Convexity of 2-convex translating and expanding solitons to the mean curvature flow in \(\mathbb{R}^{n+1}\)

In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdziński [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in \(\mathbb{R}^{n+1}\). More precisely, for \(n\geq 3\), we show that any \(n\)-dimensional complete 2-convex translating solitons are convex, and any \(n\)-dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.