# Graphical translators for mean curvature flow

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## Abstract

In this paper we provide a full classification of complete translating graphs in \({\mathbf {R}}^3\). We also construct \((n-1)\)-parameter families of new examples of translating graphs in \({\mathbf {R}}^{n+1}\).

## Mathematics Subject Classification

Primary 53C44 53C21 53C42## Notes

## References

- 1.Altschuler, S.J., Wu, L.F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differ. Equ.
**21**, 101–111 (1994)MathSciNetCrossRefGoogle Scholar - 2.Bourni, T., Langford, M., Tinaglia, G.: On the existence of translating solutions of mean curvature flow in slab regions. arXiv:1805.05173, 1-25 (2018)
- 3.Clutterbuck, J., Schnürer, O., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differ. Equ.
**29**, 281–293 (2007)MathSciNetCrossRefGoogle Scholar - 4.Colding, T.H., Minicozzi II, W.P.: Sharp estimates for mean curvature flow of graphs. J. Reine Angew. Math.
**574**, 187–195 (2004). https://doi.org/10.1515/crll.2004.069 MathSciNetCrossRefzbMATHGoogle Scholar - 5.Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. III. J. Geom. Anal.
**2**(2), 121–150 (1992). https://doi.org/10.1007/BF02921385 MathSciNetCrossRefzbMATHGoogle Scholar - 6.Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math.
**110**(3), 439–486 (1979). https://doi.org/10.2307/1971233 MathSciNetCrossRefzbMATHGoogle Scholar - 7.Hoffman, D., Ilmanen, T., Martín, F., White, B.: Notes on translating solitons for mean curvature flow. Preprint arXiv:1901.09101 (2019)
- 8.Hoffman, D., Martín, F., White, B.: Scherk-like translators for mean curvature flow. Preprint arXiv:1903.04617 (2019)
- 9.Hoffman, D., Martín, F., White, B.: Translating annuli for mean curvature flow. In preparation (2019)Google Scholar
- 10.Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc.
**108**(520), x+90 (1994)MathSciNetzbMATHGoogle Scholar - 11.Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. PDE’s
**54**(3), 2853–2882 (2015)MathSciNetCrossRefGoogle Scholar - 12.Morgan, F.: Geometric Measure Theory. Academic Press Inc, Boston, MA (1988)CrossRefGoogle Scholar
- 13.Radó, T.: On the Problem of Plateau. Chelsea Publishing Co., New York, NY (1951)zbMATHGoogle Scholar
- 14.Shahriyari, L.: Translating graphs by mean curvature flow. Geom. Dedic.
**175**, 57–64 (2015)MathSciNetCrossRefGoogle Scholar - 15.Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)Google Scholar
- 16.Simon, L.: A strict maximum principle for area minimizing hypersurfaces. J. Differ. Geom.
**26**(2), 327–335 (1987)MathSciNetCrossRefGoogle Scholar - 17.Solomon, B., White, B.: A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals. Indiana Univ. Math. J.
**38**(3), 683–691 (1989). https://doi.org/10.1512/iumj.1989.38.38032 MathSciNetCrossRefzbMATHGoogle Scholar - 18.Spruck, J., Xiao, L.: Complete translating solitons to the mean curvature flow in \({\bf R}^3\) with nonnegative mean curvature. Amer. J. Math. pp. 1–23 (2017). arXiv:1703.01003v2
- 19.Wang, X.J.: Convex solutions of the mean curvature flow. Ann. Math.
**173**, 1185–1239 (2011)MathSciNetCrossRefGoogle Scholar - 20.White, B.: Curvature estimates and compactness theorems in \(3\)-manifolds for surfaces that are stationary for parametric elliptic functionals. Invent. Math.
**88**(2), 243–256 (1987). https://doi.org/10.1007/BF01388908 MathSciNetCrossRefzbMATHGoogle Scholar - 21.White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc.
**16**(1), 123–138 (2003). https://doi.org/10.1090/S0894-0347-02-00406-X. (electronic)MathSciNetCrossRefzbMATHGoogle Scholar - 22.White, B.: Introduction to minimal surface theory, Geometric analysis, IAS/Park City Math. Ser., vol. 22, Amer. Math. Soc., Providence, RI, pp. 387–438 (2016)Google Scholar
- 23.White, B.: Controlling area blow-up in minimal or bounded mean curvature varieties. J. Differ. Geom.
**102**(3), 501–535 (2016)MathSciNetCrossRefGoogle Scholar

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