An extension of Ruh-Vilms Theorem for hypersurfaces in symmetric spaces and some applications

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The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to 3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface Mⁿ⁻¹ immersed in a symmetric space Nⁿ taking values in the unit pseudo sphere Sᵐ of the Lie algebra g of the isometry group of N, dim(g)=m+1, and it is proved that M has CMC if and only if its Gauss map is harmonic. As an application, it is proved that if dim(N)=3 and the image of the Gauss map of a CMC surface S immersed in N is contained in a hemisphere of Sᵐ determined by a vector X, then S is invariant by the one parameter subgroup of isometries of N of the Killing field determined by X. In particular, it is obtained an extension of HOS Theorem to the 3-dimensional hyperbolic space. In the paper it is also shown that the holomorphic quadratic form induced by the Gauss map coincides (up to a sign) with the Hopf quadratic form when the ambient space is a space form and with the Abresch-Rosenberg quadratic form when the ambient space is H² x R and S² x R providing, then, an unified way of relating Hopf's and Abresch-Rosenberg's quadratic form with the quadratic form induced by a harmonic Gauss map of a CMC surface in these 5 spaces.