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

TL;DR

This paper extends Ruh-Vilms theorem to symmetric spaces, linking constant mean curvature hypersurfaces with harmonic Gauss maps, and applies it to generalize the Hoffman-Osserman-Schoen theorem to 3D symmetric spaces, unifying quadratic forms in CMC surface theory.

Contribution

It generalizes Ruh-Vilms theorem to symmetric spaces and extends HOS theorem to 3D symmetric spaces, providing a unified framework for quadratic forms in CMC surfaces.

Findings

CMC hypersurfaces have harmonic Gauss maps in symmetric spaces.

Gauss map image in a hemisphere implies invariance under isometries.

Extension of HOS theorem to hyperbolic space.

Abstract

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 M^{n-1} immersed in a symmetric space N^n taking values in the unit pseudo sphere S^m 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 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, 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 H^2 x R and S^2 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.