The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.

Transcription

1 A General Schwarz Lea for Kahler Manifolds Author(s): Shing-Tung Yau Reviewed work(s): Source: Aerican Journal of Matheatics, Vol. 100, No. 1 (Feb., 1978), pp Published by: The Johns Hopkins University Press Stable URL: Accessed: 06/12/ :46 Your use of the JSTOR archive indicates your acceptance of the Ters & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use inforation technology and tools to increase productivity and facilitate new fors of scholarship. For ore inforation about JSTOR, please contact support@jstor.org. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to Aerican Journal of Matheatics.

2 A GENERAL SCHWARZ LEMMA FOR KAHLER MANIFOLDS. By SHING-TUNG YAU. Introduction. The classical Schwarz-Pick lea states that any holoorphic ap of the unit disk into itself decreases the Poincare etric. Later Ahlfors generalized this lea to holoorphic appings between two Rieann surfaces where curvatures of these Rieann surfaces were used in a very explicit way. More recently, Chern initiated the study of holoorphic appings between higher-diensional coplex anifold by generalizing the Ahlfors lea to these spaces. Then this lea was further extended by Kobayashi, Griffiths, Wu and others. It plays a very iportant role in their theory. In this note, we shall prove the following generalization of the Schwarz lea. THEOREM Let M be a coplete Kahler anifokl with Ricci curvature bounded fro below by a constant, and N be another Heritian anifokl with holowrphic bisectional curvature bounded fro above by a negative constant. Then any holowrphic apping fro M into N decreases distances up to a constant depending only on the curvatures of M and N. The ain point here is that the doain M is a very general anifold. The best known result in this direction is due to Kobayashi [3], who assued that M is a bounded doain whose Bergan kernel function should behave well near the boundary. The ethod eployed previously in proving the Schwarz lea depends largely on a nice exhaustion of the anifold M. This is assued in order to assure the existence of a axial point of a certain function. In this note, we eliinate these hypotheses by applying a ethod that we developed in [5]. We would like to thank Professor H. Wu for his interest and encourageent in this work. Professor H. Royden infored us that he is able to iprove the estiate in our ain theore. Naely, he is able to replace K2 by the upper bound of the holoorphic sectional curvature of N. 1. Notation and Forulas for Heritian Manifolds. Let M be a Heritian anifold of diension. Let el, e2,..., e be a unitary frae field in an open set of M. Let 91, 02,... O be its cofrae field. Then there are Manuscript received Septeber 22, AericanjJoirnal of Mat/neatics, Vol. 100, No. 1, pp /78/ $01.50 Copyright? 1978 by The Johns Hopkins University Piress.

3 198 SHING-TUNG YAU. coplex-valued linear differential fors of type (1,0) such that the Heritian etric is given by ds2 = It is known that there are connection fors 9ij such that i=l i oi with d9i= j=l O,AOij9+ E (2) oij + ij = and Ei A=k- E (3) Tiek, The tensor Tiik is called the torsion tensor. The curvature fors Ejj are defined by E) d9ij + 9 Oik A,O (5) k and we have &.ij = -e. = 2 E Ri,klUkAol. (6) The skew-heritian syetry of (i, expressed by the first equation of (6) is equivalent to R'ikl = ilk (7) If = j4 ei and = iqi ei are two tangent vectors, then the holoorphic bisectional curvature deterined by t and q is defined [3] by i, j,k, l (E iti(ew nini) If =,q, the above quantity is called the holoorphic sectional curvature in directioa {. The Ricci tensor is defined as Rk= Riikl= Rlk, (8) i=l

4 GENERAL SCHWARZ LEMMA. 199 and the scalar curvature is defined as R=ERkk. (9) k Let Nn be another Heritian anifold with diension n. Then we can define the corresponding frae field Wa, curvature tensor Safiy8, Ricci tensor Sao and scalar curvature S. Let f: M'->Nn be any holoorphic apping. Then we can define f wa E aaii (10) and Clearly, we have U = E qj 5-.j (11) a,i f* dsn? uds. (12) In order to relate things to curvature, one has to copute the Laplacian of u. It is defined as follows. Let du= (uiji + Ui9) (13) Then the Laplacian of u is - d (E ui6ti )=E uii-6tiaoi (14) Au = E if- (15) The Chern-Lu forula [1] states the following:,au> E Rjiaaijaj-X X aaiafiaykaqksafyq. (16) asi,1~~~~i ik as,,o,y, B [In applying (16), one also has to use the eleentary fact that the bisectional curvature of a coplex subanifold is not greater than that of the abient anifold.]

5 200 SHING-TUNG YAU. 2. Schwarz Lea for General Coplex Manifolds. We shall apply the following theore of [5] and [6]. THEOREM 1. Let M be a coplete Rieannian anifold with Ricci curvature bounded fro below. Let f be a C2-function which is bounded fro below on M. Then for all E > 0, there exists a p in M such that at p, IgradfI <, \f > -- and f ( p) < inff+ e. (17) Now consider the function u defined in Section 1. Let c be any positive nuber. Then direct coputation shows '\( u-+c 2(u+ C )3/3 (u+c)( Hence applying (16), we have A I( u'+ )(u3c)5 ui vu(+ )s (u +Cs5/2 E 2(u+C)3/ (18) X [ E aaaajrji E E a..iaoia-yka7,ksaby71 (19 i, k a13iyk,qk,1 Let > 0 be any nuber. Then, by Theore 1, there is a point p such that at p, 4(u+c)3 A E i I I UiJ2 <, ) (20) 1 <inf 1 +e. u+c u+c Dividing (20) by /u+ c and coparing it with (21)8 we obtain (u [ E aia// E aaiaf3iaykyksa/3th] 2(u+c)2 La,i,j i, k a,,83, y,rq

6 GENERAL SCHWARZ LEMMA. 201 Let K1 be the greatest lower bound of the Ricci curvature of M, and K2 be the least upper bound of the holoorphic bisectional curvature of N. Then it follows fro (21) that -U K + u K2 )-24 (22) When c--0, 1/ u + c goes to its infiu and u goes to its supreu. Therefore, if K2 is negative and u is not identically zero, then K1 is non-positive and K1 0<supu < K,. (23) THEOREM 2. Let M be a coplete Kahler anifold with Ricci curvature bounded fro below by K1. Let N be another Heritian anifold with holoorphic bisectional curvature bounded fro above by a negative constant K2. Then if there is a non-constant holoaphic apping f fro M into N, we have K1?0 and f*dsn < # ds. (24) In particular, if K1 > 0, every holonorphic apping fro M into N is constant. Since the unit disk has a Kiihler etric with constant negative holoorphic sectional curvature, we have the following COROLLARY. Let M be a coplete Kahler anifold with non-negative Ricci curvature. Then M does not adit any bounded holoorphic function. In case dim= 1, one can weaken the hypothesis on N. THEOREM 2'. Let M be a coplete Rieann surface with curvature bounded fro below by a constant K1. Let N be another Heritian anifold with holoorphic sectional curvature bounded fro above by a negative constant K2. Then for any constant holoorphic apping f fro M into N, (24) holds. 3. Other Generalizations. Instead of taking the trace of the tensor f* dsn, we can also consider the other eleentary function of this tensor. Since

7 202 SHING-TUNG YAU. forulas corresponding to (16) still exist [4], one can derive corresponding properties for these eleentary functions. For siplicity, we shall only state the following THEOREM 3. Let M be a coplex Kahler anifold with scalar curvature bounded fro below by K1. Let N be another Heritian anifold with Ricci curvature bounded fro above by a negative constant K2. Suppose the Ricci curvature of M is bounded fro below and dim= din. Then the existence of a non-degenerate holowrphic ap f fro M into N iplies that K1 < 0 and K1 f* dvn? -dv, (25) K2 M'(5 where dvm, dvn are volue eleents of M and N respectively. We can partially generalize Theore 3 in the following sense: Let dvn be a non-negative top-diensional for on a coplex anifold N such that the Ricci curvature of dvn is bounded fro above by a negative constant K2. Let P be the ball whose Poincare' etric has scalar curvature K1 and whose diension is equal to din. Let f be a eroorphic ap (in the sense of Reert) apping the polydisk P into N. Then we have f*dvn < (K1/K2) dvp. The proof of this assertion follows fro the fact that f is holoorphic outside a subvariety of codiension two, so that f*dvn can be extended through this subvariety. The standard proof of the Ahlfors lea can be applied to prove our clai. Finally, we reark that Eells and Sapson [2] have studied haronic appings between two Rieannian anifolds. One can also deduce a forula siilar to (16) for this class of appings. However, in order to draw a useful conclusion, it sees that one has to assue the apping is quasi-conforal. STANFORD UNIVERSITY STANFORD, CALIFORNIA REFERENCES. [1] S. S. Chern, On holoorphic appings of Heritian anifolds of the sae diension, in: Proc. Syp. Pure Math. 11, Aer. Math. Soc., Providence, R.I., 1968, pp

8 GENERAL SCHWARZ LEMMA. 203 [2] J. Eells and J. H. Sapson, Haronic appings of Rieannian anifolds, Aer. J. Math. 86 (1964), pp [3] S. Kobayashi, Hyperbolic anifolds and holoorphic appings, in Pure and Applied Matheatics, Vol. 2, Marcel Dekker, New York, [4] V. C. Lu, Holoorphic appings of coplex anifolds, J. Differential Geoety 2 (1968), pp [5] S. T. Yau, "Haronic functions on coplete Rieannian anifolds," Co. Pure and Appl. Math., 28, pp (1975). [6] H. Oori, Isoetric iersions of Rieannian anifolds, J. Math. Soc. Japan 19 (1967), pp [7] S. Y. Cheng and S. T. Yau, "Differential equations on Rieannian anifolds and their geoetric applications," Co. Pure and Appl. Math. 28, pp (1975).