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Abstract.We prove that if the graph Tf = {(x,f(x)): x e M} of a map /: (M, g) -> (TV, h) between Riemannian manifolds is a submanifold of (M x N,gxh) with parallel mean curvature H , then on a compact domain D C M , \\H\\ is bounded from above by ^ ffiff . In particular, ry is minimal provided M is compact, or noncompact with zero Cheeger constant. Moreover, if M is the m-hyperbolic space-thus with nonzero Cheeger constant-then there exist real-valued functions the graphs of which are nonminimal submanifolds of M x R with parallel mean curvature.

We prove the mean curvature flow of a spacelike graph in ( 1 × 2 , g 1 − g 2 ) of a map f : 1 → 2 from a closed Riemannian manifold ( 1 , g 1 ) with Ricci 1 > 0 to a complete Riemannian manifold ( 2 , g 2 ) with bounded curvature tensor and derivatives, and with sectional curvatures satisfying K 2 ≤ K 1 , remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption K 2 ≤ K 1 , that if K 1 > 0, or if Ricci 1 > 0 and K 2 ≤ −c, c > 0 constant, any map f : 1 → 2 is trivially homotopic provided f * g 2 < ρg 1 where ρ = min 1 K 1 / sup 2 K + 2 ≥ 0, in case K 1 > 0, and ρ = +∞ in case K 2 ≤ 0. This largely extends some known results for K i constant and 2 compact, obtained using the Riemannian structure of 1 × 2 , and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.

We present a construction of a canonical G 2 structure on the unit sphere tangent bundle S M of any given orientable Riemannian 4-manifold M . Such structure is never geometric or 1-flat, but seems full of other possibilities. We start by the study of the most basic properties of our construction. The structure is co-calibrated if, and only if, M is an Einstein manifold. The fibres are always associative. In fact, the associated 3-form φ results from a linear combination of three other volume 3-forms, one of which is the volume of the fibres. We also give new examples of co-calibrated structures on well known spaces. We hope this contributes both to the knowledge of special geometries and to the study of 4-manifolds.

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form Σ 1 × R, to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f : Σ 1 → Σ 2 between two Riemannian manifolds (Σ m 1 , g 1 ) and (Σ n 2 , g 2 ) of sectional curvatures K 1 and K 2 , respectively. We take on Σ 1 × Σ 2 the pseudoRiemannian product metric g 1 −g 2 . Under the curvature conditions, Ricci 1 ≥ 0 and K 1 ≥ K 2 , we prove that, if the second fundamental form of M satisfies an integrability condition, then M is totally geodesic, and it is a slice if Ricci 1 (p) > 0 at some point. For bounded K 1 , K 2 and hyperbolic angle θ , we conclude M must be maximal. If M is a maximal surface andwe show M is totally geodesic with no need for further assumptions. Furthermore, M is a slice if at some point p ∈ Σ 1 , K 1 (p) > 0, and if Σ 1 is flat and K 2 < 0 at some point f (p), then the image of f lies on a geodesic of Σ 2 . 0 MSC 2000: Primary: 53C21, 53C42, 53C50