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Open maps: Small and large holes with unusual properties
Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow
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 |
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 [
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. |
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