Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension

Let f:Σ₁↦Σ₂ be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ₁×Σ₂ by the mean curvature flow. Under suitable conditions on the curvature of Σ₁ and Σ₂ and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f _t and f _t converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschitz initial data. Oblatum 30-I-2001 & 24-X-2001?Published online: 1 February 2002