C hyperbolic manifold. The study of subgroups of lemma. As elegant and important as his applica- PSL(2, ) was successful because of its connection tions were, I believe that they have long ago been R to classical function theory and to 2-dimensional dwarfed by the impact of the lemma itself, which topology and geometry, about which a lot was has proved its value in countless other applications known, including the uniformization theorem clas- and has served as the underlying insight and model sifying all simply connected Riemann surfaces. for vast parts of modern complex manifold theory, Poincaré’s program was to take advantage of the including Kobayashi’s introduction of the metric connection of PSL(2, ) to 3-dimensional topology that now bears his name and Griffiths’s geomet- C ric approach to higher-dimensional Nevanlinna and geometry to study groups of Möbius trans- theory. It demonstrates perhaps more strikingly formations. However, in 1965 very little was known than anywhere else the power that Ahlfors was able about hyperbolic 3-manifolds. Research in the field to derive from his unique skill in melding the com- seemed to be stuck and going nowhere. Ahlfors plex analysis of Riemann surfaces with the metric completely ignored Poincaré’s program and took approach of Riemannian geometry. a different route to prove the finiteness theorem. He used complex analytic methods, and his result described the Riemann surfaces that can be represented by a Kleinian group. About fifteen years Kleinian Groups later in the mid-seventies, as a result of the fun- Irwin Kra damental contributions of W. Thurston [34], 3-dimensional topology came to the forefront in the Of the many significant contributions of Lars study of Kleinian groups. However, in the interest Ahlfors to the modern theory of Kleinian groups, of keeping this presentation to reasonable length, I will discuss only two closely related contribu- I ignore this subject. tions: the finiteness theorem (AFT) and the use The history of Ahlfors’s work on Kleinian groups of Eichler cohomology as a tool for proving this is also part of the history of a remarkable collab- and related results. Both originated in the semi- oration between Lars Ahlfors and Lipman Bers.2 nal paper [2]. Ahlfors’s finiteness theorem says that the ordinary 1It may be noted that Ahlfors’s lower bound for the Bloch set of a (finitely generated) Kleinian group G fac- constant stood for many years. The bound was shown to tored by the action of the group is an orbifold be strict by M. Heins [21] in 1962; C. Pommerenke [30] con- /GΩof finite type (finitely many “marked” points tributed some refinements in 1970; the lower estimate was and compactifiable as an orbifold by adding a fi- 14 improved by 10− by M. Bonk [16] in 1990; the related niteΩ number of points). 335 Landau constant was improved by 10− by H. Yanag- Bers [11], in approximately 1965, reproved an ihara [35] in 1995; and finally the lower bound for the 4 equivalent known result in the Fuchsian case, a Bloch constant was improved by 10− by H. Chen and much simpler case to handle. The finiteness the- P. Gauthier [17] in 1996. Irwin Kra is Distinguished Service Professor at the State 2Although they co-authored only one paper [9], their work University of New York, Stony Brook. His e-mail address and the work of many of their students was intertwined. is [email protected]. See [27].

236 NOTICES OF THE AMS VOLUME 45, NUMBER 2 orem for PSL(2, R) had been known for a long time, (z a1) ...(z a2q 2) F(z)=F 2 2q (z)= − − − and Bers reproved it using modern methods, Eich- λ − ϕ¯ 2πi ler cohomology. Bers constructs Eichler cohomol- λ2 2q()ϕ() d d ogy classes from analytic potentials (by integrat- − ∧ . × ( z)( a1) ...( a2q 2) ing cusp forms sufficiently many times, using ZZ − − − − methods developed by Eichler [18] for number Ω Then as before, (1) defines an element of 2q 2, theory). To be specific, let G be a Fuchsian group − (finitely generated) operating on the upper half and these polynomials satisfy (2). There is no 2 analogue of the Riemann-Roch theorem. ΠLet plane H . Fix an integer q 2. Let ϕ be a holo- Rb ≥2 be the span of the rational functions R morphic q-form for G on H . Choose a holomor- b 2 st with b a1,a2,...,a2q 2, , where R () phic function F on H whose (2q 1) derivative − b a∈1) (b\{a2q 2) ∞} − = − ··· − − . Ahlfors needs to establish is ϕ. Then, for every element g G, ( b)( a1) ( a2q 2) ∈ − − Λ ··· − − the density of in the Banach space of inte- 1 q A (1) χg =(F g)(g0) − F R ◦ − grable holomorphic functions on . He needs to use Stokes’s theorem. However, the functions con- is the restriction to H2 of a polynomial of degree fronting him are not smooth at theΩ boundary, and at most 2q 2; we denote the vector space of such − the boundary is not even rectifiable. The outline polynomials by . The polynomials con- 2q 2 of Ahlfors’s argument follows. Assume that q =2. structed satisfy the cocycle− condition It is easily seen that α(ϕ)=0if and only if F 2 2q Π λ − ϕ¯ 1 q vanishes on . A bounded measurable function3 µ χg1 g2 =(χg1 g2)(g20) − + χg2 , (2) ◦ ◦ induces a bounded linear functional lµ on A by the all g1 and g2 G. ∈ formula Λ lµ()= (z)µ(z)dz dz, A. One obtains in this way a holomorphic potential ∧ ∈ F for the automorphic form ϕ and a cohomology ZZ 1 class [χ]=β(ϕ) in H (G, 2q 2). If G is finitely Hence the injectivityΩ of α is equivalent to the den- 1 − generated, then H (G, 2q 2) is a finite-dimen- sity of in A. It suffices by Hahn-Banach and the − R sional vector space. Now ifΠ G is of the first kind Riesz representation theorem to prove that for and ϕ is a cusp form, then,Π as a consequence of every bounded measurable function3 µ on the the Riemann-Roch theorem, β(ϕ)=0 if and only condition if ϕ =0. The space of cusp forms for G on H2 is Ω finite dimensional if and only if H2/G is a finitely µ(z)r(z)dz dz =0, all r , ∧ ∈R punctured compact surface. From the injectivity ZZ of the linear map β, Bers concludes the finiteness is enoughΩ to guarantee that µ =0, a.e.4 The hy- result in this case. Bers’s argument for groups of pothesis on µ tells us that Fµ vanishes on = ∂ . the second kind is more complicated but proceeds A fake “proof” of the density, using divergent in- along similar lines. tegrals, is provided by Λ Ω Ahlfors generalized Bers’s methods to a much wider class of subgroups of PSL(2, C). This gener- lµ()= µdz dz alization was completely nontrivial. It required ZZ ∧ the passage from holomorphic potentials to ∂Fµ smooth potentials. This involved a conceptual = Ω dz dz ∂z¯ ∧ jump forward—a construction of Eichler coho- ZZ mology classes via an integral operator, producing Ω ∂ a conjugate linear map α that assigns an Eichler = Fµ dz dz ZZ ∂z¯ ∧ cohomology class α(ϕ) to a bounded holomorphic q-form ϕ for the group G. In addition, there ap- = Ω ∂ Fµ dz − peared a very difficult technical obstacle that ZZ Ahlfors had to surmount to prove the injectivity = ΩFµ dz =0, all A. of α. To surmount this obstacle, Ahlfors introduces − Z∂ ∈ a “mollifier”, a function used to construct an ap- To convert the Ωabove fake proof into a real one, proximate identity. Ahlfors works only with the choose a smooth function j on R with values in case q =2. Using a modified Cauchy kernel, he the closed interval [0, 1] with the properties that 2 2q constructs a potential for λ − ϕ¯ , a continuous 2 2q function F on whose z¯-derivative is λ − ϕ¯ . 3 2 C In particular, λ− ϕ¯ with ϕ a bounded holomorphic Here λ is a weight function whose exact form need 2-form on for G. not concern us. To be more specific, without loss 4Bers’s paper [11] shows that every bounded linear func- of generality, we can assume that . Choose q ∞∈ tional on AΩ is of this form with µ = λ− ϕ¯ and ϕ a 2q 2 distinct finite points a1,...,a2q 2 . holomorphic bounded 2-form. But this observation does − 2 2q − ∈ Form the potential F 2 2q for λ ϕ¯ , λ − ϕ¯ − Λ not simplify the argument. Λ

FEBRUARY 1998 NOTICES OF THE AMS 237 it vanishes on ( , 1] and takes on the value 1 on significance. Perhaps more significantly, it was −∞ [2, ). Then for n Z+, define Ahlfors’s style to make the pioneering contribu- ∞ ∈ tions to a field and leave plenty of room for oth- ers to continue in the same area. In this particu- n ωn(z)=j ,z , lar case much remained to be done.  1  ∈ log log δ(z) Bers [12] saw that if he studied the more gen-   Ω eral case of q-differentials, he would be able to im- where δ(z) is the distance from z to ∂ . Let R>0. prove on the results of Ahlfors and get quantita- tive versions of the finiteness theorem that have We now let R and R be the intersection of with the disc and circle centered at theΩ origin and become known as the Bers area theorems. The first of these theorems [12] states: If G is gener- radius R, respectively.Ω Λ Because ωn vanishes inΩ a ated by N-motions, then Area( /G) 4π(N 1). neighborhood of , we can use integration by ≤ − parts to conclude that This paper by Bers led, in turn, to investigations Λ of the structure of (the Eichler cohomologyΩ groups of) Kleinian groups by Ahlfors [4], based on his ear- ωnµdz dz lier paper [3], and this author [23, 24]. Whereas ZZ R ∧ Ahlfors used meromorphic Eichler integrals (in a ∂ωn = ω FΩ F dz dz, sense going back to Bers’s studies [11]) to describe n µ µ ¯ − Z R − ZZ R ∂z ∧ the structure of the cohomology groups, I relied Λ Ω on smooth potentials (in a sense combining meth- and from here conclude that ods of Ahlfors [2] and Bers [12]). Ahlfors’s gen- erosity, as evidenced by the footnote in [4] re- µdz dz garding the relation between his and my approach ∧ ZZ R to this problem, was very much appreciated; his remarks were most encouraging to a young math- Ω Fµ dz const. R log R dz . ematician. ≤ ≤ | | Z R Z R Since the minimal area of a hyperbolic orbifold

Λ Λ is π/21, Bers’s area theorem gives an upper bound Standard potential-theoretic estimates for the on the number of (connected) Riemann surfaces growth of Fµ are used to obtain the last in- represented by a nonelementary Kleinian group as | | equality. The finiteness of dz dz now 84(N 1). Ahlfors [3] lowered that bound to | ∧ | − implies that the last integral in the series of in- 18(N 1). Even after some important work of RR − equalities tends to zero as R becomesΩ large. This Abikoff [1], there is still no satisfactory bound on completes the proof of the injectivity of α for the number of surfaces that a Kleinian group rep- q =2. It is important to observe that the argu- resents, especially if one insists on using only ment did not require G to be finitely generated. 2-dimensional methods. Bers’s paper [12] also Ahlfors’s “mollifier” is so delicate that it has not showed that the thrice-punctured spheres issue can found any other uses even though it has been be resolved “without new ideas.” It, together with around for more than thirty-three years. In par- Ahlfors’s discoveries on Kleinian groups, led fif- ticular, it is an open problem to determine neces- teen years later to work on the vanishing6 of Poin- sary and sufficient conditions on G for α to be in- caré and relative Poincaré series [25, 26]. jective for a fixed q>2. For finitely generated G, The so-called measure zero problem first sur- once one has the injectivity for q =2, it is easy to faced during the 1965 Tulane conference.7 In his obtain the same conclusion for bigger q. Alterna- 1964 paper Ahlfors remarked that perhaps of tive approaches to the finiteness theorem are found greater interest than the theorems (AFT) that he in [33, 14, 28].5 has been able to prove were the ones he was not In his proof of the finiteness theorem [2], Ahlfors able to prove. First of these was the assertion that made a small mistake: he left out the possibility the limit set of a finitely generated Kleinian group of infinitely many thrice-punctured spheres ap- has two-dimensional Lebesgue measure zero.8 This pearing in /G. (Such surfaces admit no moduli (deformations) and alternatively carry no nontriv- 6A different, more classical, approach to the problem of ial integrableΩ quadratic differentials.) That defi- deciding when a relative Poincaré series vanishes identi- ciency was remedied in subsequent papers by work cally is found in the earlier work of Hejhal [22]. of Bers [13], Greenberg [20], and Ahlfors himself 7The first of the periodic meetings, roughly every four [3]. Ahlfors initially limited his work to quadratic years, of researchers in fields related to the mathemati- differentials (which include squares of “classical” cal interests of Lars Ahlfors and Lipman Bers. The tradi- abelian differentials), in part because this case and tion continues; the next meeting of the Ahlfors-Bers Col- the abelian case are the only ones with geometric loquium will take place in November 1998. 8Although quasiconformal mappings do not appear in the 5This paper deals with the topological aspects of AFT. proof of AFT, Ahlfors’s motivation and ideas came from

238 NOTICES OF THE AMS VOLUME 45, NUMBER 2 has become known as the Ahlfors measure zero papers [6, 7, 9, 10], which have had great impact conjecture. It is still unsolved, although impor- on contemporary analysis. tant work on it has been done by Ahlfors, Maskit, Thurston, Sullivan, and Bonahan. In some sense the On Quasiconformal Mappings [6] problem has been solved for analysts by Sullivan In his commentary to this paper Ahlfors wrote [31], who showed that a nontrivial deformation of that “It had become increasingly evident that a finitely generated Kleinian group cannot be sup- Teichmüller’s ideas would profoundly influence ported entirely on its limit set; topologists are still analysis and especially the theory of functions of interested in the measure zero problem. Formulae one complex variable.…The foundations of the in Ahlfors’s nonsuccessful attempt [5] to prove theory were not commensurate with the loftiness the measure zero conjecture led Sullivan [32] to a of Teichmüller’s vision, and I thought it was time finiteness theorem on the number of maximal con- to re-examine the basic concepts.” jugacy classes of purely parabolic subgroups of a The quasiconformal mappings considered by Kleinian group. The measure zero problem not Grötzsch and Teichmüller were assumed to be continuously differentiable except for isolated only opened up a new industry in the Kleinian points or small exceptional sets. Teichmüller’s the- groups “industrial park”, it also revived the con- orem concerned the nature of the quasiconformal nection with 3-dimensional topology following the mappings between two Riemann surfaces S and S fundamental work of Marden [29] and Thurston 0 that have minimum maximal dilatation. This and [34]. It showed that Poincaré was not at all wrong the fact that any useful theory that generalizes con- when he thought that we could study Kleinian formal mappings should have compactness and re- groups by 3-dimensional methods. The second flection properties led Ahlfors to formulate a geo- “theorem” that Ahlfors “wanted” to establish for metric definition that was free of all a priori his 1964 paper can be rephrased in today’s lan- smoothness hypotheses. guage to say that a finitely generated Kleinian A quadrilateral Q is a Jordan domain Q with group is geometrically finite; a counterexample four distinguished boundary points. The conformal was produced shortly thereafter [19]. modulus of Q, denoted mod(Q), is defined as the side ratio of any conformally equivalent rectangle R. Grötzsch showed that if f : D D is K-quasi- → 0 Quasiconformal conformal in the classical sense, then 1 (1) mod(Q) mod(f (Q)) Kmod(Q) Mappings K ≤ ≤ Frederick Gehring for each quadrilateral Q D. Ahlfors used this in- ⊂ In 1982 Birkhäuser published two fine volumes of equality to define his new class of quasiconformal mappings: a homeomorphism f : D D is K-qua- Lars Ahlfors’s collected papers [8] and his fasci- → 0 nating commentaries on them. Volume 2 contains siconformal if (1) holds for each quadrilateral Q D. Ahlfors then established all of the basic forty-three articles: twenty-one of these are di- ⊂ rectly concerned with quasiconformal mappings properties of conformal mappings for this general class of homeomorphisms, including a uniform and Teichmüller spaces, twelve with Kleinian Hölder estimate, a reflection principle, a com- groups, and ten with topics in geometric function pactness principle, and an analogue of the Hurwitz theory. This distribution shows clearly the domi- theorem. He did all of this in nine pages. nant role that quasiconformal mappings played in The major part of this article was, of course, con- this part of Ahlfors’s work. Moreover, quasicon- cerned with a statement, several interpretations, formal mappings play a key role in several other and the first complete proof of Teichmüller’s the- papers, for example, the important finiteness theorem. orem for Kleinian groups. For this reason I have In his commentary on this paper Ahlfors mod- chosen quasiconformal mappings as the subject of estly wrote that “My paper has serious shortcom- this survey. In particular, I will consider the four ings, but it has nevertheless been very influential and has led to a resurgence of interest in quasi- his work on quasiconformality. Ahlfors knew that a finitely conformal mappings and Teichmüller theory.” generated Kleinian group had a finite-dimensional de- This is an understatement! Ahlfors’s exposition formation space and for a family of such groups a de- made Teichmüller’s ideas accessible to the math- formation should be induced by a quasiconformal map whose Beltrami coefficient is supported on the ordinary ematical public and resulted in a flurry of activity set. These assertions would follow if the limit set had mea- and research in the area by scientists from many sure zero. They follow from Sullivan’s theorem [31]. different fields, including analysis, topology, al- Frederick Gehring is Distinguished University Professor gebraic geometry, and even physics. Emeritus, University of Michigan, Ann Arbor. His e-mail Next, Ahlfors’s geometric approach to quasi- address is [email protected]. conformal mappings stimulated analysts to study

FEBRUARY 1998 NOTICES OF THE AMS 239 this class of mappings quasiconformal self-mappings of H. The suffi- in the plane, in higher- ciency part consisted in showing that the remark- dimensional euclidean able formula spaces, and now in ar- 1 y bitrary metric spaces. f (z)= φ(x + t)+φ(x t) dt His inspired idea to 2y Z0 " − # (3) drop all analytic hy- i y potheses eventually led + φ(x + t) φ(x t) dt 2y 0 " − − # to striking applications Z of these mappings in yields a K-quasiconformal self-mapping of H with other parts of complex K = K(λ) whenever φ satisfies (2). Moreover, f is analysis, such as dis- a hyperbolic quasi-isometry of H, a fact that turns continuous groups, out to have many important consequences. classical function theory, complex iteration, In 1962 it was observed that a self-quasi-con- and in other fields of formal mapping of the n-dimensional upper half space Hn induces an (n 1)-dimensional self-map- mathematics, including − harmonic analysis, par- ping φ of the (n 1)-dimensional boundary plane n − tial differential equa- ∂H ; this fact, for the case n =3, was an impor- tions, differential tant step in the original proof of Mostow’s rigid- geometry, and topol- ity theorem. It was then natural to ask if every qua- ogy. siconformal self-mapping φ of ∂Hn admits a quasiconformal extension to Hn. This question The Boundary was eventually answered in the affirmative by Correspondence Ahlfors in 1963 for n =3, by Carleson in 1972 for under n =4, and by Tukia-Väisälä in 1982 for all n 3. Ahlfors in the classroom. Quasiconformal ≥ Mappings [10] Riemann’s Mapping Theorem for Variable Metrics [9] In the previous paper Ahlfors proved that a quasiconformal mapping f : D D between Jordan If f : D D0 is K-quasiconformal according to (1), → 0 → domains has a homeomorphic extension to their then f is differentiable with fz =0a.e. in D, and 6 closures. A classical theorem due to F. and M. fz Riesz implied that the induced boundary corre- (4) µ = f f spondence φ is absolutely continuous with re- z spect to linear measure whenever ∂D and ∂D0 are is measurable with rectifiable and f is conformal. Mathematicians asked whether this conclusion holds when f is K- K 1 (5) µf k = − quasiconformal. | |≤ K +1 By composing f with a pair of conformal map- a.e. in D. The complex dilatation µf determines f pings, one can reduce the problem to the case uniquely up to postcomposition with a conformal where D = D = H, where H is the upper half plane 0 mapping. and φ( )= . Next, if x and t are real with t>0 The main result of this article states that for any and if Q∞is the∞ quadrilateral with vertices at x t, − µ that is measurable with µ k a.e. in D, there x, x + t, , then mod(Q)=1and inequality (1) im- | |≤ exists a K-quasiconformal mapping f that has µ plies that∞ as its complex dilatation. Moreover, if f is suitably 1 φ(x + t) φ(x) normalized, then f depends holomorphically on µ. (2) − λ, λ ≤ φ(x) φ(x t) ≤ The above result, known by many as the “mea- − − surable Riemann mapping theorem”, has proved where λ = λ(K). Inequality (2) is a quasisymmetry to be an enormously influential and effective tool condition that it was thought would imply that φ in analysis. It is a cornerstone for the study of is absolutely continuous. Teichmüller space; it was the key for settling out- In 1956 Ahlfors and Beurling published a paper standing questions of classical function theory, in which they exhibited for each K>1 a K-quasi- including Sullivan’s solution of the Fatou-Julia conformal mapping f : H H for which the problem on wandering domains; and it currently boundary correspondence φ→: ∂H ∂H is com- plays a major role in the study of iteration of ra- pletely singular. The importance of→ this example tional functions. Indeed, application of this theo- was, however, overshadowed by the authors’ main rem has become a verb in this area. In a lecture at theorem, which stated that inequality (2) charac- the 1986 International Congress of Mathemati- terizes the boundary correspondences induced by cians a distinguished French mathematician was

240 NOTICES OF THE AMS VOLUME 45, NUMBER 2 heard to explain that “before mating two polyno- and I were scheduled to speak the same morning. mials, one must first Ahlfors-Bers the structure.” After several hours of socializing and wine the evening before, Olli and I tried to excuse ourselves Quasiconformal Reflections [7] so that we could get some sleep before our talks. A Jordan curve C is said to be a quasicircle if it is We were told by Lars that that was “a very silly idea the image of a circle or line under a quasiconfor- indeed” and that it would be far better to relax and mal self-mapping of the extended plane; a domain drink with the pleasant company. The next day D is a quasidisk if ∂D is a quasicircle. Quasicircles Lars’s talk went extremely well, and he was sub- can be very wild curves. Indeed, for 0

FEBRUARY 1998 NOTICES OF THE AMS 241 [14] ———, On Sullivan’s proof of the finiteness theorem and the eventual periodicity theorem, Amer. J. Math. 109 (1987), 833–852. [15] W. BLASCHKE, Vorlesungen über Differentialgeome- trie und Geometrische Grundlagen von Einstein’s Relativitätstheorie, Springer-Verlag, Berlin, 1921–1929. [16] M. BONK, On Bloch’s constant, Proc. Amer. Math. Soc. 110 (1990), 889–894. [17] H. CHEN and P. M. GAUTHIER, On Bloch’s constant, J. Analyse Math. 69 (1996), 275–291. [18] M. EICHLER, Eine Verallgemeinerung der Abelschen In- tegrale, Math. Z. 67 (1957), 267–298. [19] L. GREENBERG, Fundamental polyhedra for Kleinian groups, Ann. of Math. 84 (1966), 433–441. [20] ———, On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, Amer. J. Math. 89 (1967), 56–68. [21] M. HEINS, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60. [22] D. A. HEJHAL, On monodromy groups and Poincaré series, Bull. Amer. Math. Soc. 84 (1978), 339–376. [23] I. KRA, On cohomology of Kleinian groups, Ann. of Math. 89 (1969), 533–556. [24] ———, On cohomology of Kleinian groups: II, Ann. of Math. 90 (1969), 575–589. [25] ———, On the vanishing of and spanning sets for Poincaré series for cusp forms, Acta Math. 153 (1984), 47–116. [26] ———, Cusp forms associated to loxodromic elements of Kleinian groups, Duke Math. J. 52 (1985), 587–625. [27] ———, Creating an American mathematical tradi- tion: The extended Ahlfors-Bers family, A Century of Mathematical Meetings, Amer. Math. Soc., Provi- dence, RI, 1996, pp. 265–280. [28] R. S. KULKARNI, Some topological aspects of Kleinian groups, Amer. J. Math. 100 (1978), 897–911. [29] A. MARDEN, The geometry of finitely generated Klein- ian groups, Ann. of Math. 99 (1974), 383–462. [30] C. POMMERENKE, On Bloch functions, J. London Math. Soc. 2 (1970), 689–695. [31] D. SULLIVAN, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Rie- mann Surfaces and Related Topics, Ann. of Math. Stud., vol 97, Princeton Univ. Press, Princeton, NJ, 1980, pp. 465–496. [32] ———, A finiteness theorem for cusps, Acta Math. 147 (1981), 289–299. [33] ———, Quasiconformal homeomorphisms and dy- namics I, Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. 122 (1985), 401–418. [34] W. P. THURSTON, Three dimensional manifolds, Klein- ian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381. [35] H. YANAGIHARA, On the locally univalent Bolch constant, J. Analyse Math. 65 (1995), 1–17. [36] Commentationes in honorem Lars V. Ahlfors, LXXX Annos NATO, Ann. Acad. Sci. Fenn. Ser. A I Math. Dis- sertationes 13 (1988).

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