Summary: Let \(M=\Sigma_1\times\Sigma_2\) be the product of two compact Riemannian manifolds of dimension \(n\geq 2\) and two, respectively. Let \(\Sigma\) be the graph of a smooth map \(f:\Sigma_1\to \Sigma_2\); \(\Sigma\) is an \(n\)-dimensional submanifold of \(M\). Let \({\mathfrak S}\) be the Grassmannian bundle over \(M\) whose fiber at each point is the set of all \(n\)-dimensional subspaces of the tangent space of \(M\). The Gauss map \(\gamma:\Sigma\to{\mathfrak S}\) assigns to each point \(x\in\Sigma\) the tangent space of \(\Sigma\) at \(x\).
This article considers the mean curvature flow of \(\Sigma\) in \(M\). When \(\Sigma_1\) and \(\Sigma_2\) are of the same non-negative curvature, we show that a sub-bundle \({\mathfrak S}\) of the Grassmannian bundle is preserved along the flow, i.e., if the Gauss map of the initial submanifold \(\Sigma\) lies in \({\mathfrak S}\), then the Gauss map of \(\Sigma_t\) at any later time \(t\) remains in \({\mathfrak S}\). We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity. As an application, we show that if \(f\) is any map from \(S^n\) to \(S^2\) and if at each point, the restriction of \(df\) to any two dimensional subspace is area decreasing, then \(f\) is homotopic to a constant map.