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November  2018, 38(11): 5897-5919. doi: 10.3934/dcds.2018256

Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Daehwan Kim 1,2, and  Juncheol Pyo 1,2,,

1. 

Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
2. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

* Corresponding author

Received  March 2018 Revised  April 2018 Published  August 2018

Fund Project: This research was supported in part by the National Research Foundation of Korea (NRF-2017R1E1A1A03070495 and NRF-2017R1A5A1015722).

Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type Ⅱ singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in $\mathbb{R}^{3}$ and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [10]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in $\Bbb R^{n+1}$ and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.

Keywords: Mean curvature flow, translating soliton, helicoidal surface, hypersurface of revolution.
Mathematics Subject Classification: Primary: 53C44, 37C10, 53A10.
Citation: Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
References:
[1]

H. Alencar, A. Barros, O. Palmas, J. G. Reyes and W. Santos, O(m) × O(n)-Invariant Minimal Hypersurfaces in $\mathbb{R}^{m+n}$, Ann. Global Anal. Geom., 27 (2005), 179-199.  doi: 10.1007/s10455-005-2572-7.

[2]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.

[3]

E. Barbosa and Y. Wei, A compactness theorem of the space of free boundary f-minimal surfaces in three-dimensional smooth metric measure space with boundary, J. Geom. Anal., 26 (2016), 1995-2012.  doi: 10.1007/s12220-015-9616-4.

[4]

C. C. Beneki, G. Kaimakamis and B. J. Papantoniou, Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 275 (2002), 586-614.  doi: 10.1016/S0022-247X(02)00269-X.

[5]

E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[6]

X. Cheng, T. Mejia and D. Zhou, Stability and compactness for complete f-minimal surfaces, Trans. Amer. Math. Soc., 367 (2015), 4041-4059.  doi: 10.1090/S0002-9947-2015-06207-2.

[7]

J. Clutterbuck, O. C. Schnürer and F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations, 29 (2007), 281-293.  doi: 10.1007/s00526-006-0033-1.

[8]

J. Dávila, M. del Pino and X. H. Nguyen, Finite topology self-translating surfaces for the mean curvature flow in $\mathbb{R}^3$, Adv. Math., 320 (2017), 674-729.  doi: 10.1016/j.aim.2017.09.014.

[9]

M. P. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tôhoku Math. J.(2), 34 (1982), 425-435.  doi: 10.2748/tmj/1178229204.

[10]

H. P. Halldorsson, Helicoidal surfaces rotating/translating under the mean curvature flow, Geom. Dedicata, 162 (2013), 45-65.  doi: 10.1007/s10711-012-9716-2.

[11]

R. Haslhofer, Uniqueness of the bowl soliton, Geom. Topol., 19 (2015), 2393-2406.  doi: 10.2140/gt.2015.19.2393.

[12]

H. Hopf, Differential Geometry in the Large: Seminar Lectures New York University 1946 and Stanford University 1956. Vol. 1000, Springer, 2003.

[13]

G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom., 31 (1990), 285-299.  doi: 10.4310/jdg/1214444099.

[14]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations, 8 (1999), 1-14.  doi: 10.1007/s005260050113.

[15]

T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994), ⅹ+90 pp. doi: 10.1090/memo/0520.

[16]

W. C. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J., 38 (1991), 255-270.  doi: 10.1307/mmj/1029004332.

[17]

W. C. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math., 28 (1998), 983-1015.  doi: 10.1216/rmjm/1181071750.

[18]

D. Kim and J. Pyo, Translating solitons foliated by spheres, Internat. J. Math., 28 (2017), 1750006, 11 pp. doi: 10.1142/S0129167X17500069.

[19]

C. Kuhns and B. Palme, Helicoidal surfaces with constant anisotropic mean curvature, J. Math. Phys., 52 (2011), 073506, 14 pp. doi: 10.1063/1.3603816.

[20]

F. Martín, A. Savas-Halilaj and K. Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations, 54 (2015), 2853-2882.  doi: 10.1007/s00526-015-0886-2.

[21]

A. Martínez, J. P. dos Santos and K. Tenenblat, Helicoidal flat surfaces in hyperbolic 3-space, Pacific J. Math., 264 (2013), 195-211.  doi: 10.2140/pjm.2013.264.195.

[22]

P. Mira and J. A. Pastor, Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math., 140 (2003), 315-334.  doi: 10.1007/s00605-003-0111-9.

[23]

X. H. Nguyen, Translating trident, Comm. Partial Differential Equations, 34 (2009), 257-280.  doi: 10.1080/03605300902768685.

[24]

X. H. Nguyen, Complete embedded self-translating surfaces under mean curvature flow, J. Geom. Anal., 23 (2013), 1379-1426.  doi: 10.1007/s12220-011-9292-y.

[25]

S.-H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math., 32 (2002), 1019-1044.  doi: 10.1216/rmjm/1034968429.

[26]

C. Peñafiel, Screw motion surfaces in $\widetilde{PSL}_2(\mathbb{R},τ)$, Asian J. Math., 19 (2015), 265-280.  doi: 10.4310/AJM.2015.v19.n2.a4.

[27]

O. M. Perdomo, Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104. 

[28]

J. Pyo, Foliations of a smooth metric measure space by hypersurfaces with constant f-mean curvature, Pacific J. Math., 271 (2014), 231-242.  doi: 10.2140/pjm.2014.271.231.

[29]

J. Pyo, Compact translating solitons with non-empty planar boundary, Differential Geom. Appl., 47 (2016), 79-85.  doi: 10.1016/j.difgeo.2016.03.003.

[30]

J. B. Ripoll, Helicoidal minimal surfaces in hyperbolic space, Nagoya Math. J., 114 (1989), 65-75.  doi: 10.1017/S0027763000001409.

[31]

R. Sa Earp and E. Toubiana, Screw motion surfaces in $\mathbb{H}^2 × \mathbb{R}$ and $\mathbb{S}^2 × \mathbb{R}$, Illinois J. Math., 49 (2005), 1323-1362. 

[32]

J. Sato and V. F. de Souza Neto, Complete and stable O(p + 1) × O(q + 1)-invariant hypersurfaces with zero scalar curvature in Euclidean space $\mathbb{R}^{p+q+2}$, Ann. Global Anal. Geom., 29 (2006), 221-240.  doi: 10.1007/s10455-005-9006-4.

[33]

X.-J. Wang, Convex solutions to the mean curvature flow, Ann. of Math.(3), 173 (2011), 1185-1239.  doi: 10.4007/annals.2011.173.3.1.

show all references

References:
[1]

H. Alencar, A. Barros, O. Palmas, J. G. Reyes and W. Santos, O(m) × O(n)-Invariant Minimal Hypersurfaces in $\mathbb{R}^{m+n}$, Ann. Global Anal. Geom., 27 (2005), 179-199.  doi: 10.1007/s10455-005-2572-7.

[2]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.

[3]

E. Barbosa and Y. Wei, A compactness theorem of the space of free boundary f-minimal surfaces in three-dimensional smooth metric measure space with boundary, J. Geom. Anal., 26 (2016), 1995-2012.  doi: 10.1007/s12220-015-9616-4.

[4]

C. C. Beneki, G. Kaimakamis and B. J. Papantoniou, Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 275 (2002), 586-614.  doi: 10.1016/S0022-247X(02)00269-X.

[5]

E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[6]

X. Cheng, T. Mejia and D. Zhou, Stability and compactness for complete f-minimal surfaces, Trans. Amer. Math. Soc., 367 (2015), 4041-4059.  doi: 10.1090/S0002-9947-2015-06207-2.

[7]

J. Clutterbuck, O. C. Schnürer and F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations, 29 (2007), 281-293.  doi: 10.1007/s00526-006-0033-1.

[8]

J. Dávila, M. del Pino and X. H. Nguyen, Finite topology self-translating surfaces for the mean curvature flow in $\mathbb{R}^3$, Adv. Math., 320 (2017), 674-729.  doi: 10.1016/j.aim.2017.09.014.

[9]

M. P. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tôhoku Math. J.(2), 34 (1982), 425-435.  doi: 10.2748/tmj/1178229204.

[10]

H. P. Halldorsson, Helicoidal surfaces rotating/translating under the mean curvature flow, Geom. Dedicata, 162 (2013), 45-65.  doi: 10.1007/s10711-012-9716-2.

[11]

R. Haslhofer, Uniqueness of the bowl soliton, Geom. Topol., 19 (2015), 2393-2406.  doi: 10.2140/gt.2015.19.2393.

[12]

H. Hopf, Differential Geometry in the Large: Seminar Lectures New York University 1946 and Stanford University 1956. Vol. 1000, Springer, 2003.

[13]

G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom., 31 (1990), 285-299.  doi: 10.4310/jdg/1214444099.

[14]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations, 8 (1999), 1-14.  doi: 10.1007/s005260050113.

[15]

T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994), ⅹ+90 pp. doi: 10.1090/memo/0520.

[16]

W. C. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J., 38 (1991), 255-270.  doi: 10.1307/mmj/1029004332.

[17]

W. C. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math., 28 (1998), 983-1015.  doi: 10.1216/rmjm/1181071750.

[18]

D. Kim and J. Pyo, Translating solitons foliated by spheres, Internat. J. Math., 28 (2017), 1750006, 11 pp. doi: 10.1142/S0129167X17500069.

[19]

C. Kuhns and B. Palme, Helicoidal surfaces with constant anisotropic mean curvature, J. Math. Phys., 52 (2011), 073506, 14 pp. doi: 10.1063/1.3603816.

[20]

F. Martín, A. Savas-Halilaj and K. Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations, 54 (2015), 2853-2882.  doi: 10.1007/s00526-015-0886-2.

[21]

A. Martínez, J. P. dos Santos and K. Tenenblat, Helicoidal flat surfaces in hyperbolic 3-space, Pacific J. Math., 264 (2013), 195-211.  doi: 10.2140/pjm.2013.264.195.

[22]

P. Mira and J. A. Pastor, Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math., 140 (2003), 315-334.  doi: 10.1007/s00605-003-0111-9.

[23]

X. H. Nguyen, Translating trident, Comm. Partial Differential Equations, 34 (2009), 257-280.  doi: 10.1080/03605300902768685.

[24]

X. H. Nguyen, Complete embedded self-translating surfaces under mean curvature flow, J. Geom. Anal., 23 (2013), 1379-1426.  doi: 10.1007/s12220-011-9292-y.

[25]

S.-H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math., 32 (2002), 1019-1044.  doi: 10.1216/rmjm/1034968429.

[26]

C. Peñafiel, Screw motion surfaces in $\widetilde{PSL}_2(\mathbb{R},τ)$, Asian J. Math., 19 (2015), 265-280.  doi: 10.4310/AJM.2015.v19.n2.a4.

[27]

O. M. Perdomo, Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104. 

[28]

J. Pyo, Foliations of a smooth metric measure space by hypersurfaces with constant f-mean curvature, Pacific J. Math., 271 (2014), 231-242.  doi: 10.2140/pjm.2014.271.231.

[29]

J. Pyo, Compact translating solitons with non-empty planar boundary, Differential Geom. Appl., 47 (2016), 79-85.  doi: 10.1016/j.difgeo.2016.03.003.

[30]

J. B. Ripoll, Helicoidal minimal surfaces in hyperbolic space, Nagoya Math. J., 114 (1989), 65-75.  doi: 10.1017/S0027763000001409.

[31]

R. Sa Earp and E. Toubiana, Screw motion surfaces in $\mathbb{H}^2 × \mathbb{R}$ and $\mathbb{S}^2 × \mathbb{R}$, Illinois J. Math., 49 (2005), 1323-1362. 

[32]

J. Sato and V. F. de Souza Neto, Complete and stable O(p + 1) × O(q + 1)-invariant hypersurfaces with zero scalar curvature in Euclidean space $\mathbb{R}^{p+q+2}$, Ann. Global Anal. Geom., 29 (2006), 221-240.  doi: 10.1007/s10455-005-9006-4.

[33]

X.-J. Wang, Convex solutions to the mean curvature flow, Ann. of Math.(3), 173 (2011), 1185-1239.  doi: 10.4007/annals.2011.173.3.1.

Figure 1.  Two integral curves of vector field V (h = 1)
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Figure 2.  Two profile curves of vector field $V$ ($h = 1$)
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Figure 4.  Profile curves (n = 2, 5)
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Figure 3.  Two integral curves of vector field $V$ ($n = 2$)
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