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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.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
Zbl 0181.49702
Full Text: DOI
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