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ICAl 2002 • Vol. II • 681-690Value Distribution and Potential Theory*A. Eremenko^AbstractWe describe some results of value distribution theory of holomorphic curvesand quasiregular maps, which are obtained using potential theory. Among theresults discussed are: extensions of Picard's theorems to quasiregular mapsbetween Riemannian manifolds, a version of the Second Main Theorem ofNevanlinna for curves in projective space and non-linear divisors, descriptionof extremal functions in Nevanlinna theory and results related to Cartan's1928 conjecture on holomorphic curves in the unit disc omitting hyperplanes.2000 Mathematics Subject Classification: 30D35, 30C65.Keywords and Phrases: Holomorphic curves, Quasiregular maps, Meromorphicfunctions.1. IntroductionClassical value distribution theory studies the following question: Let / bea meromorphic function in the plane. What can one say about solutions of theequation f(z) = a as a varies? The subject was originated in 1880-s with twotheorems of Picard (Theorems 1 and 4 below). An important contribution wasmade by E. Borei in 1897, who gave an "elementary proof" of Theorem 1, whichopened a way to many generalizations. Borel's result (Theorem 12 below) alsogives an extension of Picard's theorem to holomorphic curves C —¥ P". In 1925, R.Nevanlinna (partially in cooperation with F. Nevanlinna) created what is called nowthe Nevanlinna Theory of meromorphic functions, which was subject of intensiveresearch [5]. A good elementary introduction to the subject is [18]. Griffiths andKing [16] extended Nevanlinna theory to non-degenerate holomorphic maps / :C" —¥ Y, where Y is a compact complex manifold of dimension n. In modern timesthe emphasis has shifted to two multi-dimensional generalizations: holomorphiccurves in complex manifolds and quasiregular mappings between real Riemannianmanifolds. This survey is restricted to a rather narrow topic: generalizations of* Supported by NSF grant DMS 0100512 and by the Humboldt Foundation.1 Department of Mathematics, Purdue University, West Lafayette IN 47907, USA. E-mail:eremenko@math.purdue.edu
682 A. EremenkoPicard's theorem that are obtained with potential-theoretic methods. Some otherapplications of potential theory to value distribution can be found in [14, 20, 27].Recent accounts of other methods in the theory of holomorphic curves are [21, 29].We begin with Picard's Little Theorem:Theorem 1 Every entire function which omits two values in C is constant.To prove this by contradiction, we suppose that / is a non-constant entirefunction which omits 0 and 1. Then « 0 = log|/| and «i = log|/ — 1| are nonconstantharmonic functions in the plane satisfying\UQ—U{\—C, (1.1)where V stands for the pointwise sup, u + = u V 0, and c is a constant. Thereare several ways to obtain a contradiction from (1.1). They are based on rescalingarguments that permit to remove the c terms in (1.1). To be specific, one can findsequences z^ £ C, rj, > 0 and Aj, —^ +oo such that A^1v,j(zk+ rj,z) —^ vfiz), k —^oo, |z|0, Vj(0) = 0, (1.2)and Vj ^ 0. This gives a contradiction with the uniqueness theorem for harmonicfunctions. The idea to base a proof of Picard's theorem on (1.2) comes from thepaper [13] (the main result of this paper is described in Section 3 below). Twoversions of the rescaling argument (existence of appropriate z%,r^ and A\A are givenin [7, 12] and [19], respectively. The second version has an advantage that it usesonly one result from potential theory, Harnack's inequality. Thus Picard's theoremcan be derived from two facts: Harnack's inequality and the uniqueness theoremfor harmonic functions. This makes the argument suitable for generalizations.2. Quasiregular maps of Riemannian manifoldsWe recall that a non-constant continuous map / between regions in R n is calledif-quasiregular if it belongs to the Sobolev class W lo ' c n (first generalized derivativesare locally L"-summable), and in additionll/'IP < KJf almost everywhere, (2.1)where J is the Jacobian determinant and K > 1 is a constant. The standardreferences are [24, 25]. If n = 2, every quasiregular map can be factored as g o ,where g is analytic and a quasiconformal homeomorphism. It follows that Picard'sTheorems 1 and 4 (below) extend without any changes to quasiregular maps ofsurfaces. For the rest of this section we assume that n > 3, and that all manifoldsare connected. The weak smoothness assumption / £ Wfi" is important: if werequire more smoothness, the maps satisfying (2.1) will be local homeomorphisms(and even global homeomorphisms if the domain is R n ). A fundamental theoremof Reshetnyak says that all quasiregular maps are open and discrete, that is they
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Beijing 2002August 20-28Proceedings
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ContentsSection 1. LogicE. Bouscare
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R. Pandharipande: Three Questions i
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Section 1. LogicE. Bouscaren: Group
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4 E. Bouscaren2. Different notions
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6 E. BouscarenWe now consider an al
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8 E. Bouscarendimension would need
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10 E. Bouscaren1. Either G is not o
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12 E. Bouscaren[7] E. Bouscaren & F
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14 J. Denef F. Loeserthis theory to
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16 J. Denef F. Loeserwill require m
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18 J. Denef F. Loeserseries P P (T)
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20 J. Denef F. Loeserfor all m> n.
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22 J. Denef F. Loeseri > 1. One ver
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ICM 2002 • Vol. II • 25-33Autom
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Automorphism Groups of Saturated St
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ICM 2002 • Vol. II • 37-45Repre
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3. The Blanchfield pairingRepresent
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Representations of Braid Groups 41n
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Representations of Braid Groups 43A
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Representations of Braid Groups 45T
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48 A.Bondal D.Orlovis believed to b
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50 A.Bondal D.OrlovDefinition 4 [BK
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52 A.Bondal D.OrlovLet Y be a smoot
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54 A.Bondal D.OrlovIn particular, w
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56 A.Bondal D.Orlov[BVdB] Bondal A.
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58 M. Levine(PB) Let £ be a rank r
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60 M. Levine2. Let 1Z d%m (X) be th
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62 M. LevineNow, if P = P(ci,. •
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64 M. LevineProof. For CH*, this us
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66 M. Levineis an isomorphism. Sinc
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68 Cheryl E. Praegeranalysis. Analo
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70 Cheryl E. Praegers-arc-transitiv
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72 Cheryl E. Praegercase a good kno
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74 Cheryl E. Praeger5. Simple group
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76 Cheryl E. Praeger[15] C. H. Li,
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78 Markus Rost1. Norm varietiesAll
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80 Markus RostWe apply the degree f
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82 Markus Rost4. Hubert's 90 for sy
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84 Markus RostFor n = 2 one can tak
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ICM 2002 • Vol. II • 87^92Dioph
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Diophantine Geometry over Groups an
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Diophantine Geometry over Groups an
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ICM 2002 • Vol. II • 93^103None
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Noncommutative Projective Geometry
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106 Dimitri Tamarkinalgebra and def
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108 Dimitri TamarkinThe product on
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110 Dimitri TamarkinProof. Let F =
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112 Dimitri Tamarkin2.5.9.Similarly
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114 Dimitri TamarkinComputeJ2,jl(a
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116 Dimitri Tamarkin3.2.1.To descri
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ICM 2002 • Vol. II • 119-128Con
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Converse Theorems, Functoriality, a
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ICM 2002 • Vol. II • 129^138Con
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Constructing and Counting Number Fi
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ICM 2002 • Vol. II • 139-148Ana
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Analyse p-adique et Représentation
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ICM 2002 • Vol. II • 149-162Equ
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Equiv. Bloch-Kato Conjecture and No
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[17;[is;[19;[20;[21[22[23;[24;[25;[
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ICM 2002 • Vol. II • 163-171Tam
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Tamagawa Number Conjecture for zeta
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174 S. S. Kudlaseries with represen
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t.176 S. S. Kudlais the exponential
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178 S. S. Kudlawhere Z = J2ì n ìP
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180 S. S. Kudla(ii) T £ Sym 2 (Z)>
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182 S. S. KudlaTheorem 6. ([13], [9
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ICM 2002 • Vol. II • 185-195Ell
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Elliptic Curves and Class Field The
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198 E. UllmoSoit X une variété al
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200 E. UllmoThéorème 2.4 Soit X u
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202 E. UllmoQuestion 4.1 Soit x n =
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204 E. UllmoNotons que cet énoncé
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206 E. Ullmo[17] H. Oh. Uniform Poi
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208 T. D. WooleyWaring's problem as
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210 T. D. Wooleycircumstances one h
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212 T. D. Wooleywith 1 < z,w < F, M
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214 T. D. Wooleyand also byl f \K(a
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216 T. D. Wooleytions involving nor
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Section 4. Differential GeometryB.
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222 B. Andrewspotentially applicabl
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224 B. Andrewssurface can look metr
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226 B. Andrewsboundary of the set {
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228 B. Andrewseach t > 0, to either
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230 B. AndrewsUSSR Izv. 20 (1983).[
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232 Robert Bartnikprovides an impor
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234 Robert BartnikTheorem 2 If (M,
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236 Robert BartnikThe horizon condi
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238 Robert BartnikTheorem 8 Suppose
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240 Robert BartnikPhysics, LNP 212.
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242 P. Biran2. Various intersection
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244 P. BiranTheorem E'. Let (M,oj)
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246 P. Biransymplectic packings in
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248 P. Birangroup of Hamiltonian di
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250 P. Birannumber of steps it take
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252 P. Biranbe used to obtain many
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254 P. Biran[6] P. Biran, A stabili
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ICM 2002 • Vol. II • 257-271Bla
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Black Holes and the Penrose Inequal
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[23;[24;[25;[26;[27[28;[29'[30^Blac
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274 Xiuxiong ChenFor simplicity, in
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276 Xiuxiong ChenTheorem 0.1. [14]
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278 Xiuxiong ChenQuestion 0.6. (Don
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280 Xiuxiong Chencurvature in S 2 c
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282 Xiuxiong Chen[12] X. X. Chen an
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284 Weiyue DingSehrödinger flows a
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286 Weiyue DingIn order to prove Th
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288 Weiyue DingFor the first term i
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290 Weiyue Dingwhich satisfies the
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ICM 2002 • Vol. II • 293^302Dif
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Differential Geometry via Harmonic
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then eitherorDifferential Geometry
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ICM 2002 • Vol. II • 303-313Ind
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Index Iteration Theory for Symplect
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316 Anton PetruninTheorem A. Given
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318 Anton PetruninRiemannian manifo
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320 Anton Petruninspace. In the cas
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ICM 2002 • Vol. II • 323-338Col
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Collapsed Riemannian Manifolds with
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ICM 2002 • Vol. II • 339-349Com
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Complex Hyperbolic Triangle Groups
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352 Paul Seideldeformation spaces.
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354 Paul Seidelforgetting all the p
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356 Paul SeidelTheorem 3. Si,...,S
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358 Paul Seidelwhich must be of ord
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360 Paul Seidel[10] E. Getzler, Bat
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362 Weiping Zhang(iii) The heat ker
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364 Weiping Zhang3. The index theor
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366 Weiping Zhangwhere ch(g) is the
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- T p ( P d M , > 0 , 9 P d M , > 0
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Section 5. TopologyMladen Bestvina:
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374 Mladen Bestvinaof Out(F n ) coc
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376 Mladen Bestvina3. Culler-Vogtma
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378 Mladen BestvinaDefinition 8. A
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380 Mladen Bestvinaand then gluing
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382 Mladen BestvinaAnal. 10 (2000),
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384 Mladen Bestvina63 John R. Stall
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386 Yu. V. Chekanovplu I,Figure 1:
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388 Yu. V. Chekanovwithout loss of
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390 Yu. V. ChekanovFigure 2: Lagran
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392 Yu. V. ChekanovFigure 5: Local
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394 Yu. V. Chekanovderived from the
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396 M. Furatadimensional vector spa
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398 M. FurataHowever in the Seiberg
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400 M. Furata6. Seiberg-Witten-Floe
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402 M. Furata[14] K. Fukaya & K. On
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ICM 2002 • Vol. II • 405-114Gé
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Géométrie de Contact 407- la sph
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Géométrie de Contact 4093) chaque
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412 E. Giroux- en tout point p où
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414 E. Giroux[EU] Y. ELIASHBERG, Cl
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416 L. Hesselholt1. Algebraic üC-t
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418 L. Hesselholt(i) a pro-log diff
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420 L. HesselholtThe canonical map
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422 L. Hesselholttogether with a mu
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424 L. Hesselholtwhere \'- GK —^
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ICM 2002 • Vol. II • 427-436Sym
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Symplectic Sums and Gromov-Witten I
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ICM 2002 • Vol. II • 437-146Kno
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Knots, von Neumann Signatures, and
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ICM 2002 • Vol. II • 447-156Str
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Strings and the Stable Cohomology o
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ICM 2002 • Vol. II • 457-168Non
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Non-zero Degree Maps between 3-Mani
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Section 6. Algebraic and Complex Ge
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472 Hélène Esnaultto ^klX)/k carr
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474 Hélène Esnaultf*c 2 (E, V) is
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476 Hélène EsnaultTheorem 3.1 ([5
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478 Hélène Esnaultno longer the c
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480 Hélène EsnaultQuestion 5.4. W
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ICM 2002 • Vol. II • 483-494Hil
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Hilbert Schemes of Points on Surfac
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and use this to define operatorsHil
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ICM 2002 • Vol. II • 495-502Vec
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Vector Bundles on a K3 Surface 497s
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Vector Bundles on a K3 Surface 499A
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Vector Bundles on a K3 Surface 501a
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ICM 2002 • Vol. II • 503-512Thr
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Three Questions in Gromov-Witten Th
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ICM 2002 • Vol. II • 513^524Upd
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Update on 3-folds 515in the hyperbo
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Update on 3-folds 517In many contex
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Update on 3-folds 5193.3. Explicit
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Update on 3-folds 521The modem view
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Update on 3-folds 523is entertainin
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ICM 2002 • Vol. II • 525-532Sur
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Sur les Algèbres Vertex Attachées
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ICM 2002 • Vol. II • 533-541Top
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Topology of Singular Algebraic Vari
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ICM 2002 • Vol. II • 545-554Har
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Harmonie Analysis on Real Reductive
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[BS3][BS4][Be]Harmonie Analysis on
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ICM 2002 • Vol. II • 555-570On
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On the Dynamical Yang-Baxter Equati
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ICM 2002 • Vol. II • 571-582Geo
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Geometrie Langlands Correspondence
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ICM 2002 • Vol. II • 583-597On
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On the Local Langlands Corresponden
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600 A. KlyachkoAmong commonly known
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602 A. KlyachkoBy Theorem 2.3 this
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604 A. Klyachkois stable iff for ev
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606 A. KlyachkoUnitary spectral pro
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608 A. Klyachkospectra. By Metha-Se
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610 A. KlyachkoAgain our experience
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612 A. Klyachko[9] G. Faltings, Mum
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ICAl 2002 • Vol. II • 615-627Br
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Branching Problems of Unitary Repre
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Page 604 and 605: Branching Problems of Unitary Repre
Page 606 and 607: 630 Vikram Bhagvandas AlehtaDefinit
Page 608 and 609: 632 Vikram Bhagvandas Alehtais in t
Page 610 and 611: 634 Vikram Bhagvandas AlehtaTheorem
Page 612 and 613: ICAl 2002 • Vol. II • 637^642Cl
Page 614 and 615: Clifford Algebras and the Duflo Iso
Page 616 and 617: Clifford Algebras and the Duflo Iso
Page 618 and 619: ICAl 2002 • Vol. II • 643^654Re
Page 620 and 621: Representations of Yangians 6451.2.
Page 622 and 623: Representations of Yangians 6478 9
Page 624 and 625: Representations of Yangians 6492. T
Page 626 and 627: Representations of Yangians 651irre
Page 628 and 629: Representations of Yangians 6532.4.
Page 630 and 631: ICAl 2002 • Vol. II • 655-666Au
Page 632 and 633: Automorphic L-functions and Functor
Page 642 and 643: ICAl 2002 • Vol. II • 667^677Mo
Page 644 and 645: Modular Representations 669(l.b.2)
Page 646 and 647: Modular Representations 671e) Any p
Page 648 and 649: Modular Representations 673conjugac
Page 650 and 651: Modular Representations 675A genera
Page 652 and 653: Modular Representations 677^-repres
Page 656 and 657: Value Distribution and Potential Th
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Page 666 and 667: The Branch Set of a Quasiregular Al
Page 674 and 675: ICAl 2002 • Vol. II • 701-709Ha
Page 676 and 677: Harmonie Aleasure and "Locally Flat
Page 684 and 685: 712 N. Lerner1. From Hans Lewy to N
Page 686 and 687: 714 N. Lernergood cases in (1.5) (w
Page 688 and 689: 716 N. Lerner2. Counting the loss o
Page 690 and 691: 718 N. LernerSolvability with loss
Page 692 and 693: 720 N. Lerner6] L.Hörmander, On th
Page 694 and 695: 722 C. ThieleA basic point of singu
Page 696 and 697: 724 C. ThieleFigure 1: "Cone"< en I
Page 698 and 699: 726 C. Thieleis negative. If all of
Page 700 and 701: 728 C. Thielealso in these degenera
Page 702 and 703: 730 C. ThieleThus he needed good co
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732 C. Thiele[6] Gilbert J., Nahmod
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I p : — \ J ( Z i , . . . , Z m )
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736 S. ZelditchTheorem 1 [2] The pa
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738 S. Zelditchfor U C
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740 S. Zelditch[Pi,Pj] = 0 and whos
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742 S. Zelditch[11] V. F. Lazutkin,
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744 Xiangyu Zhou+00 at the boundary
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746 Xiangyu Zhouconnected component
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748 Xiangyu ZhouWightman [32], Jost
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750 Xiangyu ZhouBrief proof is as f
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752 Xiangyu Zhou[11] Al. Jarnicki,
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Section 9. Operator Algebras andFun
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758 Semyon Alesker0.1.1 Definition,
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760 Semyon Alesker2. Translation in
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762 Semyon Alesker2.3. Valuations i
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764 Semyon AleskerReferences[1] Ale
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766 P. Bianewith classical cumulant
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768 P. BianeObserve thatr(ai ... a
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770 P. Biane4. Noncrossing cumulant
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772 P. Bianethis gives the order of
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774 P. Biane[20] R. Speicher, Combi
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776 D. Bischsubfactors has develope
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778 D. Bisch(for all k > 0), where
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780 D. BischObserve that by constru
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782 D. BischIt should be evident th
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784 D. Bisch[4] D. Bisch, Bimodules
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ICAl 2002 • Vol. II • 787^794Fr
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Free Probability, Free Entropy and
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ICAl 2002 • Vol. II • 795^812Ba
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Banach KK-theory and the Baum-Conne
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ICAl 2002 • Vol. II • 813-822On
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On Some Inequalities for Gaussian A
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Author IndexAlesker, Semyon 757Andr
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