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We consider the graphical mean curvature flow of strictly area decreasing maps f : M → N , where M is a compact Riemannian manifold of dimension m > 1 and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature BRic M of M is bounded from below by the sectional curvature σ N of N. In addition, we obtain smooth convergence to a minimal map if Ric M ≥ sup { 0 , sup N σ N } . These results significantly improve known results on the graphical mean curvature flow in codimension 2. [ABSTRACT FROM AUTHOR]

In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal maps. Moreover, their corresponding graphs have second fundamental form of constant norm. We prove that a minimal submersion from S 3 to S 2 whose Gauss map satisfies a suitable pinching condition must be weakly conformal with totally geodesic fibers. As a consequence, we obtain that an equivariant minimal submersion from S 3 to S 2 coincides with the Hopf fibration. Furthermore, we prove that a minimal map f : U ⊂ S 3 → S 2 with constant singular values and constant norm of the second fundamental form is either constant or, up to isometries, coincides with the Hopf fibration. [ABSTRACT FROM AUTHOR]

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space. [ABSTRACT FROM AUTHOR]

In this article we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article. [ABSTRACT FROM AUTHOR]