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SOME RESULTS of F-BIHARMONIC MAPS 1. Introduction Harmonic Maps Play a Central Roll in Variational Problem

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SOME RESULTS of F-BIHARMONIC MAPS 1. Introduction Harmonic Maps Play a Central Roll in Variational Problem SOME RESULTS OF F -BIHARMONIC MAPS YINGBO HAN and SHUXIANG FENG Abstract. In this paper, we give the notion of F -biharmonic maps, which is a generalization of bi- harmonic maps. We derive the first variation formula which yields F -biharmonic maps. Then we investigate the harmonicity of F -biharmonic maps under the curvature conditions on the target mani- fold (N; h). We also introduce the stress F -bienergy tensor SF;2. Then, by using the stress F -bienergy tensor SF;2, we obtain some nonexistence results of proper F -biharmonic maps under the assumption that SF;2 = 0. Moreover, we derive some monotonicity formulas for the special case of the biharmonic map, i.e., where F -biharmonic map with F (t) = t. Then, by using these monotonicity formulas, we obtain new results on the non existence of proper biharmonic isometric immersions from complete manifolds. 1. Introduction Harmonic maps play a central roll in variational problems for smooth maps between manifolds 1 R 2 u:(M; g) ! (N; h) as the critical points of the energy functional E(u) = 2 M kduk dvg. On the other hand, in 1981, J. Eells and L. Lemaire [7] proposed the problem to consider the k-harmonic JJ J I II maps which are critical maps of the functional Z k 2 Go back k(d + δ) uk Ek(u) = dvg M 2 Full Screen Close Received January 10, 2013; revised July 15, 2013. 2010 Mathematics Subject Classification. Primary 58E20, 53C21. Key words and phrases. F -biharmonic map; stress F -bienergy tensor; harmonic map. Quit for smooth maps u : M ! N. G. Y. Jiang [9] studied the first and second variation formulas of the bienergy E2 where critical maps of E2 are called biharmonic maps. There have been extensive studies on biharmonic maps (for instance, see [9, 13, 14, 15, 16, 18, 19]). Let F : [0; 1) ! [0; 1) be a C3 function such that F 0 > 0 on (0; 1). For a smooth map u:(M; g) ! (N; h) between Riemannian manifolds (M; g) and (N; h), we define the F -k-energy EF;k(u) of u by Z k(d + δ)kuk2 EF;k(u) = F ( )dvg; M 2 which is Ek(u) if F (t) = t. When k = 1, we have Z kduk2 EF;1(u) = F dvg = EF (u); M 2 which was introduced by M. Ara in [1]. The critical maps of EF (u) are called F -harmonic maps which are the generalization of harmonic maps, p-harmonic maps or exponentially harmonic maps. There have been extensive studies in this area (for instance, [4,5, 11, 12 ]). When k = 2, we have Z kτ(u)k2 JJ J I II EF;2(u) = F dvg; M 2 Go back where τ(u) = −δdu = trace re (du). It is the bienergy of G.Y. Jiang [9], the p-bienergy of P. p t Full Screen Hornung and R. Moser [6] or exponentially bienergy when F (t) = t, F (t) = (2t) 2 or F (t) = e . We say that u is an F -biharmonic map if Close d EF;2(ut)jt=0 = 0 Quit dt for any compactly supported variation ut : M ! N with u0 = u. In this note, we derive the first variation formula which yields F -biharmonic maps. Then we investigate the harmonicity of F - biharmonic maps under the curvature conditions on the target manifold (N; h). We also introduce the stress F -bienergy tensor SF;2. Then, by using the stress F -bienergy tensor SF;2, we obtain some non existence results of proper F -biharmonic maps under the assumption SF;2 = 0. Also, we derive some monotonicity formulas for the special case of a biharmonic map, i.e., an F -biharmonic map with F (t) = t. Then, by using these monotonicity formulas, we investigate the harmonicity of biharmonic isometric maps from complete manifolds. Remark 1.1. In [17], the authors introduced f-biharmonic maps which are critical points of the bi-f-energy functional Z 2 1 2 Ef (u) = kτf (u)k dvg; 2 M 1 where τf (u) = fτ(u) + du(grad f) and f 2 C (M). We think that it is more reasonable to call JJ J I II them \bi-f-harmonic maps" as parallel to \biharmonic maps". Go back 2. The first variation formula Full Screen Let r and N r always denote the Levi-Civita connections of M and N, respectively. Let re be the Close −1 N induced connection on u TN defined by re X W = rdu(X)W , where X is a tangent vector of M −1 and W is a section of u TN. We choose a local orthonormal frame field feig on M. We define Quit the F -bitension field τF;2(u) of u by kτ(u)k2 τ (u) = −J(F 0 τ(u)) F;2 2 2 2 0 kτ(u)k X N 0 kτ(u)k = −4e (F )τ(u) − R (du(ei);F τ(u))du(ei); 2 2 i 1 R 2 where J is the Jacobi operator of the second variation for the energy E(u) = 2 M kduk dvg, 4 = − P (r r − r ) is the rough Laplacian on the section of u−1TN and RN (X; Y ) = e i e ei e ei e rei ei N N N [ rX ; rY ]− r[X;Y ] is the curvature operator on N. Under the notation above we have the following theorem Theorem 2.1 (The first variation formula). Let u: M ! N be a smooth map. Then d Z (1) EF;2(ut) jt=0 = h(τF;2(u);V )dvg; dt M d where V = dt utjt=0. Proof. Let Ψ: (−"; ") × M ! N be defined by Ψ(t; x) = ut(x), where (−"; ") × M is equipped JJ J I II @ with the product metric. We extend the vector fields @t on (−"; "), X on M naturally on (−"; ") × @ Go back M, and denote those also by @t , X. Then @ d Full Screen dΨ = u j = V: @t dt t t=0 Close We shall use the same notations r and re for the Levi-Civita connection on (−"; ") × M and the induced connection on Ψ−1TN, respectively. Quit We compute @ kτ(u )k2 F t @t 2 kτ(u )k2 1 @ = F 0 t kτ(u )k2 2 2 @t t 2 0 kτ(ut)k = F h re @ τ(ut); τ(ut) 2 @t 2 X 0 kτ(ut)k = F h re @ [(re ei dΨ)(ei)]; τ(ut) (2) 2 @t i 2 X 0 kτ(ut)k = h re @ re ei dΨ(ei) − re @ dΨ(rei ei ;F )τ(ut) @t @t 2 i 2 X @ kτutk = h(RN dΨ ; dΨ(e ))dΨ(e );F 0 τ(u ) @t i i 2 t i 2 X @ @ 0 kτ(ut)k + h re ei re ei dΨ − re re ei dΨ( ;F )τ(ut) ; @t i @t 2 JJ J I II i where we use Go back @ @ re @ dΨ(ei) − re ei dΨ = dΨ ; ei = 0 @t @t @t Full Screen and @ @ Close re @ dΨ(rei ei) − re re ei dΨ = dΨ ; rei ei = 0 @t i @t @t for the fifth equality. Quit @ Let Xt and Yt be two compactly supported vector fields on M such that g(Xt;Z)=h(re Z dΨ @t ; 2 2 0 kτutk @ 0 kτ(ut)k F 2 )τ(ut) and g(Yt;Z) = h dΨ @t ; re Z F 2 τ(ut) for any vector field Z on M. Then the divergence of Xt and Yt are given by the following: X X X div(Xt) = g(rek Xt; ek) = ekg(Xt; ek) − g(Xt; rek ek) k k k 2 X @ 0 kτ(ut)k = ekh re e dΨ ;F τ(ut) k @t 2 k 2 X @ 0 kτ(ut)k − h re re e dΨ ;F ( )τ(ut) k k @t 2 (3) k X @ = h re e re e dΨ k k @t k 2 @ 0 kτ(ut)k − re re e dΨ ;F τ(ut) k k @t 2 2 X @ 0 kτ(ut)k + h re e dΨ ; re e F τ(ut) k @t k 2 JJ J I II k Go back Full Screen Close Quit and X X X div(Yt) = g(rek Yt; ek) = ekg(Yt; ek) − g(Yt; rek ek) k k k 2 X @ 0 kτ(ut)k = ekh dΨ( ; re e F )τ(ut) @t k 2 k 2 X @ 0 kτ(ut)k − h(dΨ( ); re re e (F )τ(ut)) @t k k 2 k (4) 2 X @ 0 kτutk = h dΨ ; re e re e F τ(ut) @t k k 2 k 2 0 kτ(ut)k − re re e F τ(ut) k k 2 2 X @ 0 kτ(ut)k + h re e dΨ ; re e F τ(ut) : k @t k 2 k From (2), (3) and (4), we have JJ J I II 2 2 @ kτ(ut)k X N @ 0 kτ(ut)k F = h R dΨ ; dΨ(ei) dΨ(ei);F τ(ut) Go back @t 2 @t 2 i @ kτ(u )k2 Full Screen X 0 t (5) + h dΨ ; re ei re ei F τ(ut) @t 2 i Close kτ(u )k2 − r F 0 t τ(u ) + div(X ) − div(Y ): e rei ei t t t Quit 2 By (5) and Green's theorem, we have d E (u )j dt F;2 t t=0 Z 2 @ kτ(ut)k = F dvg M @t 2 t=0 Z kτ(u)k2 = h −4e F 0 τ(u) M 2 ! X kτ(u)k2 − RN du(e ); F 0 τ(u) du(e );V dv i 2 i g i Z = h(τF;2(u);V )dvg: M This proves Theorem 2.1.
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