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

2020 ◽  
Author(s):  
Joel Spruck ◽  
Liming Sun
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
The Mean ◽  
Translating Solitons

scholarly journals On the topology of translating solitons of the mean curvature flow

2015 ◽  
Vol 54 (3) ◽  
pp. 2853-2882 ◽  
Author(s):  
Francisco Martín ◽  
Andreas Savas-Halilaj ◽  
Knut Smoczyk
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
The Mean ◽  
Translating Solitons

scholarly journals Rigidity results and topology at infinity of translating solitons of the mean curvature flow

2017 ◽  
Vol 19 (06) ◽  
pp. 1750002 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Minimal Hypersurfaces ◽  
Rigidity Results ◽  
And Topology ◽  
The Mean ◽  
Translating Solitons

In this paper, we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of [Formula: see text]-minimal hypersurfaces.

Translating Solitons of the Mean Curvature Flow Asymptotic to Hyperplanes in ℝn+1

2018 ◽  
Vol 2020 (24) ◽  
pp. 10114-10153 ◽  
Author(s):  
Eddygledson S Gama ◽  
Francisco Martín
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Normal Component ◽  
Vertical Cylinder ◽  
Curvature Flow ◽  
Grim Reaper ◽  
The Family ◽  
The Mean ◽  
Translating Solitons

Abstract A translating soliton is a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ such that the family $M_t= M- t \,\mathbf e_{n+1}$ is a mean curvature flow, that is, such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf e_{n+1}^{\perp }.$ In this paper we obtain a characterization of hyperplanes that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the 2nd author, Perez-Garcia, Savas-Halilaj, and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

scholarly journals Simply connected translating solitons contained in slabs

2020 ◽  
Vol 5 (1) ◽  
pp. 102-120
Author(s):  
Francesco Chini
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Grim Reaper ◽  
Rotationally Symmetric ◽  
The Mean ◽  
Simply Connected ◽  
Mean Convex ◽  
Translating Solitons

AbstractIn this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

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