Mean curvature flow of spacelike graphs
نویسنده
چکیده
We prove the mean curvature flow of a spacelike graph in (Σ1 ×Σ2,g1 −g2) of a map f : Σ1 → Σ2 from a closed Riemannian manifold (Σ1,g1) with Ricci1 > 0 to a complete Riemannian manifold (Σ2,g2) with bounded curvature tensor and derivatives, and with K2 ≤ K1, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption K2 ≤ K1, that if K1 > 0, or if Ricci1 > 0 and K2 ≤ −c, c > 0 constant, any map f : Σ1 → Σ2 is trivially homotopic provided f ∗g2 < ρg1 where ρ = minΣ1 K1/supΣ2 K + 2 ≥ 0. This largely extends some known results for Ki constant and ρ = 1.
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