Geodesic
The gap theorems for some extremal submanifolds in a unit sphere
Communications in Mathematics, Tome 23 (2015) no. 1, pp. 85-93.

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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.
Classification : 53C24, 53C40
Keywords: Extremal functional; Mean curvature; Totally umbilical 
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Wu, Xi Guo and Lan. The gap theorems for some extremal submanifolds in a unit sphere. Communications in Mathematics, Tome 23 (2015) no. 1, pp. 85-93. http://geodesic.mathdoc.fr/item/COMIM_2015__23_1_a5/