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.


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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