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Reference Ic/86/6 REFERENCE IC/86/6 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS a^. STABLE HARMONIC MAPS FROM COMPLETE .MANIFOLDS Y.L. Xin INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1986 Ml RAM ARE-TRIESTE if ic/66/6 I. INTRODUCTION In [9] the author shoved that there is no non-.^trioLHri-t Vuliip harmonic maps from S with n > 3. Later Leung and Peng ['i] ,[T] proved a similar result for stable harmonic maps into S . These results remain true when the sphere is replaced by certain submanifolds in the Euclidean space [5],[6]. It should be International Atomic Energy Agency pointed out that these results are valid for compact domain manifolds. Recently, and Schoen and Uhlenbeck proved a Liouville theorem for stable harmonic maps from United Nations Educational Scientific and Cultural Organization Euclidean space into sphere [S]. The technique used In this paper inspires us INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS to consider stable harmonic maps from complete manifolds and obtained a generalization of previous results. Let M be a complete Riemannian manifold of dimension m. Let N be an immersed submanifold of dimension n in ]Rn * with second fundamental form B. Consider a symmetric 2-tensor T = <B(-,e.), B( •,£.)> in N, where {c..} Is a local orthonormal frame field. Here and in the sequel we use the summation STABLE HAKMOIIIC MAPS FROM COMPLETE MANIFOLDS * convention and agree the following range of Indices: Y.L. Xin ** o, B, T * n. International Centre for Theoretical Physics, Trieste, Italy. We define square roots of eigenvalues X. of the tensor T to be generalized principal curvatures of S in TEn <1. Let k (x) be a function on H defined by It. (x)= the maximum of the generalized principal curvature squared at x in ABSTRACT H, and kg(x) a function on fl given by k (x) = the minimum of all sectional By choosing distinguished cross-sections in the second variational formula curvature of N at x. We assume that the function k + (l-n)k is bounded above for harmonic maps from manifolds with not too fast volume growth into certain by a negative constant - k. submanifolds in the Euclidean space some Liouville type theorems have been proved in this article. A Riemannian manifold is called strongly parabolic if it admits no non- constant positive superharmonic function. Thus we have the following results. Theorem 1 MIRAMARE - TRIESTE If M is a compact or complete noncompact strongly parabolic and N is January 1986 an immersed submanifold in Euclidean space TR satisfying the conditions mentioned above, then any stable harmonic map f : M •+ N has to be constant. As is known 1R is strongly parabolic, while B , n 1 3> is not. In fact, dimension plays no role here, and the decisive criterion is the rate of volume growth. Karp [3] introduced that a Riemannian manifold has moderate volume m growth if there is such that sup vol B Cxn) ** for some * To be submitted for publication. F(r) ** Permanent address: Institute of Mathematics, Fudan University, Shanghai,. People's Republic of China. -2- XQ«M, here ^~= {F : (0,») + (0,»); r is increasing on (0,-) andJ: II. PRELIKINAEIES in H He is a geodesic tall of radius y and centered at xQ - Let H and H be Riemannian manifolds and f : M + N is a smooth map. showed that if M has moderate vojiane growth then it is strongly parabolic. Thus A map f : M -+ N is harmonic if it is a critical point of energy functional we have the following conclusion. Corollary 2 If M is a complete Riemannian manifold with moderate volume growth and M N is the manifold mentioned above, then any stable harmonic map from M into N where {e } is a local orthonormal frame field, < > means Riemannian metric Ct N has to be constant. in H. Assume that Ricci curvature of M is bounded below by a nonpositive In the case when M is Euclidean space the above result answered the case constant - A, A > 0, sectional curvature of N is bounded above by a positive of dimension 2, What would happen for a higher dimensional case. We have a slight constant k' and f i M •+ N is harmonic map with rank f .j y. Thus we have improvement of Schoen-Uhlenbeck's result. the following Bochner type formula [l]; *= | vdf |*+ Theorem 3 Let M be a complete Riemannian manifold whose Ricci curvature is bounded below by a nonpositive constant -A<A > 0). Let M be a manifold whose sectional curvature is bounded above by a positive constant k1 besides satisfying the conditions mentioned above. Assume f : M •+• H is a stable harmonic map with rank f then for any nonnegative smooth function f with Y 5 nk'-k compact support in M, we have (2) where c is a constant depending on n, m and k. On the other hand, Theorem (3) Let H be a complete Riemannian manifold with nonnegative Ricci curvature. If the volume of geodesic ball B (xn) M satisfies From (2) and (3) it follows R 0 0 m* H goes to infinity, then the harmonic map in Theorem 3 is constant. If M is By a computation in the paper [S] Iftmlidean space of dimension U, f is constant. 14 ^ (5) -3- -k- Substituting (5) into CO, we have which is a quadratic form in IR , Thus (6) Harmonic maps as solutions of the variational problem have second variational formula [l] (T) ( vhere v is a cross-section of f*TIf with compact support and 7*7 is the trace Laplace operator on f*TN Therefore, we have the stability inequality III. PROOF OF THEOREM 1 (9) Let u be a constant vector field in IR ^. Consider a vector field M on f(M) as follows where 41 is any smooth function on H with compact support. (8) If M is compact without boundary by choosing f E 1 in {9) f has to be constant which is the result of Leung and Peng. where {E.} is a local orthonormal frame field in N and is a smooth function If M is complete and noncompact (9) means the first eigenvalue of the in H with compact support. We then have operator L = 4 + — |df| on M is positive. By using Fischer-Colbrie and Schoen's theorem [2] there is a positive solution h > 0 to Lh = Ah + — |df| h = 0. Thus, (10) namely h is a positive superharmonic function on M. When M is strongly where {e } is a local orthonormal frame field in M. Substituting them into the parabolic, h has to be constant, which together with (10) implies |df| =0 second variational formula (Y)> we obtain Q.E.D. J IV. PROOFS Or THEOREM 3 MD THEOREM 1* K Replacing 4. by |df | if in (9), we have (11) 71 Jjv( H -6- -5- On the other hand if we multiply the inequality (6} by $ and integrate over Beplacing t 'y • in (16) we have the desired inequality. Thus, we complete the M we obtain proof of Theorem 3. When A = 0 (1) becomes .i •! i (17) M (12) Let B (Xn) be a geodesic ball of radius R and centered at vm£M, Choose J 11 U 0 .M the cutoff function By adding (11) and (12) 1 in t i: out of BR(XQ) J Prom .M (13) with I' <P I 5 f~ - (!T) it follows Y 1 It f 4 here we have considered the condition J— kr ^ —„ By using Cauchy inequality R goes to infinity. This means f is constant. For It-dimensional Euclidean space M = Ht choosing for any e > 0 (11) 'becomes Uh) 0 Substituting (13) into (lit) gives in (17) we have \dj\A*i * c (JfyR) -1 (15) Letting R go infinity we conclude that f is constant. This proves Theorem h. C is dependent on m and n. By using Cauchy inequality again we have for any e > 0 Remark (Tbviously Theorem It is valid for n ^ *t dimensional Euclidean space as domain manifolds. ACKNOWLEDGMENTS Thus (15) becomes The author would lite to thank Professor Atdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for -Z\ Theoretical Physics, Trieste. L (16) M -8- -7- REFERENCES [lj Bells and L. Lemaire, "A report on harmonic maps", Bull. London Math. Soc, 1£ (1978) 1-68. [2] D. Fischer-Coltrie and E. Schc-en, "The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature", Comm. Pure Appl. Math. 33_ (I960) 199-211. {33 L. Karp, "Subharmonic functions, harmonic mappings and isometric immersions11 Seminar on Differential Geometry Ann. of Math. Studies 102, (Princeton University Press), [M P.P. Leung, "On the stability of harmonic maps"LNM 9^9, 122-129, (Springer- Verlag). [5] P.F. Leung, "A note on stable harmonic maps", J. London Math. Soc. 29(2)> 380-38U. [6] Pan Yanglian, "Some nonexistence theorems on stable harmonic mappings", Chin. Ann. Math. 3.(10(1982) 515-518.. [7] Peng Jia Oui, "Some relations between minimal submanifolds and harmonic maps", Chin. Ann. Hath. Ser A 5(l)(19810, 85-9 . [8] R. Schoen and K. Uhlenbeck, "Regularity of minimizing harmonic maps into the sphere", Invent. Math. 78^ (1981*) 89-100. [9] Y.L. Xin, "Some results on stable harmonic maps", Duke Math. J. hj_ (1980), 609-613. -9- •'t \r jr.
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