Pacific Journal of Mathematics GAUSS MAP HARMONICITY AND MEAN CURVATURE OF A HYPERSURFACE IN A HOMOGENEOUS MANIFOLD FIDELIS BITTENCOURT AND JAIME RIPOLL Volume 224 No. 1 March 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 224, No. 1, 2006 GAUSS MAP HARMONICITY AND MEAN CURVATURE OF A HYPERSURFACE IN A HOMOGENEOUS MANIFOLD FIDELIS BITTENCOURT AND JAIME RIPOLL We define a Gauss map of an orientable hypersurface in a homogeneous manifold with an invariant Riemannian metric. Our main objective is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface of an Euclidean space, namely the Ruh–Vilm the- orem relating the harmonicity of the Gauss map and the constancy of the mean curvature, and the Hoffman–Osserman–Schoen theorem characteriz- ing the plane and the circular cylinder as the only complete constant mean curvature surfaces whose Gauss image is contained in a closed hemisphere of the sphere. 1. Introduction In this article we define a Gauss map of an orientable hypersurface in a homo- geneous manifold with an invariant Riemannian metric. Our main objective is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface of an Euclidean space. We focus our investigation on the extension of two well known theorems: Theorem (Ruh–Vilm). A hypersurface in ޒn has constant mean curvature if and only if the Gauss map of the hypersurface is harmonic. Theorem (Hoffman–Osserman–Schoen). If the Gauss map of a complete constant mean curvature surface in ޒ3 is contained in a closed hemisphere of the sphere, the surface is a plane or a circular cylinder. Extensions of these results were already given in [Esp´ırito-Santo et al. 2003] in the case that the ambient space is a Lie group with a bi-invariant metric; the main theorems in the present work extend those of that reference to the case of homogeneous spaces. We also obtain certain other results. Section 2 introduces notations and basic facts about homogeneous manifolds which will be used throughout the paper. We introduce some of these notations MSC2000: 53A10. Keywords: Gauss map, harmonic map, mean curvature. 45 46 FIDELISBITTENCOURTANDJAIMERIPOLL now. Let އ be a Lie group with a fixed bi-invariant Riemannian metric, ވ a com- pact Lie subgroup of އ. Set k = dim ވ and n +k +1 = dim އ. In the quotient އ/ވ of left residue classes we consider a homogeneous Riemannian metric in such a way that the projection π : އ → އ/ވ becomes a Riemannian submersion. Denote by Ᏻ the Lie algebra of އ, by Ᏼ the Lie algebra of ވ, and let ޓn+k be the unit sphere centered at the origin of Ᏻ. Let M be an orientable hypersurface in އ/ވ oriented by a unit normal vector field η. In Section 3 we define, prove some general properties of, and give examples of the Gauss map N : M → ޓn+k of M; this map is defined by taking the horizontal lifting η˜ of η to އ followed by the right translation of η˜ to the identity of އ, that is, the right translation of η˜ to Ᏻ. In the Euclidean space, taking އ = ޒn and ވ = {e}, N coincides with the usual Gauss map of M. In Section 4 we study the Gauss map of constant mean curvature hypersurfaces of އ/ވ. Our main result asserts that M has constant mean curvature if and only if N is harmonic (Corollary 4.4). This fact has many implications and, to state some of them, we first introduce a definition. A codimension one vector subspace of Ᏻ divides Ᏻ in two connected compo- nents; the closure of any of these components is called a half-space of Ᏻ and its n+k n+k l intersection with ޓ is a half-sphere, denoted by ޓ+ . Given l ∈ ގ, a (1/2 )- sphere ޓn+k,l of ޓn+k is the intersection of l linearly independent half spheres n+k n+k n+k (ޓ1 )+,... , (ޓl )+ of ޓ (by “linearly independent” we mean that the nor- mal vectors to the bounding half-spaces are linearly independent). We prove: Let M be a compact constant mean curvature hypersurface of އ/ވ and N : M → ޓn+k the Gauss map of M. Assume that either M is not totally geodesic or the Ricci curvature of އ is positive. Given l ∈ ގ, let l n+k,l \ n+k ޓ = (ޓi )+ i=1 be a (1/2l )-sphere of ޓn+k. Then there is equivalence between: (1) N(M) ⊂ ޓn+k,l . T l n+k (2) N(M) ⊂ i=1 ∂(ޓi )+. T l n+k ⊥ (3) ᏷ := i=1 ∂(ޓi )+ is a Lie subalgebra of Ᏻ and M is invariant under the Lie subgroup ދ of އ whose Lie algebra is ᏷. It follows under the same general hypotheses that if l ≥ n, then M is an extrinsi- cally homogeneous manifold (that is, there is a Lie subgroup of isometries of އ/ވ acting transitively on M). If M is assumed only to be complete, a similar result holds when dim(އ/ވ) = 3. In this case, more explicit applications are given in Corollaries 4.13 and 4.14. In GAUSS MAP AND MEAN CURVATURE IN HOMOGENEOUS MANIFOLDS 47 particular, we obtain the following result related to a conjecture of B. Lawson (which asserts that an embedded minimal torus in ޓ3 has to be a Clifford torus, that is, the Riemannian product of two circles): A complete minimal surface M in ޓ3 = SO(4)/ SO(3) is a Clifford torus if and only if the Gauss image N(M) of M lies in a certain hemisphere of the unit sphere through the origin in so(4) (Corollary 4.14). The harmonicity of the Gauss map of a constant mean curvature surface can also be used to establish a stability criterion: If D is a domain of a constant mean curvature surface M of އ/ވ (dim(އ/ވ) = 3) such that N(D) is contained in a half space of Ᏻ, then D is stable (Theorem 4.10). 2. Preliminaries Let އ be an (n+k+1)-dimensional connected Lie group, for n ≥ 2, k ≥ 0, with a bi-invariant Riemannian metric h , i. Denote by ∇e the Riemannian connection on އ determined by h , i. Let Ᏻ be the Lie algebra of އ. Given x ∈ އ, let L x and Rx denote the left and right translations on އ, that is, L x (y) = xy and Rx (y) = yx. As usual, exp : Ᏻ → އ denotes the Lie exponential map. Given g ∈ އ and X ∈ Ᏻ, we have the maps Adg, adX : Ᏻ → Ᏻ defined by −1 Adg = d(Rg ◦ Lg)e and adX (Y ) = [X, Y ]. We note that d adX (Y ) = Adexp t X (Y ) , dt t=0 and that Adg is a Lie group isomorphism and an isometry. We recall that if X, Y , Z are both left (or both right) invariant vector fields of އ, then ∇e and the curvature tensor R on އ are given by ∇ = 1 [ ] = 1 [[ ] ] eX Y 2 X, Y and R(X, Y )Z 4 X, Y , Z . We have h[X, Y ], Zi = h[Z, Y ], Xi. The Ricci tensor of އ is given by n+k+1 X Ric(u, v) = trace X 7→ R(u, X)v = R(u, E j )v, E j , i=1 where {E j } is an orthonormal basis of tangent vectors of އ. The Ricci curvature of އ on the u-direction is Ric(u) = Ric(u, u). Throughout this article, Ric will always mean the Ricci curvature of އ. Let ވ be a k-dimensional closed Lie subgroup of އ (recall that k ≥ 0), with Lie algebra Ᏼ, and let އ/ވ be the smooth (n+1)-dimensional manifold of left residue classes of ވ: އ/ވ = {xވ : x ∈ އ}. 48 FIDELISBITTENCOURTANDJAIMERIPOLL We consider on އ/ވ a Riemannian metric, also denoted by h , i, induced by the projection π : އ → އ/ވ, π(x) = xވ, as follows. Given z ∈ އ/ވ, take x ∈ π −1(z) and define an inner product h , i in Tz(އ/ވ) in such a way that −1 ⊥ lx := dπx −1 ⊥ : Tx (π (z)) → Tz(އ/ ވ) Tx (π (z)) ⊥ is a linear isometry, where denotes the orthogonal complement in Tx (އ). It is easy to see that h , i is well defined in Tz(އ/ވ) (that is, it does not depend on the choice of x ∈ π −1(z)) and defines a Riemannian homogeneous (bi-invariant) metric on އ/ވ. Denote by ∇ the Riemannian connection associated to h , i. Denote by ᐄ(އ/ވ) and by ᐄ(އ) the spaces of smooth vector fields on އ/ވ and ⊥ އ. A vector field eX in އ is called horizontal if eX(x) ∈ Tx (xވ) for x ∈ އ, and is called vertical if eX(x) ∈ Tx (xވ) for x ∈ އ. Given eX ∈ ᐄ(އ), let eX v be the vertical vector field on އ given as the orthogonal h projection of eX(x) on Tx (xވ), and let eX be the horizontal vector field on އ given ⊥ as the orthogonal projection of eX(x) on Tx (xވ) , x ∈ އ. It follows forthwith from the definition of h , i that the projection π : އ → އ/ވ is a Riemannian submersion. It is not difficult to see that given X, Y ∈ ᐄ(އ/ވ), we have ∇ = −1∇ + 1 [ ]v eeX Ye(x) lx X Y (π(x)) 2 eX, Ye (π(x)), −1 −1 where eX, Ye∈ ᐄ(އ) are defined by eX(x) =lx (X (π(x))) and Ye(x) =lx (Y (π(x))) for x ∈ އ, and [ , ] is the Lie bracket in އ.