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$r$-minimal submanifolds in space forms. (English) Zbl 1168.53029
Let $x: M\rightarrow \Bbb R^{n+p}(c)$ be an $n$-dimensional compact, possibly with boundary, submanifold in an ($n+p$)-dimensional space form $\Bbb R^{n+p}(c)$. 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 $\bold{S}_{r+1}$. A hypersurface is called an $r$-minimal submanifold if $\bold{S}_{r+1}\equiv0$, 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 $x: M\rightarrow \Bbb R^{n+p}(c)$. 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 ${\bold S}_r> 0$ in ${\bold S}^{n+p}$. When $r = 0$, noting that $S_0= 1$, this result reduces to {\it 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.

53C42Immersions (differential geometry)
53C40Global submanifolds (differential geometry)
Full Text: DOI
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