Mean curvature flow of spacelike graphs

作者：Li Guanghan; Salavessa Isabel M C^*

来源：Mathematische Zeitschrift, 2011, 269(3-4): 697-719.

DOI：10.1007/s00209-010-0768-4

• Mean curvature flow
• Spacelike submanifold
• Maximum principle
• Homotopic maps

We prove the mean curvature flow of a spacelike graph in (Sigma(1) x Sigma(2), g(1) - g(2)) of a map f : Sigma(1) -> Sigma(2) from a closed Riemannian manifold (Sigma(1), g(1)) with Ricci(1) > 0 to a complete Riemannian manifold (Sigma(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 : Sigma(1) -> Sigma(2) is trivially homotopic provided f*g(2) < rho g(1) where rho = min(Sigma 1) K(1)/sup(Sigma 2) K(2)(+) >= 0, in case K(1) > 0, and rho = +infinity in case K(2) <= 0. This largely extends some known results for K(i) constant and Sigma(2) compact, obtained using the Riemannian structure of Sigma(1) x Sigma(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.

• 出版日期2011-12
• 单位湖北大学