Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Extremal functional; Mean curvature; Totally umbilical
Extremal functional; Mean curvature; Totally umbilical
Summary:
Let $M$ be an $n$-dimensional submanifold in the unit sphere $S^{n+p}$, we call $M$ a $k$-extremal submanifold if it is a critical point of the functional $\int _M\rho ^{2k}\,\mathrm{d}v $. In this paper, we can study gap phenomenon for these submanifolds.
References:
[1] Guo, Z., Li, H.: A variational problem for submanifolds in a sphere. Monatsh. Math., 152, 2007, 295-302, DOI 10.1007/s00605-007-0476-2 | MR 2358299 | Zbl 1186.53066
[2] Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure. Appl. Math., 27, 1974, 715-727, DOI 10.1002/cpa.3160270601 | MR 0365424 | Zbl 0295.53025
[3] Kenmotsu, K.: Some remarks on minimal submanifolds. Tohoku. Math. J., 22, 1970, 240-248, DOI 10.2748/tmj/1178242819 | MR 0268791 | Zbl 0202.21003
[4] Li, A.-M., Li., J.-M.: An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math., 58, 1992, 582-594, DOI 10.1007/BF01193528 | MR 1161925
[5] Li., H.: Willmore hypersurfaces in a sphere. Asian. J. Math., 5, 2001, 365-378, DOI 10.4310/AJM.2001.v5.n2.a4 | MR 1868938
[6] Li., H.: Willmore submanifolds in a sphere. Math. Res. Letters, 9, 2002, 771-790, DOI 10.4310/MRL.2002.v9.n6.a6 | MR 1906077
[7] Simons., J.: Minimal varieties in Riemannian manifolds. Ann. of Math., 88, 1968, 62-105, DOI 10.2307/1970556 | MR 0233295
[8] Xu, H.-W., Yang., D.: The gap phenomenon for extremal submanifolds in a Sphere. Differential Geom and its Applications, 29, 2011, 26-34, MR 2784286
[9] Wu., L.: A class of variational problems for submanifolds in a space form. Houston J. Math., 35, 2009, 435-450, MR 2519540
Search
Partner of
