## A new characterization of $$r$$-stable hypersurfaces in space forms.(English)Zbl 1249.53081

The $$r$$-stability of closed hypersurfaces with constant $$r$$-mean curvature in Riemannian manifolds of constant sectional curvature is studied. $$r$$-stable hypersurfaces are characterized through the first eingenvalue of an operator naturally attached to the $$r$$-mean curvature.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53Z05 Applications of differential geometry to physics 83C99 General relativity

### Keywords:

space form; $$r$$-th mean curvature; $$r$$-stability
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