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

Let x:M n+p (c) be an n-dimensional compact, possibly with boundary, submanifold in an (n+p)-dimensional space form n+p (c). Assume that r is even and r{0,1,,n-1}, the authors introduce the r-th mean curvature function S r and (r+1)-th mean curvature vector field 𝐒 r+1 . A hypersurface is called an r-minimal submanifold if 𝐒 r+1 0, a 0-minimal submanifold is nothing but an ordinary minimal submanifold.

The authors define a functional J r (x)= M F r (S 0 ,S 2 ,,S r )dv of x:M 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 𝐒 r >0 in 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:
53C42Immersions (differential geometry)
53C40Global submanifolds (differential geometry)
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 · doi:10.1007/BF03025712
[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 · doi:10.1007/s00014-002-8351-1
[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 · doi:10.1515/crll.2003.006
[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 · doi:10.1090/S0002-9939-04-07493-3
[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 · doi:10.1023/A:1006514303828
[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 · doi:10.1007/s10455-005-0039-5
[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 · doi:10.1007/BF01161634
[8]Cheng S.Y., Yau S.T. (1977). Hypersurfaces with constant scalar curvature. Math. Ann. 225: 195–204 · Zbl 0349.53041 · doi:10.1007/BF01425237
[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 · doi:10.2748/tmj/1113246548
[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 · doi:10.2140/pjm.2002.206.93
[12]Guo, Z., Li, H.: A variational problem for submanifolds in a sphere. To appear in Monatsh. Mat. (2007).
[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
[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
[16]Li A.M. (1985). A class of variational problems and integral formula in Riemannian manifold. Chinese Acta Math. Sinica 28(2): 145–153
[17]Li H. (1996). Hypersurfaces with constant scalar curvature in space forms Math. Ann. 305: 665–672
[18]Li H. (1997). Global rigidity theorems of hypersurface. Ark. Math. 35: 327–351 · Zbl 0920.53028 · doi:10.1007/BF02559973
[19]Li H. (2001). Willmore hypersurfaces in a sphere. Asian J. Math. 5: 365–378
[20]Li H. (2002). Willmore submanifolds in a sphere. Math. Res. Lett. 9: 771–790
[21]Reilly R. (1973). Variation properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential. Geom. 8: 465–477
[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 · doi:10.1007/BF02567385
[23]Rosenberg H. (1993). Hypersurfaces of constant curvature in space forms. Bull. Soc. Math. 117: 211–239
[24]Simons J. (1968). Minimal varieties in Riemmannian manifolds. Ann. Math. 2nd Ser. 88(1): 62–105 · Zbl 0181.49702 · doi:10.2307/1970556
[25]Takahashi T. (1966). Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18: 380–385 · Zbl 0145.18601 · doi:10.2969/jmsj/01840380