Convexity of 2-Convex Translating Solitons to the Mean Curvature Flow in Rn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {\varvec{{\mathbb {R}}}}^{n+1}$$\end{document}

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton Σ⊂Rn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma \subset {\mathbb {R}}^{n+1}$$\end{document} for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in Rn+1,n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n+1},\, n\ge 3 $$\end{document} is the axisymmetric “bowl soliton.”