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From Schwarz to Pick to Ahlfors and Beyond Robert Osserman

n his pivotal 1916 paper [P], Georg Pick famous “reflection principle” for analytic func- begins somewhat provocatively with the tions.) He goes on to give a general formula, not- phrase, “The so-called Schwarz Lemma ing that Christoffel developed it independently. He says…”, followed by a reference to a 1912 credits Weierstrass for filling in the details of paper of Carathéodory. Pursuing that lead, one showing that the arbitrary constants involved in findsI a reference to the original source in the ex- the integral expression can be chosen to give the panded notes from a lecture course at the Eid- desired mapping for a polygon of any number of genössische Polytechnische Schule in Zürich given sides, whereas Schwarz himself had succeeded for by Hermann Amandus Schwarz during 1869–70 just four sides. ([S], 108–132). The lecture notes start by stating the Once in possession of a mapping for polygons, Riemann Mapping Theorem from Riemann’s doc- Schwarz proceeds to approximate an arbitrary toral dissertation of 1851 and noting that Rie- convex domain by domains bounded by polygons mann’s argument did not provide a fully rigorous proof. The goal of the lecture course is to provide and to show that the corresponding mappings the first complete proof for a general class of do- converge to a limit mapping with the desired prop- mains. The Riemann mapping theorem states that erties. This proof has long been forgotten, since any simply connected plane domain other than the the result was superseded by more general re- entire plane can be mapped one-to-one and con- sults, leading eventually to a proof of the full the- formally onto the interior of the unit circle. (The orem. However, the first step in his argument for plane domains considered at the time appear to convex domains is precisely the statement and have been domains bounded by a simple closed proof of an early version of what eventually became curve.) The lecture notes prove the theorem for do- known as the “Schwarz Lemma”. mains bounded by a closed convex curve. (The Schwarz Lemma) f (z) Schwarz’s proof is based on his earlier work on Lemma 1 . Let be ana- {| | } domains bounded by polygons and what is now lytic on a disk D = z unit disk. He decides to start with the simplest case of R (2) f (z)= 2 eiz for some real . a square. (It is in that context that he proves his R1

Robert Osserman is professor emeritus of mathematics at As immediate corollaries, one has: Stanford University and is special projects director of the Corollary 1 (Liouville’s Theorem). A bounded an- Mathematical Sciences Research Institute (MSRI), Berke- ley, CA. His e-mail address is [email protected]. alytic function in the entire plane is constant.

This article is an expanded version of a talk given at the Proof. R2 is fixed, and R1 may be chosen arbi- Symposium in Honor of Lars Ahlfors held at Stanford trarily large. University, September 19–21, 1997. The research at MSRI was supported in part by NSF grant no. DMS-9701755. Corollary 2. If R1 = R2, then

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0 (3) |f (0)|1. 1988 articles in French and English by Henri Car- tan and Jacqueline Ferrand). A slightly less obvious, but still elementary Bloch’s Principle is more a heuristic device than corollary is a result. It states, in essence, that whenever one has a global result such as Liouville’s Theorem, there Corollary 3. If R = R and if f maps the bound- 1 2 should be a stronger, finite version from which the ary to the boundary, then at any point b with general result will follow. The move from Liouville’s |b| = R where f 0(b) exists, one has 1 Theorem to the Schwarz Lemma is a perfect ex- ample of such a result. Bloch himself gave another | 0 | (4) f (b) 1. example, by proving a finite result that not only The proof follows immediately from the fact that implies Picard’s Theorem, but was a simple “ele- distances to the origin are shrunk under f, and mentary proof” of Picard’s Theorem, not relying therefore distances from the boundary are on the use of the elliptic modular function or the stretched. More precisely, for t real, 0 0, with the property that for | | | | f (tb) f (b) R1 tR1 = tb b every value of Rhyperbolic geometry f (b) 1+ 0 . 1+|f (0)| which one might have thought would have been made earlier by Klein or Poincaré. A proof may be found in [O3]. The standard proof of the Schwarz Lemma—not Lemma 2 (Schwarz-Pick Lemma). Let f (z) be a the proof that Schwarz himself originally gave— holomorphic map of the unit disk D into the unit consists of observing that the condition f (0) = 0 im- disk. Then plies that the function g(z)=f (z)/z is a regular an- (5) ˆ(f (z1),f(z2)) ˆ(z1,z2) for all z1,z2 ∈ D, alytic function in the disk |z|

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But hyperbolic distance to the origin, given by Over the course of the twentieth century, a (8), is a monotonic function of Euclidean distance, whole line of investigations has extended the ap- so that (9) is equivalent to (1) when R1 = R2 =1. proach of Pick and Ahlfors to the Schwarz Lemma, Furthermore, equality holds in the Schwarz Lemma where a key factor is that the unit disk is complete if and only if distances to the origin are preserved; in the hyperbolic metric. (See, for example, Theo- hence the same holds for hyperbolic distances. It rems 3 and 4 below.) However, in the past decade follows that equality holds in the Schwarz-Pick another approach has allowed a return to the orig- Lemma if and only if f is an isometry of the hy- inal Schwarz Lemma, in which one has a map of a perbolic plane, which is to say, a linear fractional finite disk into a finite disk. Let us start with some transformation of the unit disk onto itself. examples of such results. An equivalent formulation of the Schwarz-Pick A geodesic disk of radius R on a surface is the Lemma states that every holomorphic map of the diffeomorphic image of a Euclidean disk of radius unit disk into itself is either linear fractional— R under the exponential map. Equivalently, one has hence a non-Euclidean isometry—or else it shrinks geodesic polar coordinates: the hyperbolic length of every curve. Among the many generalizations of the (12) ds2 = d2 + G(,)2 d2, Schwarz-Pick Lemma, probably the single most in- where represents distance to the center of the fluential one was that of Ahlfors [A1] in 1938 (or [A2], pp. 350–355). disk, and ∂G Lemma 3 (Schwarz-Pick-Ahlfors Lemma). Let f be (13) G(0,)=0, (0,)=1,G(,) > 0 ∂ a holomorphic map of the unit disk D into a Rie- mann surface S endowed with a Riemannian met- for 0 <

(15) ˆ(z) = distance on DR from 0 to z. (11) kdfzk1 everywhere, | | where the norm is taken with respect to the hy- Example 1. Let f be a holomorphic map of z conformal map f, one obtains a conformal metric on the unit disk, to be Corollary 1. Any holomorphic map of the entire compared with the original hyperbolic metric. plane into a geodesic disk on a surface with K 0 When his collected papers [A2] were published must be constant. in 1982, Ahlfors had the opportunity to evaluate Corollary 2. If R2 R1 , then kdf0k1. his earlier work with the wisdom of hindsight. He confesses that his generalization of the Schwarz Corollary 3. If R2 = R1 and if at some point z with Lemma “has more substance than I was aware of,” |z| = R1, (f (z)) = R1 and dfz exists, then but still says that “without applications my lemma would have been too lightweight for publication” (17) kdfzk1. ([A2], p. 341). Of the applications he gave, the most striking is an elementary new proof of Bloch’s The- Remarks. (1) This example is a slightly more gen- orem (Theorem 1 above) with an explicit lower√ eral form of the first part of Lemma 6 of Ros [R]; 3 bound for Bloch’s constant B, namely B 4 . his proof goes through without change. (2) Corol- Despite many attempts over the years, only slight lary 1 is false for K>0; the inverse of stereo- improvements on this lower bound have been ob- graphic projection is a nonconstant conformal tained; a more detailed discussion of all these mat- map of the entire plane onto a geodesic disk con- ters can be found in the article [O1]. sisting of the sphere minus a point.

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Example 2. Let f be a holomorphic map of paring two metrics on the same domain: the orig- 2 2 {|z|

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where for any point P in Db , ˆ(P) is the distance be- lar region near the boundary of Db . We are then back tween P and the center O of the disk. We map Db to our original situation on a disk of smaller radius conformally by f into a surface S with metric ds2 in Db , and we may expect the same kind of con- and assume that the image lies in a geodesic disk traction (28) to extend. D of the same radius centered at the point f (0). In brief, the heuristic connection is that an Under suitable curvature restrictions we wish to equality like (29) on Gauss curvature implies an ex- show that pansion of the boundary ˆ = 1 to = 1, which by conformality of f implies an expansion in the ra- (28) (f (P)) ˆ(P) for all P in D, dial direction from the boundary, or a movement of points toward the center, and therefore a con- where (Q) is the distance on S from f (0) to Q. traction in the sense of (28). We introduce geodesic polar coordinates We have not been able to turn this heuristic ar- ds2 = d2 + G(,)2 d2, gument into a complete proof under the full gen- erality of (29), but we have been able to obtain the with 0 <1 and 0 <2, on the image, result for a very broad class of metrics, including and we assume that the curvature relation is those of Examples 1 and 2, namely, for all metrics dˆs2 which have circular symmetry. (29) K(,) Kb (,ˆ ) when = ˆ; Theorem 4 (General Finite Shrinking Lemma). Let b that is, for each fixed , the curvature of the image D be a geodesic disk of radius 1 with respect to a geodesic disk is at most equal to the curvature of metric dˆs2. Assume that dˆs2 is circularly symmet- the original at the same distance from the center. ric, so that Then what we want to show, inequality (28), is 2 2 b 2 2 that each geodesic disk

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Theorem 4 completes in a certain sense this is the extremal surface, giving the precise value of circle of ideas by going back to the original another Bloch-type constant, analogous to the one Schwarz-type lemma for a map of a finite disk in Bloch’s Theorem (Theorem 1 above). Further- into a finite disk, in contrast to the Schwarz-Pick- more, the authors show that their result implies Ahlfors-Yau-Troyanov-Ratto-Rigoli-Véron versions, the original Bloch Theorem, as well as its striking all of which require the domain of the map to be generalization by Ahlfors to the “five-island theo- provided with a complete Riemannian metric. At rem”, stating that for any five Jordan domains on the same time, it applies to maps in which one has the sphere whose closures are disjoint, at least one a pointwise comparison of Gauss curvatures. How- of them must be simply covered by the image of ever, whereas the hypotheses of the papers [T] any nonconstant meromorphic function in the and [RRV] described in Theorem 3 compare the cur- plane. vatures at each point of the image with that at the A key idea in the Bonk-Eremenko proof is to in- preimage of the point under a given holomorphic troduce a metric on a branched Riemann surface map, Theorem 4 compares curvatures in the image over the unit sphere that is the ordinary spherical with curvatures in the domain at comparable dis- metric away from branch points and has infinite tances from fixed points, independent of the map. negative curvature at the branch points. One con- In cases such as the original Ahlfors Lemma and siders the surface to be the union of spherical tri- the Yau generalization, where there are global angles satisfying certain conditions. Then the idea, bounds on curvatures, there is no difference be- in the authors’ words, is that “if the triangles are tween the two types of comparison. small enough, then the negative curvature con- We conclude with two final notes. centrated at the vertices dominates the positive cur- First, the Schwarz Lemma may be pictured as vature spread over the triangles. Thus on a large the progenitor of a huge family tree, branching out scale our surface looks like one whose curvature in many directions. In this article, we have fol- is bounded above by a negative constant.” lowed just one of those branches, but there are And so the fundamental insight of Ahlfors con- many others. To name just two, there are the many cerning the Schwarz-Pick Lemma continues to bear generalizations to higher dimensions and the “Dis- fruit in the most beautiful and unexpected new crete Schwarz-Pick Lemma” for circle packings ways. proved by A. F. Beardon and K. Stephenson in 1991, which was applied in 1996 by Z.-X. He and References O. Schramm to give a new proof of Thurston’s in- [A1] L. V. AHLFORS, An extension of Schwarz’s Lemma, novative approach to the Riemann mapping theo- Trans. Amer. Math. Soc. 43 (1938), 359–364. rem via circle packing. [A2] ——— , Collected Papers, Birkhäuser, Basel, 1982. Second, the particular branch we have pursued [BE] M. BONK and A. EREMENKO, Covering properties of here has blossomed in a most remarkable way in meromorphic functions, negative curvature and a recent result [BE] announced by M. Bonk and spherical geometry, preprint. A. Eremenko. We can describe their main result as [O1] R. OSSERMAN, Conformal geometry, in The Mathe- follows. matics of Lars Valerian Ahlfors, Notices Amer. Math. Consider a triangulation of the Riemann sphere Soc. 45 (1998), 233–236. consisting of four equilateral triangles with vertices [O2] ——— , A new variant of the Schwarz-Pick-Ahlfors at the points of an inscribed regular tetrahedron. Lemma, Manuscripta Math. (to appear). [O3] , A sharp Schwarz inequality on the boundary, Let f be a conformal map of a Euclidean equilat- ——— Proc. Amer. Math. Soc. (to appear). eral triangle onto one of those four spherical tri- [OR] R. OSSERMAN, and M. RU, An estimate for the Gauss angles. By successive reflections, f can be extended curvature of minimal surfaces in Rm whose Gauss to a meromorphic function in the entire plane map omits a set of hyperplanes, J. Differential Geom. whose image will be an infinite-sheeted Riemann 46 (1997), 578–593. surface over the sphere with simple branch points [P] G. PICK, Über eine Eigenschaft der konformen Ab- at each of the vertices of the triangulation. Each bildung kreisförmiger Bereiche, Math. Ann. 77 (1916), circular disk on the sphere whose boundary circle 1–6. passes through three vertices of the triangulation [R] A. ROS, The Gauss map of minimal surfaces, preprint. will have an infinite number of unbranched sheets [RRV] R. RATTO, M. RIGOLI, and L. VÉRON, Conformal im- of the surface lying over its interior. Said differ- mersions of complete Riemannian manifolds and ex- ently, every such circular disk on the sphere is the tensions of the Schwarz lemma, Duke Math. J. 74 (1994), 223–236. one-one conformal image under f of (infinitely [S] H. A. SCHWARZ, Gesammelte Mathematische Ab- many) simply connected regions in the domain. handlungen. II, Springer-Verlag, Berlin, 1890. What Bonk and Eremenko assert is that, for any [T] M. TROYANOV, The Schwarz Lemma for nonpositively smaller disk on the sphere, every meromorphic curved Riemannian surfaces, Manuscripta Math. 72 function in the plane has the property that its (1991). image contains an unbranched disk of at least that [Y] S.-T. YAU, Remarks on conformal transformations, size. In other words, the surface described above J. Differential Geom. 8 (1973), 369–381.

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