Yong Luo, Shun Maeta Biharmonic hypersurfaces in a sphere Proceedings of the American Mathematical Society DOI: /proc/13320

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1 Yong Lo, Shn aeta Biharmonic hypersrfaces in a sphere Proceedings of the American athematical Society DOI: /proc/13320 Accepted anscript This is a preliminary PDF of the athor-prodced manscript that has been peer-reviewed and accepted for pblication. It has not been copyedited, proofread, or finalized by AS Prodction staff. Once the accepted manscript has been copyedited, proofread, and finalized by AS Prodction staff, the article will be pblished in electronic form as a Recently Pblished Article before being placed in an isse. That electronically pblished article will become the Version of Record. This preliminary version is available to AS members prior to pblication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the pblication date of the Version of Record. The Version of Record is accessible to everyone five years after pblication in an isse.

2 PROCEEDINGS OF THE AERICAN ATHEATICAL SOCIETY Volme 00, Nmber 0, Pages S (XX) BIHARONIC HYPERSURFACES IN A SPHERE YONG LUO AND SHUN AETA (Commnicated by ) Abstract. In this short note we will srvey some recent development in the geometric theory of biharmonic sbmanifolds, with an emphasize on the newly discovered Lioville type theorems and applications of known Lioville type theorems in the research of nonexistence of biharmonic sbmanifolds. A new Lioville type theorem for sperharmonic fnctions on complete manifolds is proved and its applications in a kind of nonexistence of biharmonic hypersrfaces in a sphere is provided. 1. Introdction In their celebrated 1964 paper Eells and Sampson (cf. [15]) made great progress in the theory of harmonic maps. Since then many works on harmonic maps are done and they have been sed in varios fields in differential geometry. However there are non-existence reslts for harmonic maps. Therefore a generalization of harmonic maps is an important sbject. In their paper Eells and Sampson sggested to consider the (intrinsic) bi-energy of a map ϕ defined by E 2 (ϕ) = 1 (1.1) τ(ϕ) 2 dµ g, 2 where τ(ϕ) is the tension field of ϕ and dµ g is the volme element on (, g). Stationary points of the bi-energy fnctional are called biharmonic maps. Jiang (cf. [21], [22], [23]) is the first mathematician who seriosly considered the bi-energy fnctional and he compted the first and second variations of E 2. The stationary points of the fnctional E 2 satisfy the following E-L eqation (cf. [21]) (1.2) τ(ϕ) = m R N (dϕ(e i ), τ(ϕ))dϕ(e i ), where is the Laplacian, R N is the Riemann crvatre tensor of the ambient manifold N and {e i, i = 1,..., m} is a local orthonormal frame field of. If frther c XXXX American athematical Society 2010 athematics Sbject Classification. primary 53C43, secondary 58E20, 53C40. The first athor is partially spported by the Postdoctoral Science Fondation of China(No ), and the Project-sponsored by SRF for ROCS, SE. The second athor is partially spported by the Grant-in-Aid for Yong Scientists(B), No.15K17542, Japan Society for the Promotion of Science. 1 This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

3 2 YONG LUO AND SHUN AETA ϕ is an isometry, ϕ satisfies the following eqation H m (1.3) = R N (e i, H)e i, where H is the mean crvatre vector field. By decomposing H m RN (e i, H)e i into its tangential and normal parts, we see that a sbmanifold is biharmonic if and only if it satisfies (cf. [9], [10]) H m m (1.4) B(A H e i, e i ) + (R N (e i, H)e i ) = 0, (1.5) m H m m A ei H e i 4 (R N (e i, H)e i ) T = 0, where (R N (e i, H)e i ), (R N (e i, H)e i ) T denote the normal and tangential parts of R N (e i, H)e i respectively, B is the second fndamental form, A is the shape operator and is the Laplacian of the normal connection. In particlar, if N is a space form of constant sectional crvatre c, then a sbmanifold is biharmonic if and only if H m (1.6) B(A H e i, e i ) + cmh = 0, (1.7) m H m A ei H e i = 0. Frthermore if is a hypersrface in N and H is the mean crvatre of then we have (cf. [27]) (1.8) H H A 2 + HRic N (ξ, ξ) = 0. A sbmanifold satisfying eqation (1.3) is called biharmonic sbmanifold. In [22] Jiang proved that every biharmonic srface in R 3 is minimal. In the stdy of his finite type sbmanifolds Chen proposed to consider sbmanifolds with harmonic mean crvatre, which is jst biharmonic sbmanifolds whose target manifold is a Eclidean space. In [11] Chen fond Jiang s nonexistence reslt independently(in a earlier version of [11]) and conjectred that every biharmonic sbmanifolds in a Eclidean space is minimal, which is called Chen s conjectre nowadays. Several partial answers of Chen s conjectre were obtained by Dimitric, who was Chen s Ph.D. stdent, in his Ph.D. thesis (cf. [14]). In particlar, he proved that every biharmonic crve in a Eclidean space is an open part of a straight line. Another striking reslt is proved by Hasanis and Vlachos (cf. [19]) that every biharmonic hypersrface in R 4 is minimal. See also [13] for a more mathematical proof. These reslts make Chen s conjectre poplar, particlarly in the past decade. Inspired by the sccess of Chen s conjectre, Caddeo, ontaldo and Onicic (cf. [6]) proposed the generalized Chen s conjectre. Generalized Chen s Conjectre(GCC). Every biharmonic sbmanifold in a non-positively crved manifold is minimal. Till now there are many affirmative partial answers to the GCC. In particlar, there are affirmative partial answers to GCC in the case of ambient space is hyperbolic space. Caddeo, ontaldo and Onicic (cf. [7]) showed that every biharmonic This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

4 BIHARONIC HYPERSURFACES IN A SPHERE 3 srface in H 3 ( 1) is minimal. Balmş, ontaldo and Onicic (cf. [4]) showed that every biharmonic hypersrface in H 4 ( 1) is minimal. Now we know the GCC is false, by a conterexample constrcted by O and Tang (cf. [28]). See also [29] for frther examples. Hence the second athor (cf. [31]) proposed the global version of the GCC: Generalized Chen s Conjectre: global version. Every complete biharmonic sbmanifold in a non-positively crved manifold is minimal. Under the completeness assmption, things shold be easier. For example we cold se ct off fnctions and the integral by parts argment. With this method, Nakachi and Urakawa(cf. [34]) first obtained an affirmative partial answer to GCC nder the assmption of completeness and the integrability of mean crvatre vector field. After their stdies, several generalization of their reslts were obtained (cf. [24], [30], [35]). In particlar, in [24] the first athor proved among other things that every complete biharmonic sbmanifold in a non-positively crved manifold with L p (0 < p < + ) integrable mean crvatre vector field is minimal. He sed certain Lioville type theorem on complete manifolds and it was made explicitly and broadened in a sbseqent paper (cf. [25], Theorem 1.9). Here we remark that the application of ct off fnctions to biharmonic sbmanifolds was first sed by Wheeler [39] and independently sed by Nakachi and Urakawa [34]. The stdy of the GCC is a good prototype of applying Lioville type theorems and it is also very helpfl in finding new Lioville type theorems of fnctions on complete manifolds, as we showed in the last paragraph and will show in the following. Aktagawa and the second athor (cf. [1]) showed that every properly immersed biharmonic sbmanifold in a Eclidean space is minimal, where they sed essentially a new Lioville type theorem inspired by Omori-Ya s generalized maximal principle. This argment is later developed by the athors to obtain new Lioville type theorem for sbmanifolds in non-positively crved target manifolds with certain crvatre decay order at infinity (cf. [26], [31]). When the target manifold is a Eclidean sphere, in contrast to the non-positive crvatre case, there are natral examples of nonminimal biharmonic sbmanifolds. For example, S n 1 ( 1 2 ) and S n p ( 1 2 ) S p 1 ( 1 2 )(n p p 1) in S n given by Jiang (cf. [21]). Bt similar to the non-positive crvatre case, it is conjectred (cf. [3]) that biharmonic sbmanifolds in a sphere is a constant mean crvatre sbmanifold by Balmş, ontaldo and Onicic, that BO conjectre. Every biharmonic sbmanifold in a sphere has constant mean crvatre. There are affirmative partial answers to BO conjectre, if is one of the following: (i) A compact hypersrface with nowhere zero mean crvatre vector field and B 2 m by J. H. Chen (cf. [12]), (ii) A compact hypersrface with nowhere zero mean crvatre vector field and B 2 m by Balmş, ontaldo and Onicic (cf. [4]), (iii) A compact sbmanifold with H 1 by Balmş and Onicic (cf. [2], see also [30]), where B 2 is the sqared norm of the second fndamental form. There are many stdies for biharmonic sbmanifolds in sphere (cf. [5], [6], [7], [16], [36] etc.). Recently the second athor considered the complete noncompact case and he proved the BO conjectre nder the assmption of H 1, together with certain integrality condition (cf. [32]), where he sed an argment originated from [24], [30] and [34]. For the H (0, 1] case, he proved the This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

5 4 YONG LUO AND SHUN AETA BO conjectre nder the assmption (1.9) H p dµ g < +, where he sed a Lioville type theorem for drift Laplacian de to Petersen and Wylie (cf. [37]) to get this reslt. We observe that we cold se an old Lioville type theorem of Ya to get a stronger reslt. Actally we cold get the following Lioville type theorem for sperharmonic fnctions on a complete manifold and as a corollary we can get a stronger reslt for biharmonic hypersrfaces. We have Theorem 1.1. Let (, g) be a complete noncompact manifold and (0, C], (C > 0) a sperharmonic fnction on. If (log (k) Ce (k) (1.10) )p dµ g < + for some p > 0 and k N, then is a constant. Here log (k) = log(log (k 1) ) and e (k) = e e(k 1), where log (1) = log and e (1) = e. Applying this Lioville type theorem, we obtain the following theorem. Theorem 1.2. Let ϕ : ( m, g) (N m+1, h) be a complete biharmonic hypersrface. Assme that the mean crvatre H satisfies 0 < H 1. We also assme that B 2 Ric N (ξ, ξ) where B is the second fndamental form of in N, Ric N is the Ricci crvatre of N, and ξ is the nit normal vector field on. If (log (k) e (k) (1.11) H )p dµ g < + for some p > 0 and k N, then H is constant. Remark 1.3. It is easy to see that condition (1.11) is weaker than condition (1.9). As a corollary of Theorem 1.2 we have Corollary 1.4. Let ϕ : ( m, g) (S m+1, h) be a complete biharmonic hypersrface with H > 0. If B 2 m and (log (k) e (k) (1.12) H )p dµ g < + for some 0 < p < and k N, then H is constant. Recently Hornng and oser defined p-biharmonic(p > 1) maps and isometric p-biharmonic maps are called p-biharmonic sbmanifolds. In the appendix we will discss the nonexistence reslts of p-biharmonic sbmanifolds. The rest of this paper organized as follows: Or main reslt is proved in section 2. In appendix, we apply or main reslt to p-biharmonic hypersrfaces. 2. Proofs In this section, we will se notations log (k) = log(log (k 1) ) and e (k) = e e(k 1), where log (1) = log and e (1) = e. We will need the following Lioville type theorem de to Ya (cf. [38], Theorem 1). This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

6 BIHARONIC HYPERSURFACES IN A SPHERE 5 Theorem 2.1 (Ya). Sppose f is bonded from below by a constant and is Lebesge integrable with 0 < f. Assme that log v = f, v > 0, then vp = for p > 0, nless v is a constant fnction. If f is zero almost everywhere, the same conclsion holds. Proof of Theorem 1.1. (I) The case k = 1: A direct comptation shows that log log Ce = = Ce Ce log log log Ce (log Ce )2 Ce log 2 log log 2 (log Ce )2 Since log = 1 log 2 we finally get (2.1) log log Ce = 1 log Ce + log C Ce log 2 (log Ce. )2 Let s apply Theorem 2.1 to get the conclsion. Let v = log Ce = 1 + log C and f = 1 + log C Ce log Ce log 2. Then log v = f by eqation (2.1) and it is easy to see (log Ce )2 that f 0. Therefore we mst have f is zero almost everywhere or 0 < f. By Theorem 2.1 we conclde that v is a constant. (II) The case k 2: A direct comptation shows that and log (k+1) f = log(k) f log (k) f log (k) f 2 By an elementary argment we have (log (k) f) 2 log (k+1) f 2 = log(k) f 2 (log (k) f) 2. log f k ( k (2.2) log (k+1) log(i) f log f 2 ) k i=2 j=i log(j) f + 1 f = k. (log(i) f) 2 Let f = Ce(k). We obtain log (k+1) Ce (k) (2.3) k { k ( k log(i) f + 2 )} k log(i) f i=2 j=i log(j) f + 1 =. 2 k (log(i) f) 2 This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

7 6 YONG LUO AND SHUN AETA Since f = Ce(k) and C, we have k ( k log (i) f (log f k) k i=2 j=i k log (i) f i=2 (log e (k) k) ) log (j) f + 1 k log (i) e (k) 0, where in the first ineqality we sed log (i) f log (i) e (k) 1 for any 1 i k. By Theorem 2.1, we conclde that is constant. i=2 Proof of Theorem 1.2. Recall that Hence H H A 2 + HRic N (ξ, ξ) = 0. H = H A 2 HRic N (ξ, ξ) = H B 2 HRic N (ξ, ξ) 0, where we sed A 2 = B 2. That H is a sperharmonic fnction on. Let H = in Theorem 1.1 with C = 1 we complete the proof of Theorem 1.2. Proof of Theorem 1.4. Since N = S m+1, Ric N (ξ, ξ) = m and hence by assmption B 2 m. Now since mh 2 B 2, H 1 is atomatically satisfied. Therefore we prove that H is a constant fnction on. 3. Appendix We can apply or method to p-biharmonic sbmanifolds (p > 1) (cf. [20]). If an isometric immersion ϕ : (, g) (N, h) satisfies (3.1) ( H p 2 H) m R N (e i, H p 2 H)ei = 0, then is called a p-biharmonic sbmanifold. By decomposing (3.1) into its tangential and normal parts, we see that a sbmanifold is p-biharmonic if and only if it satisfies (cf. [17]) (3.2) ( H ) m m ( p 2 H B(A H p 2 H e i, e i ) + R N ( H ) p 2 H, ei, )e i = 0, (3.3) Tr g ( A H p 2 H ) + Tr g [A H p 2 H ( )] m ( R N ( H p 2 H, ei, )e i ) T = 0. For p-biharmonic sbmanifolds, it is easy to see that we can get similar reslts as in the reslts of biharmonic sbmanifolds in many cases(cf. [8] [17] [18] [26]). In fact, we have the following theorem. Theorem 3.1. Let ϕ : ( m, g) (N m+1, h) be a complete p-biharmonic hypersrface. Assme that the mean crvatre H satisfies 0 < H 1. We also assme This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

8 BIHARONIC HYPERSURFACES IN A SPHERE 7 that B 2 Ric N (ξ, ξ) where B is the second fndamental form of in N, Ric N is the Ricci crvatre of N, and ξ is the nit normal vector field on. If (i) (log (k) e (k) H p 1 )q dµ g < +, or (ii) p 2 and (log (k) e (k) H )q dµ g < +, for some q > 0 and k N, then H is constant. Proof. The proof is similar to the proof of Theorem 1.2. The case (i): Let = H p 1. Since H p 1 0, we can apply Theorem 1.1. The case (ii): Let = H. Since p 2, we have H 0. Applying Theorem 1.1, we get the reslt. References 1. K. Aktagawa and S. aeta, Biharmonic properly immersed sbmanifolds in Eclidean spaces, Geom. Dedicata 164(1)(2013), A. Balmş and C. Onicic, Biharmonic sbmanifolds with parallel mean crvatre vector field in spheres, J. ath. Anal. Appl. 386 (2012), A. Balmş, S. ontaldo and C. Onicic, Classification reslts for biharmonic sbmanifolds in spheres, Israel J. ath.168(2008), A. Balmş, S. ontaldo and C. Onicic, New reslts toward the classification of biharmonic sbmanifolds in S n, An. Stiint. Univ. Ovidis Constanta Ser. at. 20(2012), A. Balmş, S. ontaldo and C. Onicic, Biharmonic PNC sbmanifolds in spheres, Ark. at.,51(2013), R. Caddeo, S. ontaldo and C. Onicic, Biharmonic sbmanifolds of S 3, Internat. J. ath.12(8)(2001), R. Caddeo, S. ontaldo and C. Onicic, Biharmonic sbmanifolds in spheres, Israel J. ath.130(2002), X. Z. Cao and Y. Lo, On p-biharmonic sbmanifolds in nonpositively crved manifolds, Kodai ath. J., in press. 9. B. Y. Chen, Total ean Crvatre and Sbmanifolds of Finite Type, Series in Pre athematics 1, World Scientific, Singapore, B. Y. Chen, Finite type sbmanifolds in psedo-eclidean spaces and applications, Kodai ath. J. 6(1985), B. Y. Chen, Some open problems and conjectres on sbmanifolds of finite type, Soochow J. ath. 17(2)(1991), J. H. Chen, Compact 2-harmonic hypersrfaces in S n+1 (1), Acta ath. Sinica 36 (1993), F. Defever, Hypersrfaces of E 4 with harmonic mean crvatre vector, ath. Nachr.196(1998), I. Dimitric, Sbmanifolds of E m with harmonic mean crvatre vector, Bll. Inst. ath. Acad. Sinica 20(1)(1992), J. Eells and J. H. Sampson, Harmonic appings of Riemannian anifolds, Amer. J. ath. 86(1964), Y. F, Biharmonic hypersrfaces with three distinct principal crvatres in spheres, ath. Nachr. 288(2015), Y. B. Han, Some reslts of p-biharmonic sbmanifolds in a Riemannian manifold of nonpositive crvatre, J. Geom. 106(2015), Y. B. Han and W. Zhang, Some reslts of p-biharmonic maps into a non-positively crved manifold, J. Korean ath. Soc. 52(2015), This is a pre-pblication version of this article, which may differ from the final pblished version. Copyright ay 23 restrictions :21:48 may apply. EDT Version 2 - Sbmitted to PROC

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