## Stability of area-preserving variations in space forms.(English)Zbl 1156.53037

The authors study compact hypersurfaces $$M$$ of real space forms. They define inductively the functional $$\mathcal A_r = \int_M F_r dM$$, where $$F_0=1$$, $$F_1=S_1$$, $$F_r = S_r+c(n-r+1)/(r-1) F_{r-2}$$ and $$S_r$$ denotes the r-th elementary symmetric function associated to the shape operator of the immersion. They consider the problem of minimizing $$\mathcal A_r$$ while keeping the area of $$M$$ constant and show that critical points of this variational problem are hypersurfaces for which the quotient of $$S_{r+1}$$ and $$S_1$$ is a constant. It is also shown that, in case $$S_1$$ is positive and the surrounding space is the Euclidean space, a critical immersion is stable if and only if $$M$$ is a sphere immersed in the Euclidean space as a totally umbilical hypersurface. Similar results are also obtained in case $$M$$ is an hypersurface contained in the hyperbolic space or contained in an open hemisphere of the unit sphere.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A30 Conformal differential geometry (MSC2010) 53C43 Differential geometric aspects of harmonic maps
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### References:

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