In this article we study the two-dimensional complete $\lambda$-translators immersed in the Euclidean space $\mathbb{R}^3$ and Minkovski space $\mathbb{R}^3_1$. We obtain two classification theorems: one for two-dimensional complete $\lambda$-translators $x:M^2\to\mathbb{R}^3$ and one for two-dimensional complete space-like $\lambda$-translators $x:M^2\to\mathbb{R}^3_1$, with a second fundamental form of constant length.
Xingxiao LI
,
Ruina QIAO
,
Yangyang LIU
. ON THE COMPLETE 2-DIMENSIONAL λ-TRANSLATORS WITH A SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1897
-1914
.
DOI: 10.1007/s10473-020-0618-3
[1] Altschuler S J, Wu L F. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc Var, 1994, 2:101-111
[2] Chen Q, Qiu H B. Rigidity of self-shrinkers and translating solitons of mean curvature flows. Adv Math, 2016, 294:517-531
[3] Cheng Q -M, Ogata S. 2-dimensional complete self-shrinkers in $\mathbb{R}^3$. Math Z, 2016, 284:537-542
[4] Cheng Q -M, Ogata S, Wei G X. Rigidity theorems of λ-hypersurfaces. Comm Anal Geom, 2016, 24:45-58
[5] Cheng Q -M, Wei G X. Complete λ-surfaces in $\mathbb{R}^3$. arXiv:1807.06760v1[math.DG], 2018
[6] Cheng Q -M, Wei G X. Complete λ-hypersurfaces of the weighted volume-preserving mean curvature flow. Calc Var, 2018, 57(2):Art 32, DOI 10.1007/s00526-018-1303-4
[7] Clutterbuck J, Schnürer O, F Schulze. Stability of translating solutions to mean curvature flow. Calc Var, 2007, 29:281-293
[8] Gromov M. Isoperimetry of waists and concentration of maps. Geom Func Anal, 2003, 13:178-215
[9] Halldorsson H P. Helicoidal surfaces rotating/translating under the mean curvature flow. Geom Dedicata, 2013, 162:45-65
[10] Huisken G, Sinestrari C. Mean curvature flow singularities for mean convex surfaces. Calc Var, 1999, 8:1-14
[11] Ilmanen T. Elliptic regularization and partial regularity for motion by mean curvature. Mem Amer Math Soc, 1994, 108
[12] Li X X, Li X. On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane C2. Acta Math Sci, 2019, 39B(6):1695-1712
[13] López R. Invariant surfaces in Euclidean space with a log-linear density. Adv Math, 2018, 339:285-309
[14] López R. Compact λ-translating solitons with boundary. Mediterranean J Math, 2018, 15(5):Art 196
[15] Martín F, Savas-Halilaj A, Smoczyk K. On the topology of translating solitons of the mean curvature flow. Calc Var, 2015, 54:2853-2882
[16] Minh N, Hieu D T. Ruled minimal surfaces in $\mathbb{R}^3$ with density ez. Pacific J Math, 2009, 243:277-285
[17] Morgan F. Manifolds with density. Notices Amer Math Soc, 2005, 52:853-858
[18] Pyo J. Compact translating solitons with non-empty planar boundary. Diff Geom App, 2016, 47:79-85
[19] Shahriyari L. Translating Graphs by Mean Curvature Flow[D]. The Johns Hopkins University, 2013
[20] Smith G. On complete embedded translating solitons of the mean curvature flow that area of finite genus. arXiv:1501.04149[math.DG], 2015
[21] Wang X-J. Convex solutions to the mean curvature flow. Ann Math, 2011, 173:1185-1239
[22] White B. Subsequent singularities in mean-convex mean curvature flow. Calc Var Pertial Diff Equ, 2015, 54(2):1457-1468
