Shogo SUZUKI

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SUZUKI, Shogo

Chuo University Academic Information Repository / 中央大学学術リポジトリ

球面内の完備な部分多様体についてOn complete minimal submanifolds in a sphere数学専攻　鈴木　翔吾Shogo SuzukiLet Sn+p(c) be an (n+ p)-dimensional Euclidean sphere of constant curvature cand M an n-dimensional minimal submanifold isometrically immersed in Sn+p(c).We denote by A the Weingarten endomorphism associated a normal vector eld and T the tensor dened by T (; ) = traceAA.Yuan and Matsuyama [13] proved the following: Let M be an n-dimensionalcompact minimal submanifold isometrically immersed in Sn+p(c). Let  and  arethe second fundamental form ofM in Sn+p(c) and the immersion respectively. Thenjj2  np(n+ 2)2(n+ p+ 2)c and T = kh ; iif and only if one of the following conditions is satised:(A) jj2  0 and M is totally geodesic.(B) jj2 = np(n+ 2)2(n+ p+ 2)c and M is isotropic and has parallel second fundamentalform.Hence if  is full, then  is one of the following standard ones: Sn(c) !Sn(c); PR2(13c) ! S4(c); S2(13c) ! S4(c); CP 2(c) ! S7(c); QP 2(34c) ! S13(c);CP 2(43c)! S25(c).Moreover, they obtain the reseult of the case of M being complete: Let M bean n-dimensional complete minimal submanifold isometrically immersed in Sn+p(c).Thenjj2  np(n+ 2)2(n+ p+ 2)c and T = kh ; i:Then if and only if one of the following conditions is satised:(A) jj2  0 and M is totally geodesic.(B) jj2 = np(n+ 2)2(n+ p+ 2)c and M is isotropic and has parallel second fundamentalform.Rerated to these results, Li and Li[2] obtained without assumption of T = kh ; i,the following: Let A1; A2; :::; Ap be symmetric (n  n)-matrices (p  2). DenoteS = tracetAA; S = S = N(A); S = S1 +   + Sp. Then we haveX;N(AA   AA) +X;S2 32S2;and the equality holds if and only if one of the following conditions holds:11) A1 = A2 = ::: = Ap = 0,2) only two of the matrices A1; A2; :::; Ap are dierent from zero. Moreover, as-suming A1 6= 0; A2 6= 0; A3 = ::: = Ap = 0; then S1 = S2; and there exists anorthogonal(n n)-matrix T such thattTA1T =rS120B@ 1 00  1 00 01CA ;tTA2T =rS120B@ 1 00 1 00 01CA :Using the result, they proved the following: LetM be an n-dimensional compactminimal submanifold in Sn+p; p  2. If jj2  23n everywhere on M , then M iseither a totaly geodesic submanifold or a Velonese surface in S4.Now let v 2 UMx, x 2M . If e2; :::; en are orthonormal vectors in UMx orthogonalto v, then we can consider fe2; :::; eng as an orthonormal basis of Tv(UMx). Weremark that fv = e1; e2; :::; eng is an orthonormal basis of TxM . If we denote theLaplacian of UMx = Sn 1 by , then f = e2e2f +    + enenf , where f is adierentiable function on UMx.Dene functions f1(v); f2(v);    ; f16(v) on UMx, x 2M , byf1(v) =nXi=1hA(v;ei)v;A(v;v)eiif2(v) =nXi;j=1hA(ej ;ei)ej; A(v;v)eii;f3(v) =nXi=1hA(v;v)v;A(v;ei)eii;f4(v) =nXi;j=1hA(ej ;ei)ej; A(v;ei)vi;f5(v) =nXi;j=1hA(ei;v)ei; A(v;ej)eji;f6(v) =nXi=1hA(v;v)ei; A(v;v)eii;f7(v) = j(v; v)j2;f8(v) =nXi;j=1hA(v;ei)ej; A(ej ;v)eii;f9(v) =nXi;j=1hA(ej ;v)ei; A(ej ;v)eii;f10(v) =nXi=1hA(v;ei)ei; vi;f11(v) = jA(v;v)vj2:f12(v) =nXi=1hA(v;ei)v; A(v;ei)vif13(v) = j(v; v)j4f14(v) =nXi=1hA(v;ei)ei; vij(v; v)j2f15(v) = (nXi=1hA(v;ei)ei; vi)2f16(v) = jj2j(v; v)j2;2The following generalized maximum principle due to Omori [11] and Yau [18]will be used in order to prove our theorem.Generalized Maximum Principle. (Omori [11] and Yau [18])Let Mn be a com-plete Riemannian manifold whose Ricci curvature is bounded from below and f 2C2(M) a function bounded from above on Mn. Then, for any  > 0, there exists apoint p 2Mn such thatf(p)  sup f   ; jjgrad f jj < ; f(p) < :We have the following (See [7] and [8])Lemma. Let M be an n-dimensional minimal submanifold isometrically im-mersed in Sn+p(c). Then for v 2 UMx we have12nXi=1(r2f7)(ei; ei; v) =nXi=1j(r)(ei; v; v)j2 + ncj(v; v)j2+ 2nXi=1hA(v;v)ei; A(ei;v)vi   2nXi=1hA(v;ei)ei; A(v;v)vi nXi=1hA(v;v)ei; A(v;v)eii=nXi=1j(r)(ei; v; v)j2 + nf7(v) + 2f1(v)  2f3(v)  f6(v)Using this Lemma and the result [2], we obtained: Theorem 1. Let M be an n-dimensional complete minimal submanifold in Sn+p; p  2. If jj2  23n everywhereon M , then M is isotropic and either a totally geodesic submanifold or a Veronesesurface in S4On the other hand, in Yuan and Matsuyama [13], we assume codimension = 2andtraceA2 n(n+ 2)2(n+ 4)c for 8every where on M , we obtained:Theorem 2. LetM be an n-dimensional complete minimal submanifold in Sn+2.If traceA2 n(n+ 2)2(n+ 4)c for 8, then M is isotropic and either a totally geodesicsubmanifold or isotropic and has parallel second fundamental form.3Especially, n = 2 ) S2(13c)! S4(c) and n = 5 ) S5 ! S7(c):References[1] S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of asphere with second fundamental form of constant length, Functional Analysisand Related Fields (1970), 59-75.[2] A. M. Li and J. M. Li, An intrinsic rigidity theorem for minimal submanifoldsin a sphere, Arch. Math., 58(1992), 582-594.[3] Y. Matsuyama, On some pinchings of minimal submanifolds, Geometry andits applications, edited by Tadashi Nagano et al., World scie ntic, Singapore(1993), 121-134.[4] Y. Matsuyama, On submanifolds of a sphere with bounded second fundamentalform, Bull. Korean Math. Soc. 32 (1995), No. 1, pp. 103-113.[5] Y. Matsuyama, Curvature pinching for totally real submanifolds of a complexprojective space, J. Math. Soc. Japana Vol. 52, No. 1, 2000.[6] S. Montiel, A. Ros and F. Urbano, Curvature pinching and eigenvalue rigidityfor minimal submanifolds, Math. Z. 191 (1986), 537-548.[7] H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc.Japan, 19(1967), 205-214.[8] K. Sakamoto, Planar geodesic immersions, Tohoku Math. J. 29 (1977), 25-56.[9] Y. Uchida and Y. Matsuyama, Submanifolds with nonzero mean curvature ina euclidean sphere, I. J. Pure and Appl. Math., 29(2006), 119-130.[10] C. Xia, On the minimal submanifolds in CPm(c) and SN(1), Kodai Math. J.15 (1992), 143-153.[11] S. T. Yau, Submanifolds with constant mean curvature, Amer. J. Math.,96(1974), 346-366.[12] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm.Pure and Appl. Math, 28(1975), 201-228.[13] L. Yuan and Y. Matsuyama, Curvature pinching for a minimal submanifoldsof a sphere, J. Adv. Math. Stud. Vol. 7(2014), No. 1, 45-55.4

On complete minimal submanifolds in a sphere

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球面内の完備な部分多様体について

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