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\(r\)-minimal submanifolds in space forms. (English) Zbl 1168.53029

Let \(x: M\rightarrow \mathbb R^{n+p}(c)\) be an \(n\)-dimensional compact, possibly with boundary, submanifold in an (\(n+p\))-dimensional space form \(\mathbb R^{n+p}(c)\). Assume that \(r\) is even and \(r \in\{0, 1, \dots, n-1\}\), the authors introduce the \(r\)-th mean curvature function \(S_r\) and (\(r+1\))-th mean curvature vector field \(\mathbf{S}_{r+1}\). A hypersurface is called an \(r\)-minimal submanifold if \(\mathbf{S}_{r+1}\equiv0\), a 0-minimal submanifold is nothing but an ordinary minimal submanifold.
The authors define a functional \(J_r(x)=\int_M F_r(S_0, S_2, \dots, S_r)dv\) of \(x: M\rightarrow \mathbb R^{n+p}(c)\). By calculation of the first variational formula, the authors obtain that \(x\) is a critical point of \(J_r\) if and only if \(x\) is \(r\)-minimal. They also calculate the second variational formula of \(J_r\) and prove that there exists no compact without boundary stable \(r\)-minimal submanifold with \({\mathbf S}_r> 0\) in \({\mathbf S}^{n+p}\). When \(r = 0\), noting that \(S_0= 1\), this result reduces to J. Simons’ result [Ann. Math. (2) 88, 62–105 (1968; Zbl 0181.49702)]: there exists no compact without boundary stable minimal submanifold in the unit sphere \(S^{n+p}\).
In this paper, the obtained results are original and very interesting. So this paper can be recommended to everyone who is interested in the study of \(r\)-minimal submanifolds in space forms.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds

Citations:

Zbl 0181.49702
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References:

[1] Alencar H., do Carmo M., Colares A.G. (1993). Stable hypersurfaces with constant scalar curvature. Math. Z. 213: 117–131 · Zbl 0792.53057
[2] Alencar H., do Carmo M., Santos W. (2002). A gap theorem for hypersurfaces of the sphere with constant scalar curvature one. Comment. Math. Helv. 77: 549–562 · Zbl 1032.53045
[3] Alencar H., do Carmo M., Elbert M.F. (2003). Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces. J. Reine Angew. Math. 554: 201–216 · Zbl 1093.53063
[4] Alencar H., Rosenberg H., Santos W (2004). On the Gauss map of hypersurfaces with constant scalar curvature in spheres. Proc. Amer. Math. Soc. 132: 3731–3739 · Zbl 1061.53036
[5] Barbosa J.L.M., Colares A.G. (1997). Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15: 277–297 · Zbl 0891.53044
[6] Barbosa J.L.M., do Carmo M. (2005). On stability of cones in R n+1 with zero scalar curvature. Ann. Global Anal. Geom. 28: 107–127 · Zbl 1082.53061
[7] Barbosa J.L., do Carmo M., Eschenburg J. (1988). Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197: 123–138 · Zbl 0653.53045
[8] Cheng S.Y., Yau S.T. (1977). Hypersurfaces with constant scalar curvature. Math. Ann. 225: 195–204 · Zbl 0349.53041
[9] Chern, S.S.: Minimal Submanifolds in a Riemannian Manifold (mimeographed). University of Kansas, Lawrence, (1968).
[10] do Carmo M., Elbert M.F. (2004). On stable complete hypersurfaces with vanishing r-mean curvature. Tohoku Math. J. 56(2): 155–162 · Zbl 1062.53052
[11] Grosjean J.F. (2002). Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. Pacific J. Math. 206: 93–112 · Zbl 1050.58024
[12] Guo, Z., Li, H.: A variational problem for submanifolds in a sphere. To appear in Monatsh. Mat. (2007). · Zbl 1186.53066
[13] Guo Z., Li H., Wang C.P. (2001). The second variational formula for Willmore submanifolds in S n . Results in Math. 40: 205–225 · Zbl 1163.53312
[14] Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press (1989)
[15] Hu Z.J., Li H.: Willmore submanifolds in Riemannian manifolds. In: Proceedings of the Workshop, Contem. Geom., Related Topics, pp. 251–275. World Scientific, May 2002 · Zbl 1076.53071
[16] Li A.M. (1985). A class of variational problems and integral formula in Riemannian manifold. Chinese Acta Math. Sinica 28(2): 145–153 · Zbl 0587.53053
[17] Li H. (1996). Hypersurfaces with constant scalar curvature in space forms Math. Ann. 305: 665–672 · Zbl 0864.53040
[18] Li H. (1997). Global rigidity theorems of hypersurface. Ark. Math. 35: 327–351 · Zbl 0920.53028
[19] Li H. (2001). Willmore hypersurfaces in a sphere. Asian J. Math. 5: 365–378 · Zbl 1025.53031
[20] Li H. (2002). Willmore submanifolds in a sphere. Math. Res. Lett. 9: 771–790 · Zbl 1056.53040
[21] Reilly R. (1973). Variation properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential. Geom. 8: 465–477 · Zbl 0277.53030
[22] Reilly R. (1977). On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helvetici. 52: 525–533 · Zbl 0382.53038
[23] Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Soc. Math. 117: 211–239 · Zbl 0787.53046
[24] Simons J. (1968). Minimal varieties in Riemmannian manifolds. Ann. Math. 2nd Ser. 88(1): 62–105 · Zbl 0181.49702
[25] Takahashi T. (1966). Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18: 380–385 · Zbl 0145.18601
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