Notes on Translating Solitons for Mean Curvature Flow

The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in $\mathbb{R}^3$, particularly those which are complete graphs over domains in $\mathbb{R}^2$. In this paper we describe a full classification of complete translating graphs in $\mathbb{R}^3$.

[1]  Or Hershkovits.  Translators asymptotic to cylinders , 2018, Journal für die reine und angewandte Mathematik (Crelles Journal).

[2]  T. Ilmanen.  Elliptic regularization and partial regularity for motion by mean curvature , 1994 .

[3]  X. H. Nguyen.  Translating Tridents , 2009 .

[4]  Graham Smith.  On complete embedded translating solitons of the mean curvature flow that are of finite genus , 2015, 1501.04149.

[5]  X. H. Nguyen.  Complete Embedded Self-Translating Surfaces Under Mean Curvature Flow , 2010, 1004.2657.

[6]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[7]  G. Huisken.  Asymptotic-behavior for singularities of the mean-curvature flow , 1990 .

[8]  Klaus Ecker,et al.  Regularity Theory for Mean Curvature Flow , 2003 .

[9]  Steven J. Altschuler,et al.  Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle , 1994 .

[10]  J. Spruck,et al.  Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature , 2020, American Journal of Mathematics.

[11]  Robert Haslhofer.  Uniqueness of the bowl soliton , 2014, 1408.3145.

[12]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[13]  G. Huisken.  Flow by mean curvature of convex surfaces into spheres , 1984 .

[14]  Brian White,et al.  Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals , 1987 .

[15]  D. Hoffman,et al.  Graphical translators for mean curvature flow , 2018, Calculus of Variations and Partial Differential Equations.

[16]  J. Spruck,et al.  Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature , 2017, 1703.01003.

[17]  X. H. Nguyen.  Doubly periodic self-translating surfaces for the mean curvature flow , 2012, 1204.4921.

[18]  K. Smoczyk,et al.  A characterization of the grim reaper cylinder , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[19]  E. S. Gama,et al.  Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb{R}^{n+1}$ , 2018, 1802.08468.

[20]  William H. Meeks,et al.  A survey on classical minimal surface theory , 2012 .

[21]  S. Angenent.  On the formation of singularities in the curve shortening flow , 1991 .

[22]  K. Smoczyk,et al.  On the topology of translating solitons of the mean curvature flow , 2014, 1404.6703.

[23]  Mathew T. Langford,et al.  On the existence of translating solutions of mean curvature flow in slab regions , 2018, Analysis & PDE.

[24]  D. Hoffman,et al.  Scherk-like translators for mean curvature flow , 2019, Journal of Differential Geometry.

[25]  B. White.  Controlling Area Blow-up in Minimal or Bounded Mean Curvature Varieties and Some Extensions of the Allard Regularity Theorems , 2012, 1207.3448.

[26]  M. Pino,et al.  Finite topology self-translating surfaces for the mean curvature flow in $\mathbb R^3$ , 2015, 1501.03867.

[27]  Xu-jia Wang.  Convex solutions to the mean curvature flow , 2004, math/0404326.

[28]  S. Brendle,et al.  Uniqueness of convex ancient solutions to mean curvature flow in $${\mathbb {R}}^3$$R3 , 2017, Inventiones mathematicae.

[29]  William P. Minicozzi,et al.  A Course in Minimal Surfaces , 2011 .

[30]  Stability of translating solutions to mean curvature flow , 2005, math/0509372.

[31]  S. Hildebrandt,et al.  Minimal Surfaces I , 1992 .

[32]  M. Rigoli,et al.  Maximum Principles and Geometric Applications , 2016 .

[33]  L. Shahriyari,et al.  Translating graphs by mean curvature flow , 2012, 1212.6418.

[34]  L. Evans,et al.  Partial Differential Equations , 1941 .

[35]  Sakinah,et al.  Vol. , 2020, New Medit.

[36]  Carlo Mantegazza,et al.  Lecture Notes on Mean Curvature Flow , 2011 .