An extension of Ruh-Vilms’ theorem to hypersurfaces in symmetric spaces and some applications

Abstract:

This paper has two main purposes: 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 the Hoffman-Osserman-Schoen theorem (HOS theorem) to $3$-dimensional symmetric spaces. Precisely, we define a Gauss map of a hypersurface $M^{n-1}$ immersed in a symmetric space $N^{n}$ taking values in the unit pseudo-sphere $\mathbb {S}^m$ of the Lie algebra $\mathfrak {g}$ of the isometry group of $N$, $\dim {\mathfrak {g}}=m+1,$ 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 $\mathbb {S}^m$ 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, an extension of the HOS theorem to the $3$-dimensional hyperbolic space is obtained, which, as far as the authors know, has not been done.

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 $\mathbb {H}^{3}, \mathbb {R}^{3}$ and $\mathbb {S}^{3}$ and coincides with the Abresch-Rosenberg quadratic form when the ambient space is $\mathbb {H}^{2}\times \mathbb {R}$ and $\mathbb {S}^{2}\times \mathbb {R}$. This then provides a 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 five spaces.