arXiv:0804.0783v2 [math.DG] 10 Apr 2008 Mean Curvature Flow of Spacelike Graphs in Pseudo-Riemannian Manifolds Guanghan Li 1,2† and Isabel M.C. Salavessa 2‡ 1 School of Mathematics and Computer Science, Hubei University, Wuhan, 430062, P. R. China, e-mail: liguanghan@163.com 2 Centro de F´ ısica das Interac¸ c˜ oes Fundamentais, Instituto Superior T´ ecnico, Technical Univer- sity of Lisbon, Edif´ ıcio Ciˆ encia, Piso 3, Av. Rovisco Pais, 1049-001 Lisboa, Portugal; e-mail: isabel.salavessa@ist.utl.pt Abstract: Let (Σ 1 ,g 1 ) and (Σ 2 ,g 2 ) be two compact Riemannian manifolds with sec- tional curvatures K 1 and K 2 , and a smooth map f :Σ 1 → Σ 2 . On Σ 1 × Σ 2 we consider the pseudo-Riemannian metric g 1 − g 2 , and assume the graph of f is a spacelike sub- manifold Γ f . We consider the evolution of Γ f in Σ 1 × Σ 2 by mean curvature flow and show that if K 1 (p) ≥ max{0,K 2 (q)} for any p ∈ Σ 1 and q ∈ Σ 2 then the flow remains a spacelike graph and exists for all time and converges at infinity to the graph of a totally geodesic map f ∞ . Moreover, if K 1 > 0 somewhere, f ∞ is a constant map. If K 1 > 0 everywhere we may replace the compactness assumption of Σ 2 by bounded curvature tensor and all its derivatives. As a consequence we prove that for any arbitrary compact Riemannian manifolds Σ i , i =1, 2 if K 1 > 0 everywhere then there exist a constant ρ ≥ 0 that depends only on K 1 and K 2 such that any map f :Σ 1 → Σ 2 with f ∗ g 2 <ρ −1 g 1 is homotopic to a constant one. This largely extends known results with constant K i ′ s . 1 Introduction Let M be a smooth manifold of dimension m ≥ 2, and F 0 : M → ¯ M a smooth submanifold immersed into an (m + n)-dimensional Riemannian or pseudo-Riemannian manifold ( ¯ M, ¯ g). 0 MSC 2000: Primary: 53C21, 53C40; Secondary: 58D25, 35K55 Key Words: mean curvature flow, spacelike submanifold, maximum principle, homotopic maps. † Partially supported by NSFC (No.10501011) and by Funda¸ c˜ ao Ciˆ encia e Tecnologia (FCT) through a FCT fellowship SFRH/BPD/26554/2006. ‡ Partially supported by FCT through the Plurianual of CFIF and POCI-PPCDT/MAT/60671/2004. 1