theorem for harmonic diffeomorphisms. Theorem. Let n be a complete manifold with Ricci 0, and let n be a simplyconnected manifold with nonpositive sec

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1 on-existence of Some Quasi-conformal Harmonic Diffeomorphisms Lei i Λ Department of athematics University of California, Irvine Irvine, CA lni@math.uci.edu October Introduction The property of harmonic maps between complete Riemannian manifolds has been studied extensively by many authors(cf. [Ch], [Sh], [T], etc). In the present paper we show some non-existence results for uasi-conformal harmonic diffeomorphism between complete Riemannian manifolds. In dimension two, harmonic maps are closely related to the deformation theory of Riemann surfaces. One of the uestions that arises naturally is: whether Riemann surfaces which are related by harmonic diffeomorphism are necessarily uasi-conformally related? See R. Schoen's article [S] for a general discussion on this subject, where other uestions were also discussed. The result we show in this paper provides some partial answers to the high dimension generalization of this type of uestions. In particular we prove the following result of which can be thought as a Liouville type Λ Research partially supported by SF grant #DS

2 theorem for harmonic diffeomorphisms. Theorem. Let n be a complete manifold with Ricci 0, and let n be a simplyconnected manifold with nonpositive sectional curvature, where n is the dimension of both manifolds. If there is a point p 2 such that lim r! = o(r n ), then there is no uasi-conformal harmonic diffeomorphism from into with polynomial growth energy density. It is not surprising that the growth rate of energy density plays a role here. For example in [W], T. Wan proved that a harmonic diffeomorphism between hyperbolic spaces of dimension two is uasi-conformal if and only if it has bounded energy density. The only if part of Wan's theorem was generalized to high dimension in [L-T-W]. There they proved that if the Ricci curvature of the domain manifold is bounded from below and the first eigenvalue of the target manifold is positive, then any uasi-conformal harmonic diffeomorphism into the target manifold has bounded energy density. These results and some other related results in [H-T-T-W] all indicate that the growth condition on the energy density is a natural assumption and is closely related to the study of uasi-conformal diffeomorphisms. On the other hand, we can show by examples that Theorem. will not be true if any of the assumptions is removed. We also should point out that, besides the curvature assumption, we only use the fact that the target manifold satisfies Sobolevineuality in the proof of Theorem.. So Theorem. still holds for more general target manifolds, for example, when is a minimal submanifolds of R K, since in this case, we know from [-S] Sobolev ineuality holds on. On the other hand, it is well-known that there is no non-constant holomorphic map from a complete Kähler manifold into a complex Hermitian manifold if has nonnegative Ricci curvature and has holomorphic bisectional curvature bounded from above by a negative constant. This follows easily from the generalized Schwarz lemma(cf. [Y]). Using a modified argument of Theorem. we can generalize this result to uasiconformal diffeomorphisms as follows: 2

3 Theorem.2 Let n be a complete Riemannian manifold with lim r! = o(r n ), and let n beacomplete Riemannian manifold with () > 0, where n is the dimension of both manifolds, p 2 is any fixed point, is the volume of the ball of radius r centered at p and () is the lower bound of the spectrum of the Laplacian-Beltrami operator. Then there is no uasi-conformal diffeomorphism form into. We know that if is simply-connected and has sectional curvature bounded from above by some negative constant then one has () > 0. That is why we call Theorem.2 a generalization of the above mentioned result on holomorphic maps( which is derived from the generalized Schwarz lemma). The interesting thing is that Theorem.2 is invariant under the uasi-isometries while Theorem. is not. Acknowledgements. The author would like to thank his advisor, Prof. P. Li for many valuable advices and Prof. J. Wang for helpful suggestions. The author would also like to thank the referee for many valuable comments. They were great help while improving the exposition of this paper. 2 The Proof of the Theorems Proof of Theorem.. We prove by contradiction. Assume that there is a harmonic diffeomorphism u from into. Let a 2 (x) = inf fv2txjkvk=g j(du) Λ ffi (du)(v)j 2. By choosing a suitable orthonormal frame fe i g around x in T we have that (du) Λ ffi (du) = 0 2 ::: ::: 2 n C A : We can assume that ::: 2 n = a 2 (x). By definition we have e(u) =± n i= 2 i J(u) = ::: n. and 3

4 Let ffi(x) be a smooth function with compact support defined on. Then ffi ffi u (y) is a smooth function with compact support defined on. By the assumption on we know that the Sobolev ineuality holds (Cf. [H-S]), i. e. we have a constant S such that jr'j dv S Applying to ffi ffi u we have (2.) j(rffi ffi u )(y)j dv S n j'j n n n ; for all ' 2 C0 (). On the other hand, direct calculation shows that jr (ffi ffi u )(y)j dv»» j(ffi ffi u n )(y)j n n n : j(r ffi)(x)ja (x)j(x)dv jr ffijj (x)::: n (x)jdv : Using the arithmetic-geometric ineuality ( ::: n ) 2 n» (2.2) Combining (2.) and (2.2) we have (2.3) P n i= 2 i (x) n» e(u) n jr (ffi ffi u )(y)jdv» C(n) j(r ffi)(x)j e n 2 dv : j(ffi ffi u n )(y)j n n n ow we estimate R j(ffi ffi u )(y)j n n n n» C(n; S) jr ffij e n 2 dv : we have using the uasi-conformity of uas follows. Recall the definition of the uasi-conformal constant to be ff = sup x2 n. Then we have i» ff, and J(x) = ::: n (a(x)) n ; nx e(x) = 2 i» a(x) 2 ((n )ff 2 +): i= 4

5 Combining them we have (2.4) j(ffi ffi u n )(y)j n n n dv C(n; ff) jffi(x)j n n e n n 2 dv n : (2.4) together with (2.3) implies (2.5) jffij n n e n n 2 dv n» C(n; ff; S) jr ffij e n 2 dv : The rest of the proof is just deriving a contradiction out of (2.5). Let S(r) = sup Bp(r) e(u). It is well known that e(u) isasubharmonic function under our curvature assumptions, from the Bochner formula for harmonic maps. By the mean-value ineuality for subharmonic functions on manifolds with nonnegative Ricci curvature we have By choosing ffi to be S( r 2 )» C(n) e(u)(x) dv : B p(r) ffi(x) = ( for x 2 Bp (r); 0 for x 2 n B p (2r); jrffij» C ;with C =2; r the ineuality (2.5) yields ψ! n Bp(r) e(u) n n 2 dv» C(n; ff; S) (V p(r)) n r V p (2r) V p (2r) Bp(2r) e(u) n 2 dv : Combining with Li-Schoen's mean-value ineuality and the volume doubling property of Ricci nonnegative manifolds we have (2.6) S( r 2 ) n 2» C (V p(r)) n r (S(2r)) n 2 : 5

6 ow we use the polynomial growth condition on the energy density of u. The polynomial growth assumption simply means that there exists a constant K such that S(r)» K(+r d ) for some d 0. But it is not hard to show that the polynomial growth condition implies that there exists a constant A>0 and r j!such that Applying (2.6) to r j we have S(2r j ) S( r j 2 )» A: S( r ) 0 n j 2 p (r j )) n C 2 (A) n 2 A» 0: r j Letting r j!and using the fact =o(r n )wehave lim r j! S(r j 2 )» 0: Since e(u)(x) is a subharmonic function and it achieves its maximum at infinity we have e(u)(x) 0, which is a contradiction since u is a uasi-conformal diffeomorphism..e.d. Corollary 2. There is no uasi-conformal harmonic diffeomorphism from S k R n k into R n with polynomial growth energy density. In order to prove Theorem.2 we need the following lemma which is well-known to the experts. But for the completeness we include a simple proof here. Lemma 2.2 If there exists a positive constant A p such that (2.7) j'j p dv» A p jr'j p dv ; for any ' 2 C0 (): Then there exists a positive constant A such that (2.8) jgj dv» A jrgj dv for any g 2 C0 (); provided p. In other words, L p -Poincaré implies L -Poincaré if p. 6

7 Proof. Let ψ = jgj p. Then jrψj = p p p jgj jrgj: Applying (2.7) to ψ we have jgj dv =» A p jψj p dv = A p ψ p» A p ψ p jrψj p dv! p! p jrgj p jgj p dv p jrgj p jgj p. ow we can begin to prove Theorem.2. By the assumption on we have the L2- Then we have (2.8) with A =(A p ) p Poincaré ineuality j'j 2» Applying Lemma 2.2 we have that jr'j 2 ; for any ' 2 C 0 (): j'j n dv» C( ;n) jr'j n dv n : As in the proof of Theorem. we apply above ineuality toffiffiu.then we have (2.9) j(ffi ffi u )(y)j n dv» C( ;n) Similar calculation as in the proof of (2.2) shows jr (ffi ffi u )(y)j n dv» C(ff) jr (ffi ffi u )(y)j n dv : j(r ffi)(x)j n dv : On the other hand same calculation as in the proof of (2.4) shows that jffi(x)j n e(u) n 2 dv» j(ffi ffi u )(y)j n dv : : 7

8 Combining the preceding two ineualities and choosing ffi as in the proof of Theorem. we will have (2.0) C( ;ff;n) 2: r n B p(r) e(u)n ow our assumption on the volume growth yields e(u) 0, which completes the proof. Combining the proof of Theorem. and Theorem.2 we can have the following corollary. Corollary 2.3 Let n beacomplete Riemannian manifold with nonnegative Ricci curvature, and let n beacomplete Riemannian manifold with nonpositive sectional curvature and () > 0, where n is the dimension of the both manifolds. Then there exists no uasi-conformal harmonic diffeomorphism from into. Proof. From (2.0) and the mean value ineuality of Li-Schoen we can write sup B p( r 2 ) e(u) n 2» C() B p(r) e(u) n 2» C( ;ff;n;) r n: Letting r!,wehave e(u) 0. We complete the proof. Finally we present two examples. They will show that in Theorem. both the volume growth assumption and the polynomial growth assumption of the energy density are indeed necessary. Example. This example shows that if we do not assume =o(r n )wecan't have our theorem. Let = = R n and u be the identity map between R n 's then u has bounded energy density and satisfies all the other assumptions of Theorem.. Example 2. This example shows that if we do not have the growth condition on the energy density our theorem also fails. Here we let = S R and = R 2, u mapping from into (in fact onto R 2 nf0g)isgiven by = exp(r) and! =, where dr 2 + d 2 is the metric on and d d! 2 is the metric on. We can see easily e(u) has exponential growth energy density and satisfies all the other assumptions in Theorem.. 8

9 References [Ch] [H-S] S. Y. Cheng, Liouville theorems for harmonic maps, Proc. Symp. Pure ath. 36(980), D. Hoffman, J. Spruck, Sobolev and isoperimetric ineualities for Riemannian submanifolds. Comm. Pure Appl. ath. 27 (974), [H-T-T-W]. Han, L.F. Tam, A. Treibergs and T. Wan, Harmonic aps From the Complex Plane into Surfaces with onpositive Curvature, to appear in Communications in Analysis and Geometry. [-S] J. ichael, L.. Simon, Sobolev and mean-value ineualities on generalized submanifolds of R n, Comm. Pure Appl. ath. 26(973), [L-T-W] P. Li, L.F Tam, and J. Wang Harmonic Diffeomorphisms Between Hardamard anifolds. to appear in Tran. of AS. [Sh] Y. Shen, A liouville theorem for harmonic maps, Amer. J. ath. 7(995), [S] R. Schoen, The role of harmonic mapping in rigidity and deformation problems, Collection:Complex Geometry (Osaka, 990), Lecture otes in Pure and Applied mathematics, Vol. 43, Dekker, ew York, 993, pp [T] L. F. Tam, Liouville properties of harmonic maps, ath. Res. Letters. 2(995), [W] [Y] T. Wan, Constant ean Curvature Surfaces, Harmonic aps, and Universal Teichmüller Space, J. Diff. Geom. 35(992) S. T. Yau, A generalized Schwartz lemma for Kähler manifolds, Amer. J. ath. 00(978)