A rigidity result for $ 2 $-dimensional $ \lambda $-translators

Jin Liu; Botao Wang; Jin Liu; Botao Wang

doi:10.3934/math.20231272

AIMS Mathematics

2023, Volume 8, Issue 10: 24947-24956. doi: 10.3934/math.20231272

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A rigidity result for $ 2 $-dimensional $ \lambda $-translators

Jin Liu ,
Botao Wang ^,

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

Received: 02 July 2023 Revised: 14 August 2023 Accepted: 16 August 2023 Published: 28 August 2023
MSC : 53C40, 53E10

In this paper, we will develop a different technique to study the rigidity of complete $ \lambda $-translators $ x:M^{2} \rightarrow \mathbb R^{3} $ with the non-zero constant gauss curvature in the Euclidean space $ \mathbb R^{3} $.
- second fundamental form,
- $ \lambda $-translator,
- Gauss curvature,
- non-umbilic points,
- mean curvature
Citation: Jin Liu, Botao Wang. A rigidity result for $ 2 $-dimensional $ \lambda $-translators[J]. AIMS Mathematics, 2023, 8(10): 24947-24956. doi: 10.3934/math.20231272

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Abstract

In this paper, we will develop a different technique to study the rigidity of complete $ \lambda $-translators $ x:M^{2} \rightarrow \mathbb R^{3} $ with the non-zero constant gauss curvature in the Euclidean space $ \mathbb R^{3} $.

References

[1]	S. J. Altschuler, L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differ. Equ., 2 (1994), 101–111. https://doi.org/10.1007/BF01234317 doi: 10.1007/BF01234317
[2]	Q. Chen, H. Qiu, Rigidity of self-shrinkers and translating solitons of mean curvature flows, Adv. Math., 294 (2016), 517–531. https://doi.org/10.1016/j.aim.2016.03.004 doi: 10.1016/j.aim.2016.03.004
[3]	Q. M. Cheng, G. Wei, Complete $\lambda$-surfaces in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equ., 60 (2021), 46. https://doi.org/10.1007/s00526-021-01920-y doi: 10.1007/s00526-021-01920-y
[4]	J. Clutterbuck, O. C. Schnurer, F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differ. Equ., 29 (2007), 281–293. https://doi.org/10.1007/s00526-006-0033-1 doi: 10.1007/s00526-006-0033-1
[5]	A. Derdziński, Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor, Math. Z., 172 (1980), 273–280. https://doi.org/10.1007/BF01215090 doi: 10.1007/BF01215090
[6]	H. P. Halldorsson, Helicoidal surfaces rotating/translating under the mean curvature flow, Geom. Dedicata, 162 (2013), 45–65. https://doi.org/10.1007/s10711-012-9716-2 doi: 10.1007/s10711-012-9716-2
[7]	R. Haslhofer, Uniqueness of the bowl soliton, Geom. Topol., 19 (2015), 2393–2406. https://doi.org/10.2140/gt.2015.19.2393 doi: 10.2140/gt.2015.19.2393
[8]	D. T. Hieu, N. M. Hoang, Ruled minimal surfaces in $\mathbb{R}^{3}$ with density $e^{z}$, Pacific J. Math., 243 (2009), 277–285.
[9]	G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differ. Equ., 8 (1999), 1–14. https://doi.org/10.1007/s005260050113 doi: 10.1007/s005260050113
[10]	G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45–70. https://doi.org/10.1007/BF02392946 doi: 10.1007/BF02392946
[11]	T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Am. Math. Soc., 108 (1994), 520. https://doi.org/10.1090/memo/0520 doi: 10.1090/memo/0520
[12]	D. Impera, M. Rimoldi, Rigidity results and topology at infinity of translating solitons of the mean curvature flow, Commun. Contemp. Math., 19 (2017), 1750002. https://doi.org/10.1142/S021919971750002X doi: 10.1142/S021919971750002X
[13]	X. Li, R. Qiao, Y. Liu, On the complete $2$-dimensional $\lambda$-translators with a second fundamental form of constant length, Acta Math. Sci., 40 (2020), 1897–1914. https://doi.org/10.1007/s10473-020-0618-3 doi: 10.1007/s10473-020-0618-3
[14]	R. López, Invariant surfaces in Euclidean space with a log-linear density, Adv. Math., 339 (2018), 285–309. https://doi.org/10.1016/j.aim.2018.09.029 doi: 10.1016/j.aim.2018.09.029
[15]	R. López, Compact $\lambda$-translating solitons with boundary, Mediterr. J. Math., 15 (2018), 196. https://doi.org/10.1007/s00009-018-1241-6 doi: 10.1007/s00009-018-1241-6
[16]	F. Martin, A. Savas-Halilaj, K. Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differ. Equ., 54 (2015), 2853–2882. https://doi.org/10.1007/s00526-015-0886-2 doi: 10.1007/s00526-015-0886-2
[17]	J. Pyo, Compact translating solitons with non-empty planar boundary, Differ. Geom. Appl., 47 (2016), 79–85. https://doi.org/10.1016/j.difgeo.2016.03.003 doi: 10.1016/j.difgeo.2016.03.003
[18]	J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. R. Soc. Lond. Ser. A, 264 (1969), 413–496. https://doi.org/10.1098/rsta.1969.0033 doi: 10.1098/rsta.1969.0033
[19]	L. Shahriyari, Translating graphs by mean curvature flow, Geom. Dedicata, 175 (2015), 57–64. https://doi.org/10.1007/s10711-014-0028-6 doi: 10.1007/s10711-014-0028-6
[20]	J. Spruck, L. Xiao, Complete translating solitons to the mean curvature flow in $\mathbb{R}^{3}$ with nonnegative mean curvature, Amer. J. Math., 142 (2020), 993–1015. https://doi.org/10.1353/ajm.2020.0023 doi: 10.1353/ajm.2020.0023
[21]	J. Spruck, L. Sun, Convexity of $2$-convex translating solitons to the mean curvature flow in $\mathbb{R}^{n+1}$, J. Geom. Anal., 31 (2021), 4074–4091. https://doi.org/10.1007/s12220-020-00427-w doi: 10.1007/s12220-020-00427-w
[22]	X. J. Wang, Convex solutions to the mean curvature flow, Ann. Math., 173 (2011), 1185–1239. https://doi.org/10.4007/annals.2011.173.3.1 doi: 10.4007/annals.2011.173.3.1
[23]	B. White, Subsequent singularities in mean-convex mean curvature flow, Calc. Var. Partial Differ. Equ., 54 (2015), 1457–1468. https://doi.org/10.1007/s00526-015-0831-4 doi: 10.1007/s00526-015-0831-4
[24]	J. Xie, J. Yu, Convexity of $2$-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$, J. Geom. Anal., 33 (2023), 252. https://doi.org/10.1007/s12220-023-01260-7 doi: 10.1007/s12220-023-01260-7
[25]	Y. L. Xin, Translating solitons of the mean curvature flow, Calc. Var. Partial Differ. Equ., 54 (2015), 1995–2016. https://doi.org/10.1007/s00526-015-0853-y doi: 10.1007/s00526-015-0853-y
[26]	D. Yang, Y. Fu, Y. Luo, Some classifications of $2$-dimensional self-shrinkers, J. Math. Anal. Appl., 524 (2023), 127089. https://doi.org/10.1016/j.jmaa.2023.127089 doi: 10.1016/j.jmaa.2023.127089

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