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We consider the graphical mean curvature flow of maps f:Rm→Rn$\mathbf {f}:{\mathbb {R}^{m}}\rightarrow {\mathbb {R}^{n}}$, m⩾2$m\geqslant 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well‐known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps f:Rm→R2$\mathbf {f}:{\mathbb {R}^{m}} \rightarrow {\mathbb {R}^{2}}$, m⩾2$m\geqslant 2$, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self‐expander. [ABSTRACT FROM AUTHOR]

For every complete and minimally immersed submanifold f : M n → S n + p whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / (2 ⁢ p − 1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni. [ABSTRACT FROM AUTHOR]

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 note, we prove a Schwarz–Pick-type lemma for minimal maps between negatively curved Riemann surfaces. More precisely, we prove that if f : M → N is a minimal map with bounded Jacobian determinant between two complete negatively curved Riemann surfaces M and N whose sectional curvatures σ M and σ N satisfy inf σ M ≥ sup σ N , then f is area decreasing. [ABSTRACT FROM AUTHOR]

In this article, we give a complete description of the evolution of an area decreasing map $$f{: }M\rightarrow N$$ , induced by the mean curvature of their graph, in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures $$\sigma _M$$ and $$\sigma _N$$ satisfy $$\min \sigma _M\ge \sup \sigma _N$$ . [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]

Abstract: Let be a smooth area decreasing map between two Riemannian manifolds and . Under weak and natural assumptions on the curvatures of and , we prove that the mean curvature flow provides a smooth homotopy of f to a constant map. [Copyright &y& Elsevier]