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BIHARMONIC HYPERSURFACES in a RIEMANNIAN MANIFOLD with NON-POSITIVE RICCI CURVATURE 1. Introduction and Statement of Results. In

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BIHARMONIC HYPERSURFACES in a RIEMANNIAN MANIFOLD with NON-POSITIVE RICCI CURVATURE 1. Introduction and Statement of Results. In BIHARMONIC HYPERSURFACES IN A RIEMANNIAN MANIFOLD WITH NON-POSITIVE RICCI CURVATURE NOBUMITSU NAKAUCHI AND HAJIME URAKAWA∗ Abstract. In this paper, we show that, for a biharmonic hyper- surface (M; g) ofR a Riemannian manifold (N; h) of non-positive j j2 1 Ricci curvature, if M H vg < , where H is the mean curvature of (M; g) in (N; h), then (M; g) is minimal in (N; h). Thus, for a counter example (M; g) in the case of hypersurfacesR to the gener- j j2 1 alized Chen's conjecture (cf. Sect.1), it holds that M H vg = . 1. Introduction and statement of results. In this paper, we consider an isometric immersion ' :(M; h) ! (N; h), of a Riemannian manifold (M; g) of dimension m, into another Riemannian manifold (N; h) of dimension n = m + 1. We have rN r '∗X '∗Y = '∗( X Y ) + k(X; Y )ξ; for vector fields X and Y on M, where r, rN are the Levi-Civita connections of (M; g) and (N; h), respectively, ξ is the unit normal vector field along ', and k is the second fundamental form. Let A : TxM ! TxM (x 2 M) be the shape operator defined by g(AX; Y ) = k(X; Y ), (X; Y 2 TxM), and H, the mean curvature defined by H := 1 m Trg(A). Then, let us recall the following generalized B.Y. Chen's conjecture (cf. [3], [2]): Let ' :(M; g) ! (N; h) be an isometric immersion, and the sec- tional curvature of (N; h) is non-positive. If ' is biharmonic (cf. See Sect. 2), then, it is minimal. Oniciuc ([8]) and Ou ([10]) showed this is true if H is constant. 2000 Mathematics Subject Classification. 58E20. Key words and phrases. harmonic map, biharmonic map, isometric immersion, minimal, Ricci curvature. ∗ Supported by the Grant-in-Aid for the Scientific Research, (C), No. 21540207, Japan Society for the Promotion of Science. 1 2 NOBUMITSU NAKAUCHI AND HAJIME URAKAWA∗ In this paper, we show Theorem 1.1. Assume that (M; g) is complete and the Ricci tensor RicN of (N; h) satisfies that RicN (ξ; ξ) ≤ jAj2: (1.1) If ' :(M; g) ! (N; h) is biharmonic (cf. Sect. 2) and satisfies that Z 2 H vg < 1; (1.2) M then, ' has constant mean curvature, i.e., H is constant. As a direct corollary, we have Corollary 1.2. Assume that (M; g) is a complete Riemannian mani- fold of dimension m and (N; h) is a Riemannian manifold of dimension m+1 whose Ricci curvature is non-positive. If an isometricR immersion ! 2 1 ' :(M; g) (N; h) is biharmonic and satisfies that M H vg < , then, ' is minimal. By our Corollary 1.2, if there would exist a counter example (cf. [10]) in the case dim N = dim M + 1, then it must hold that Z 2 H vg = 1; (1.3) M which imposes the strong condition on the behaviour of the boundary of M at infinity. Indeed, (1:3) implies that either H is unbounded on M, or it holds that H2 ≥ C on an open subset Ω of M with infinite volume, for some constant C > 0. 2. Preliminaries. In this section, we prepare general materials about harmonic maps and biharmonic maps of a complete Riemannian manifold into another Riemannian manifold (cf. [4]). Let (M; g) be an m-dimensional complete Riemannian manifold, and the target space (N; h) is an n-dimensional Riemannian manifold. For every C1 map ' of M into N, and relatively compact domain Ω in M, the energy functional on the space C1(M; N) of all C1 maps of M into N is defined by Z 1 2 EΩ(') = jd'j vg; 2 Ω BIHARMONIC HYPERSURFACES 3 1 1 and for a C one parameter deformation 't 2 C (M; N)(− < t < ) of ' with '0 = ', the variation vector field V along ' is defined by d −1 1 V = 't. Let ΓΩ(' TN) be the space of C sections of the dt t=0 induced bundle '−1TN of the tangent bundle TN by ' whose sup- −1 ports are contained in Ω. For V 2 ΓΩ(' TN) and its one-parameter deformation ' , the first variation formula is given by t Z d − h i EΩ('t) = τ(');V vg: dt t=0 Ω The tension field τ(') is defined globally on M by Xm τ(') = B(')(ei; ei); (2.1) i=1 where rN − r B(')(X; Y ) = '∗(X)'∗(Y ) '∗( X Y ) for X; Y 2 X(M). Then, a C1 map ' :(M; g) ! (N; h) is harmonic if τ(') = 0. For a harmonic map ' :(M; g) ! (N; h), the second variation formula of the energy functional E (') is Ω Z d2 h i 2 EΩ('t) = J(V );V vg dt t=0 Ω where J(V ) := ∆V − R(V ); m ∗ X ∆V := r rV = − fr (r V ) − rr V g; ei ei ei ei i=1 Xm N R(V ) := R (V; '∗(ei))'∗(ei): i=1 Here, r is the induced connection on the induced bundle '−1TN, and RN is the curvature tensor of (N; h) given by RN (U; V )W = rN rN − rN 2 [ U ; V ]W [U;V ]W (U; V; W X(N)). The bienergy functional is defined by Z 1 2 E2;Ω(') = jτ(')j vg; 2 Ω and the first variation formula of the bienergy is given (cf. [6]) by Z d − h i E2;Ω('t) = τ2(');V vg dt t=0 Ω where the bitension field τ2(') is defined globally on M by τ2(') = J(τ(')) = ∆τ(') − R(τ(')); (2.2) 4 NOBUMITSU NAKAUCHI AND HAJIME URAKAWA∗ and a C1 map ' :(M; g) ! (N; h) is called to be biharmonic if τ2(') = 0: (2.3) 3. Some Lemma for the Schrodinger¨ type equation In this section, we prepare some simple lemma of the Schr¨odinger type equation of the Laplacian ∆g on an m-dimensional non-compact complete Riemannian manifold (M; g) defined by Xm − r 2 1 ∆gf := ei(eif) ei eif (f C (M)); (3.1) i=1 f gm where ei i=1 is a locally defined orthonormal frame field on (M; g). Lemma 3.1. Assume that (M; g) is a complete non-compact Riemann- ian manifold, and L is a non-negative smooth function on M. Then, every smooth L2 function f on M satisfying the Schr¨odingertype equa- tion ∆gf = L f (on M) (3.2) must be a constant. Proof. Take any point x0 in M, and for every r > 0, let us consider the following cut-off function η on M: 8 > 0 ≤ η(x) ≤ 1 (x 2 M); > > <> η(x) = 1 (x 2 Br(x0)); 62 (3.3) > η(x) = 0 (x B2r(x0)); > > 2 :> jrηj ≤ (on M); r where Br(x0) = fx 2 M : d(x; x0) < rg, and d is the distance of (M; g). Multiply η2 f on (3:2), and integrale it over M, we have Z Z 2 2 2 (η f) ∆gf vg = L η f vg: (3.4) M M By the integration by part for the left hand side, we have Z Z 2 2 (η f) ∆gf vg = − g(r(η f); rf) vg: (3.5) M M Here, we have g(r(η2 f); rf) = 2η f g(rη; rf) + η2 g(rf; rf) = 2 η f hrη; rfi + η2 j rf j2; (3.6) BIHARMONIC HYPERSURFACES 5 where we use h · ; · i and j · j instead of g( · ; · ) and g(u; u) = juj2 (u 2 TxM), for simplicity. Substitute (3:6) into (3:5), the right hand side of (3:5) is equal to Z Z 2 2 RHS of (3:5) = − 2 ηf hrη; rfi vg − η jrfj vg MZ ZM 2 2 = −2 hf rη; η rf i vg − η jrfj vg: M M (3.7) Here, applying Young's inequality: for every > 0, and every vectors X and Y at each point of M, 1 ±2 hX; Y i ≤ jXj2 + jY j2; (3.8) to the first term of (3:7), we have Z Z Z 2 1 2 2 2 RHS of (3:7) ≤ jη rfj vg + jf rηj vg − η jrfj vg M Z M Z M 2 2 1 2 2 = −(1 − ) η jrfj vg + f jrηj vg: M M (3.9) Thus, by (3:5) and (3:9), we obtain Z Z Z 2 2 2 2 1 2 2 L η f vg + (1 − ) η jrfj vg ≤ f jrηj vg: M M M (3.10) 1 Now, puttig = 2 , (3:10) implies that Z Z Z 2 2 1 2 2 2 2 L η f vg + η jrfj vg ≤ 2 f jrηj vg: M 2 M M (3.11) jr j ≤ 2 ≥ Since η = 1 on Br(x0) and η r , and L 0 on M, we have Z Z Z 2 1 2 8 2 0 ≤ L f vg + jrfj vg ≤ f vg: B (x ) 2 B (x ) r2 M r 0 r 0 (3.12) Since (M; g) is non-compact and complete, r can tend to infinity, and Br(x0) goes to M. Then we have Z Z 2 1 2 0 ≤ L f vg + jrfj vg ≤ 0 (3.13) M 2 M R 2 1 2 jr j since M f vg < . Thus, we have L f = 0 and f = 0 (on M) which implies that f is a constant. 6 NOBUMITSU NAKAUCHI AND HAJIME URAKAWA∗ 4. Biharmonic isometric immersions. In this section, we consider a hypersurface M of an (m+1)-dimensional Riemannian manifold (N; h).
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