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Pacific Journal of Mathematics Vol 228 Issue 2, Dec 2006 PACIFIC JOURNAL OF MATHEMATICS Pacific Journal of Mathematics Volume 228 No. 2 December 2006 PacificPPPPPPPacificacificacificacificacificacificacificPPPPPPacificacificacificacificacificacificPacificPPacificacificPacificPacific JournalJJJJJJJournalournalournalournalournalournalournalofofofofofofofJJJJJJournalournalournalournalournalournalofofofofofofJournalJJournalournalofofJournalJournal of of of of MathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematicsMathematics 2006 Vol. 228, No. 2 Volume 228 No. 2 December 2006 PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Vyjayanthi Chari Los Angeles, CA 90095-1555 Sorin Popa Department of Mathematics pacifi[email protected] Department of Mathematics University of California University of California Riverside, CA 92521-0135 Los Angeles, CA 90095-1555 [email protected] [email protected] Robert Finn Darren Long Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University University of California University of California Stanford, CA 94305-2125 Santa Barbara, CA 93106-3080 Santa Cruz, CA 95064 fi[email protected] [email protected] [email protected] Kefeng Liu Jiang-Hua Lu Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California The University of Hong Kong University of California Los Angeles, CA 90095-1555 Pokfulam Rd., Hong Kong Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] PRODUCTION pacifi[email protected] Paulo Ney de Souza, Production Manager Silvio Levy, Senior Production Editor Alexandru Scorpan, Production Editor SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI UNIVERSIDAD DE LOS ANDES UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. OF ARIZONA UNIV. OF HAWAII INST. DE MATEMÁTICA PURA E APLICADA UNIV. OF BRITISH COLUMBIA UNIV. OF MONTANA KEIO UNIVERSITY UNIV. OF CALIFORNIA, BERKELEY UNIV. OF NEVADA, RENO MATH. SCIENCES RESEARCH INSTITUTE UNIV. OF CALIFORNIA, DAVIS UNIV. OF OREGON NEW MEXICO STATE UNIV. UNIV. OF CALIFORNIA, IRVINE UNIV. OF SOUTHERN CALIFORNIA OREGON STATE UNIV. UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF UTAH PEKING UNIVERSITY UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF WASHINGTON STANFORD UNIVERSITY UNIV. OF CALIFORNIA, SAN DIEGO WASHINGTON STATE UNIVERSITY UNIV. OF CALIF., SANTA BARBARA These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no respon- sibility for its contents or policies. See inside back cover or www.pjmath.org for submission instructions. Regular subscription rate for 2006: $425.00 a year (10 issues). Special rate: $212.50 a year to individual members of supporting institutions. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company, 11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index. The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 Evans Hall, Berkeley, CA 94720-3840 is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2007 by Pacific Journal of Mathematics PACIFIC JOURNAL OF MATHEMATICS Vol. 228, No. 2, 2006 SCHWARZIAN DERIVATIVES AND A LINEARLY INVARIANT FAMILY IN ރn RODRIGO HERNÁNDEZ R. We use Oda’s definition of the Schwarzian derivative for locally univalent holomorphic maps F in several complex variables to define a Schwarzian derivative operator ᏿F. We use the Bergman metric to define a norm k᏿Fk for this operator, which in the ball is invariant under composition with automorphisms. We study the linearly invariant family n n Ᏺα = {F : ނ → ރ | F(0) = 0, DF(0) = Id, k᏿Fk ≤ α}, estimating its order and norm order. 1. Introduction The link between the Schwarzian derivative of a locally univalent holomorphic map in one complex variable, given by f 00 0 1 f 00 2 S f = − , f 0 2 f 0 with the univalence of f and distortion problems has been studied extensively; see [Chuaqui and Osgood 1993; Epstein 1986; Kraus 1932; Nehari 1949], for example. S f vanishes identically if and only if f is a Mobius¨ mapping, and we have S( f ◦ g) = (S f ◦ g)(g0)2 + Sg. An analytic function f with Schwarzian derivative S f = 2p has the form f = u/v, where u and v are any linearly independent solutions of the equation u00 + pu = 0. If f is defined in the unit disk ބ, the norm kS f k = sup (1 − |z|2)2|S f (z)| |z|=1 is invariant under precomposition with automorphisms of the disk. Some analogues of the Schwarzian derivative in several complex variables are available, but results relating it to the aforementioned problems of univalence and MSC2000: primary 32A17, 32W50; secondary 32H02, 30C35. Keywords: Several complex varaibles, Schwarzian derivative, Linearly invariant families, Sturm comparison. 201 202 RODRIGO HERNÁNDEZ R. distortion are less satisfactory than in one variable [Molzon and Pinney Mortensen 1997]. Consider the overdetermined system of partial differential equations 2 n ∂ u X k ∂u 0 (1-1) = Pi j (z) + Pi j (z)u, i, j = 1, 2,..., n, ∂zi ∂z j ∂zk k=1 n where z = (z1, z2,..., zn) ∈ ރ . The system is called completely integrable if (1-1) has n + 1 linearly independent solutions. The system (1-1) is said to be in canonical form (see [Yoshida 1976]) if the coefficients satisfy n X j Pi j (z) = 0, i = 1, 2,..., n. j=1 k T. Oda [1974] defined the Schwarzian derivative Si j of a locally injective holomor- phic mapping F(z1, z2,..., zn) = (w1, w2, . , wn) as n 2 k X ∂ wl ∂zk 1 k ∂ k ∂ Si j F = − δi + δ j log 1, ∂zi ∂z j ∂wl n + 1 ∂z j ∂zi l=1 k where i, j, k = 1, 2,..., n, 1 = det(∂ F/∂z), and δi is the Kronecker symbol. For n > 1 these Schwarzian derivatives satisfy k Si j F = 0 for all i, j, k = 1, 2,..., n if and only if F(z) is a Mobius¨ transformation, that is, if it has the form l (z) l (z) F(z) = 1 ,..., n , l0(z) l0(z) where li (z) = ai0 + ai1z1 + · · · + ain zn with det(ai j ) 6= 0. For a composition we have n k k X r ∂wl ∂wm ∂zk (1-2) Si j (G ◦ F)(z) = Si j F(z) + Slm G(w) , w = F(z). ∂zi ∂z j ∂wr l,m,r=1 k k Thus, precomposition with a Mobius¨ transformation G leads to Si j (G ◦ F) = Si j F. 0 The coefficients Si j F are given by 2 n 0 1/(n+1) ∂ −1/(n+1) X ∂ −1/(n+1) k Si j F(z) = 1 1 − 1 Si j F(z) . ∂zi ∂z j ∂zk k=1 −1/n+1 k k The function u = 1 is always a solution of (1-1) with Si j F = Pi j . = 1 = 0 = − 1 Remark 1.1. For n 1, S11 f 0 for all locally injective f , but S11 f 2 S f . SCHWARZIAN DERIVATIVES AND A LINEARLY INVARIANT FAMILY IN ރn 203 Proposition 1.2 [Yoshida 1976]. Let (1-1) be a completely integrable system in canonical form and consider a set u0(z), u1(z), . , un(z) of linearly independent solutions. Then k k Pi j (z) = Si j F(z), i, j, k = 1, 2,..., n, where F(z) = (w1(z), . , wn(z)) and wi (z) = ui (z)/u0(z). Remark 1.3. In contrast to the one-dimensional case, when n > 1 the Schwarzian k derivatives Si j F are differential operators of order 2. One way to understand this phenomenon is through a dimensional argument: For n = 1 the Mobius¨ group has 0 00 dimension 3, which allows one to choose f (z0), f (z0) and f (z0) for a holomor- phic mapping f at a given point z0 arbitrarily. It would therefore be pointless to seek a Mobius-invariant¨ differential operator of order 2. But for n > 1 the number of parameters involved in the value and all derivatives of order 1 and 2 of a locally biholomorphic mapping is n2(n + 1)/2 + n2 + n, which exceeds the dimension 2 n k k n + 2n of the corresponding Mobius¨ group in ރ . Moreover, since Si j F = S ji F for all k and n X j Si j F = 0, j=1 k there are exactly n(n − 1)(n + 2)/2 independent terms Si j F, which is equal to the excess mentioned above. k In this paper we employ the Oda Schwarzian derivatives Si j to propose a Schwar- zian derivative operator ᏿F. Using the Bergman metric, we will define a norm for ᏿F, which for mappings defined in the ball ނ turns out to be invariant under the group of automorphisms. We then focus on the study of geometric properties of the linearly invariant family given by bounded Schwarzian norm. We will appeal to the relationship with the completely integrable system (1-1) and Sturm comparison techniques adapted to this special situation. 2. The Schwarzian derivative operator n n For ⊂ ރ open, let F : → ރ , F(z1,..., zn) = (w1, . , wn), be a locally univalent holomorphic mapping, and set 1 = det(∂ F/∂z). For k = 1,..., n, define an n × n matrix k k ޓ F = (Si j F), i, j = 1,..., n.
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