Mean Curvature Flow of Spacelike Graphs in Pseudo-Riemannian Manifolds

Abstract

Abstract: Let (Σ1, g1) and (Σ2, g2) be two compact Riemannian manifolds with sectional curvatures K1 and K2, 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 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 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 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−1g1 is homotopic to a constant one. This largely extends known results with constant Ki ′ s.

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