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e-mail: Abstract: Let (Σ1,g1) and (Σ2,g2) be two compact Riemannian manifolds with sectional curvatures K1 and K2, Σ1 orientable, and a smooth map f: Σ1 → Σ2. On Σ1 ×Σ2 we consider the pseudo-Riemannian metric g1 −g2, and assume the graph of f is a spacelike submanifold Γf. We consider the evolution of Γf in Σ1 × Σ2 by mean curvature flow and show that if K1(p) ≥ max{0,K2(q)} for any p ∈ Σ1 and q ∈ Σ2 then the flow remains a spacelike graph and exists for all time and a sequence converges at infinity to the graph of a totally geodesic map f∞. Moreover, if K1> 0 somewhere, f ∞ is a constant map. If K1> 0 everywhere all the flow converges to a slice, and we may replace the compactness assumption of Σ2 by bounded curvature tensor and all its derivatives. As a consequence we prove that for any arbitrary compact Riemannian manifolds Σi, i = 1,2, Σ1 orientable, if K1> 0 everywhere then there exist a constant ρ ≥ 0 that depends only on K1 and K2 such that any map f: Σ1 → Σ2 with f ∗ g2 < ρ −1 g1 is homotopic to a constant one. This largely extends known results with constant Ki ′ s.

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Key Words, mean curvature flow, spacelike submanifold, maximum principle, homotopic maps

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