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Convexity of 2-Convex Translating Solitons to the Mean Curvature Flow in \(\pmb {\varvec{{\mathbb {R}}}}^{n+1}\)

Abstract

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton \(\Sigma \subset {\mathbb {R}}^{n+1}\) for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in \({\mathbb {R}}^{n+1},\, n\ge 3 \) is the axisymmetric “bowl soliton.”

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Correspondence to Liming Sun.

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Spruck, J., Sun, L. Convexity of 2-Convex Translating Solitons to the Mean Curvature Flow in \(\pmb {\varvec{{\mathbb {R}}}}^{n+1}\). J Geom Anal 31, 4074–4091 (2021). https://doi.org/10.1007/s12220-020-00427-w

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  • DOI: https://doi.org/10.1007/s12220-020-00427-w

Keywords

  • Mean curvature flow
  • Entire translating soliton
  • Uniform 2-convexity
  • Bowl soliton
  • Convexity
  • Fully nonlinear elliptic

Mathematics Subject Classification

  • Primary 53C44
  • 53C21
  • Secondary 53C42
  • 35J60