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A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3 ) with constant ( r + 1 ) -th mean curvature H r + 1 immersed into the Euclidean sphere 𝕊 n + 1 , where r ∈ { 1 , ... , n - 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of 𝕊 n + 1 , a region that is determined by a totally umbilical sphere of 𝕊 n + 1 . We also provide a rigidity result for such hypersurfaces.

A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Maria Robaszewska (2002)

Annales Polonici Mathematici

We study the complex hypersurfaces f : M ( n ) → ℂ n + 1 which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of ℂ n + 1 .

Currently displaying 221 – 240 of 8747