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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
Zbl 0181.49702
Full Text:
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