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Stability of \(O(p+1) \times O(p+1)\)-invariant hypersurfaces with zero scalar curvature in Euclidean space. (English) Zbl 1036.53039

The author classifies \(O(p+1)\times O(p+1)-\)invariant hypersurfaces with zero scalar curvature in \({\mathbb R}^{2p+2},p>1\) according to their profile curves and shows that there are complete and embedded examples. Then, by studying the Morse indices of complete examples, the author shows that there exists a nonplanar globally stable, embedded complete hypersurface with zero scalar curvature in \({\mathbb R}^{2p+2}.\) This stable example provides a counterexample in odd dimensions \(\geq 9\) to a Bernstein-type conjecture for immersions with zero scalar curvature.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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