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doi: 10.3934/dcdsb.2022011
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Minimal surface generating flow for space curves of non-vanishing torsion

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic

* Corresponding author: Jiří Minarčík

Received  October 2020 Revised  December 2021 Early access January 2022

Fund Project: Short acknowledgement.
This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under the OP RDE grant number CZ.02.1.01/0.0/0.0/16 019/0000753 "Research centre for low-carbon energy technologies"

This article introduces new geometric flow for space curves with positive curvature and torsion. Curves evolving according to this motion law trace out a zero mean curvature surface. First, the geometry of surfaces defined as trajectories of moving curves is analyzed and the minimal surface generating flow is derived. Then, an upper bound for the area of the generated minimal surface and for the terminating time of the flow is provided.

Citation: Jiří Minarčík, Michal Beneš. Minimal surface generating flow for space curves of non-vanishing torsion. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022011
References:
[1]

N. H. Abdel-AllR. A. Hussien and T. Youssef, Hasimoto Surfaces, Life Science Journal, 9 (2012), 556-560. 

[2]

O. Al-Ketan, R. K. A. Al-Rub and R. Rowshan, Mechanical properties of a new type of architected interpenetrating phase composite materials, Advanced Materials Technologies, 2 (2016).

[3]

D. J. AltschulerS. J. AltschulerS. B. Angenent and L. F. Wu, The zoo of solitons for curve shortening in $\mathbb{R}^n$, Nonlinearity, 26 (2013), 1189-1226.  doi: 10.1088/0951-7715/26/5/1189.

[4]

S. J. Altschuler, Singularities for the curve shortening flow for space curves, Journal of Differential Geometry, 34 (1991), 491-514. 

[5]

S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry, 35 (1992), 283-298. 

[6]

S. AnderssonS. T. Hyde and J. O. Bovin, On the periodic minimal surfaces and the conductivity mechanism of $\alpha$-AgI, Zeitschrift für Kristallographie, 173 (1985), 97-99. 

[7]

S. AnderssonS. T. HydeK. Larsson and S. Lidin, Minimal surfaces and structures: From inorganic and metal crystals to cell membranes and biopolymers, Chemical Reviews, 88 (1988), 221-242. 

[8]

J. Arroyo, O. Garay and A. Pampano, Binormal motion of curves with constant torsion in 3-spaces, Advances in Mathematical Physics, 2017 (2017), Art. ID 7075831, 8 pp. doi: 10.1155/2017/7075831.

[9]

L. M. Bates and O. M. Melko, On curves of constant torsion I, Journal of Geometry, 104 (2013), 213-227.  doi: 10.1007/s00022-013-0166-2.

[10]

M. BergouM. WardetzkyS. RobinsonB. Audoly and E. Grinspun, Discrete elastic rods, ACM SIGGRAPH, 63 (2008), 231-264. 

[11]

A. BobenkoT. Hoffmann and B. Springborn, Minimal surfaces from circle patterns: Geometry from combinatorics, Annals of Mathematics, 164 (2006), 231-264.  doi: 10.4007/annals.2006.164.231.

[12]

H. L. Bray and J. L. Jauregui, On curves with nonnegative torsion, Archiv der Mathematik, 104 (2015), 561-575.  doi: 10.1007/s00013-015-0767-0.

[13]

D. L. Chopp, Computing minimal surfaces via level set curvature flow, Journal of Computational Physics, 106 (1993), 77-91.  doi: 10.1006/jcph.1993.1092.

[14]

K. Corrales, Non existence of type II singularities for embedded and unknotted space curves, preprint, arXiv: 1605.03100v1, 2016.

[15]

G. Dziuk, An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1990), 603-611.  doi: 10.1007/BF01385643.

[16]

G. DziukE. Kuwert and R. Schätzle, Evolution of elastic curves in $\mathbb{R}^n$: existence and computation, SIAM Journal on Mathematical Analysis, 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709.

[17]

M. Erdoǧdu and M. Özdemir, Geometry of hasimoto surfaces in minkowski 3-space, Mathematical Physics, Analysis and Geometry, 17 (2014), 169-181.  doi: 10.1007/s11040-014-9148-3.

[18]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, 23 (1986), 69-96. 

[19]

R. E. GoldsteinJ. McTavishH. K. Moffatt and A. I. Pesci, Boundary singularities produced by the motion of soap films, Proceedings of the National Academy of Sciences of the United States of America, 111 (2014), 8339-8344.  doi: 10.1073/pnas.1406385111.

[20]

R. E. GoldsteinH. K. MoffattA. I. Pesci and R. L. Ricca, Soap-film Möbius strip changes topology with a twist singularity, Proceedings of the National Academy of Sciences of the United States of America, 107 (2010), 21979-21984.  doi: 10.1073/pnas.1015997107.

[21]

S. He, Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow, preprint, arXiv: 1209.5146v1, 2012.

[22]

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, Journal of Differential Geometry, 59 (2001), 353-437. 

[23]

R. A. Hussien and S. G. Mohamed, Generated surfaces via inextensible flows of curves in $\mathbb{R}^3$, Journal of Applied Mathematics, 2016 (2016), Art. ID 6178961, 8 pp. doi: 10.1155/2016/6178961.

[24]

T. A. Ivey, Integrable geometric evolution equations for curves, Contemporary Mathematics, 285 (2001), 71-84.  doi: 10.1090/conm/285/04734.

[25]

J. P. Keener, The dynamics of three-dimensional scroll waves in excitable media, Physica D, 31 (1988), 269-276.  doi: 10.1016/0167-2789(88)90080-2.

[26]

G. Khan, A condition ensuring spatial curves develop type-II singularities under curve shortening flow, preprint, arXiv: 1209.4072v3, 2015.

[27]

M. Larsson and K. Larsson, Periodic minimal surface organizations of the lipid bilayer at the lung surface and in cubic cytomembrane assemblies, Advances in Colloid and Interface Science, 205 (2014), 68-73. 

[28]

F. Maucher and P. Sutcliffe, Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings, Physical Review Letters, 116 (2016), 178101. 

[29]

J. Minarčík and M. Beneš, Long-term behavior of curve shortening flow in $\mathbb{R}^3$, SIAM Journal on Mathematical Analysis, 52 (2020), 1221-1231.  doi: 10.1137/19M1248522.

[30]

J. MinarčíkM. Kimura and M. Beneš, Comparing motion of curves and hypersurfaces in $\mathbb{R}^m$, Discrete and Continuous Dynamical Systems-Series B, 24 (2019), 4815-4826.  doi: 10.3934/dcdsb.2019032.

[31]

T. Mura, Micromechanics of Defects in Solids, Springer Netherlands, 1987.

[32]

P. J. Olver, Invariant submanifold flows, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 344017, 22pp. doi: 10.1088/1751-8113/41/34/344017.

[33]

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Experimental Mathematics, 2 (1993), 15-36.  doi: 10.1080/10586458.1993.10504266.

[34]

H. PottmannA. Schiftner and J. Wallner, Geometry of architectural freeform structures, Internationalen Mathematischen Nachrichten, 209 (2008), 15-28. 

[35]

R. L. Ricca, Rediscovery of da Rios equations, Nature, 352 (1991), 561-562. 

[36]

G. Richardson and J. R. King, The evolution of space curves by curvature and torsion, Journal of Physics A: Mathematical and General, 35 (2002), 9857-9879.  doi: 10.1088/0305-4470/35/46/310.

[37]

S. Rodrigues Costa and M. D. C. Romero-Fuster, Nowhere vanishing torsion closed curves always hide twice, Geometriae Dedicata, 66 (1997), 1-17.  doi: 10.1023/A:1004987511985.

[38]

W. K. Schief and C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, Proceedings of The Royal Society A-Mathematical Physical and Engineering Sciences, 455 (1999), 3163-3188.  doi: 10.1098/rspa.1999.0445.

[39]

H. Schumacher and M. Wardetzky, Variational convergence of discrete minimal surfaces, Numerische Mathematik, 141 (2019), 173-213.  doi: 10.1007/s00211-018-0993-z.

[40]

L. E. Scriven, Equilibrium bicontinuous structure, Nature, 263 (1976), 123-125. 

[41]

H. Terrones, Computation of minimal surfaces, Journal de Physique Colloques, 51 (1990), 345-362. 

[42]

S. Wang and A. Chern, Computing Minimal Surfaces with Differential Forms, ACM Transactions on Graphics, 40 (2021), 1-14. 

[43]

J. L. Weiner, Closed curves of constant torsion. II, Proceedings of the American Mathematical Society, 67 (1977), 306-308.  doi: 10.1090/S0002-9939-1977-0461385-0.

[44]

J. J. van Wijk and A. M. Cohen, Visualization of seifert surfaces, IEEE Trans. on Visualization and Computer Graphics, 12 (2006), 485-496. 

show all references

References:
[1]

N. H. Abdel-AllR. A. Hussien and T. Youssef, Hasimoto Surfaces, Life Science Journal, 9 (2012), 556-560. 

[2]

O. Al-Ketan, R. K. A. Al-Rub and R. Rowshan, Mechanical properties of a new type of architected interpenetrating phase composite materials, Advanced Materials Technologies, 2 (2016).

[3]

D. J. AltschulerS. J. AltschulerS. B. Angenent and L. F. Wu, The zoo of solitons for curve shortening in $\mathbb{R}^n$, Nonlinearity, 26 (2013), 1189-1226.  doi: 10.1088/0951-7715/26/5/1189.

[4]

S. J. Altschuler, Singularities for the curve shortening flow for space curves, Journal of Differential Geometry, 34 (1991), 491-514. 

[5]

S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry, 35 (1992), 283-298. 

[6]

S. AnderssonS. T. Hyde and J. O. Bovin, On the periodic minimal surfaces and the conductivity mechanism of $\alpha$-AgI, Zeitschrift für Kristallographie, 173 (1985), 97-99. 

[7]

S. AnderssonS. T. HydeK. Larsson and S. Lidin, Minimal surfaces and structures: From inorganic and metal crystals to cell membranes and biopolymers, Chemical Reviews, 88 (1988), 221-242. 

[8]

J. Arroyo, O. Garay and A. Pampano, Binormal motion of curves with constant torsion in 3-spaces, Advances in Mathematical Physics, 2017 (2017), Art. ID 7075831, 8 pp. doi: 10.1155/2017/7075831.

[9]

L. M. Bates and O. M. Melko, On curves of constant torsion I, Journal of Geometry, 104 (2013), 213-227.  doi: 10.1007/s00022-013-0166-2.

[10]

M. BergouM. WardetzkyS. RobinsonB. Audoly and E. Grinspun, Discrete elastic rods, ACM SIGGRAPH, 63 (2008), 231-264. 

[11]

A. BobenkoT. Hoffmann and B. Springborn, Minimal surfaces from circle patterns: Geometry from combinatorics, Annals of Mathematics, 164 (2006), 231-264.  doi: 10.4007/annals.2006.164.231.

[12]

H. L. Bray and J. L. Jauregui, On curves with nonnegative torsion, Archiv der Mathematik, 104 (2015), 561-575.  doi: 10.1007/s00013-015-0767-0.

[13]

D. L. Chopp, Computing minimal surfaces via level set curvature flow, Journal of Computational Physics, 106 (1993), 77-91.  doi: 10.1006/jcph.1993.1092.

[14]

K. Corrales, Non existence of type II singularities for embedded and unknotted space curves, preprint, arXiv: 1605.03100v1, 2016.

[15]

G. Dziuk, An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1990), 603-611.  doi: 10.1007/BF01385643.

[16]

G. DziukE. Kuwert and R. Schätzle, Evolution of elastic curves in $\mathbb{R}^n$: existence and computation, SIAM Journal on Mathematical Analysis, 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709.

[17]

M. Erdoǧdu and M. Özdemir, Geometry of hasimoto surfaces in minkowski 3-space, Mathematical Physics, Analysis and Geometry, 17 (2014), 169-181.  doi: 10.1007/s11040-014-9148-3.

[18]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, 23 (1986), 69-96. 

[19]

R. E. GoldsteinJ. McTavishH. K. Moffatt and A. I. Pesci, Boundary singularities produced by the motion of soap films, Proceedings of the National Academy of Sciences of the United States of America, 111 (2014), 8339-8344.  doi: 10.1073/pnas.1406385111.

[20]

R. E. GoldsteinH. K. MoffattA. I. Pesci and R. L. Ricca, Soap-film Möbius strip changes topology with a twist singularity, Proceedings of the National Academy of Sciences of the United States of America, 107 (2010), 21979-21984.  doi: 10.1073/pnas.1015997107.

[21]

S. He, Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow, preprint, arXiv: 1209.5146v1, 2012.

[22]

G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, Journal of Differential Geometry, 59 (2001), 353-437. 

[23]

R. A. Hussien and S. G. Mohamed, Generated surfaces via inextensible flows of curves in $\mathbb{R}^3$, Journal of Applied Mathematics, 2016 (2016), Art. ID 6178961, 8 pp. doi: 10.1155/2016/6178961.

[24]

T. A. Ivey, Integrable geometric evolution equations for curves, Contemporary Mathematics, 285 (2001), 71-84.  doi: 10.1090/conm/285/04734.

[25]

J. P. Keener, The dynamics of three-dimensional scroll waves in excitable media, Physica D, 31 (1988), 269-276.  doi: 10.1016/0167-2789(88)90080-2.

[26]

G. Khan, A condition ensuring spatial curves develop type-II singularities under curve shortening flow, preprint, arXiv: 1209.4072v3, 2015.

[27]

M. Larsson and K. Larsson, Periodic minimal surface organizations of the lipid bilayer at the lung surface and in cubic cytomembrane assemblies, Advances in Colloid and Interface Science, 205 (2014), 68-73. 

[28]

F. Maucher and P. Sutcliffe, Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings, Physical Review Letters, 116 (2016), 178101. 

[29]

J. Minarčík and M. Beneš, Long-term behavior of curve shortening flow in $\mathbb{R}^3$, SIAM Journal on Mathematical Analysis, 52 (2020), 1221-1231.  doi: 10.1137/19M1248522.

[30]

J. MinarčíkM. Kimura and M. Beneš, Comparing motion of curves and hypersurfaces in $\mathbb{R}^m$, Discrete and Continuous Dynamical Systems-Series B, 24 (2019), 4815-4826.  doi: 10.3934/dcdsb.2019032.

[31]

T. Mura, Micromechanics of Defects in Solids, Springer Netherlands, 1987.

[32]

P. J. Olver, Invariant submanifold flows, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 344017, 22pp. doi: 10.1088/1751-8113/41/34/344017.

[33]

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Experimental Mathematics, 2 (1993), 15-36.  doi: 10.1080/10586458.1993.10504266.

[34]

H. PottmannA. Schiftner and J. Wallner, Geometry of architectural freeform structures, Internationalen Mathematischen Nachrichten, 209 (2008), 15-28. 

[35]

R. L. Ricca, Rediscovery of da Rios equations, Nature, 352 (1991), 561-562. 

[36]

G. Richardson and J. R. King, The evolution of space curves by curvature and torsion, Journal of Physics A: Mathematical and General, 35 (2002), 9857-9879.  doi: 10.1088/0305-4470/35/46/310.

[37]

S. Rodrigues Costa and M. D. C. Romero-Fuster, Nowhere vanishing torsion closed curves always hide twice, Geometriae Dedicata, 66 (1997), 1-17.  doi: 10.1023/A:1004987511985.

[38]

W. K. Schief and C. Rogers, Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, Proceedings of The Royal Society A-Mathematical Physical and Engineering Sciences, 455 (1999), 3163-3188.  doi: 10.1098/rspa.1999.0445.

[39]

H. Schumacher and M. Wardetzky, Variational convergence of discrete minimal surfaces, Numerische Mathematik, 141 (2019), 173-213.  doi: 10.1007/s00211-018-0993-z.

[40]

L. E. Scriven, Equilibrium bicontinuous structure, Nature, 263 (1976), 123-125. 

[41]

H. Terrones, Computation of minimal surfaces, Journal de Physique Colloques, 51 (1990), 345-362. 

[42]

S. Wang and A. Chern, Computing Minimal Surfaces with Differential Forms, ACM Transactions on Graphics, 40 (2021), 1-14. 

[43]

J. L. Weiner, Closed curves of constant torsion. II, Proceedings of the American Mathematical Society, 67 (1977), 306-308.  doi: 10.1090/S0002-9939-1977-0461385-0.

[44]

J. J. van Wijk and A. M. Cohen, Visualization of seifert surfaces, IEEE Trans. on Visualization and Computer Graphics, 12 (2006), 485-496. 

Figure 1.  Numerical solution to (11-13) with the initial condition given by (25). The curvature $ \kappa $ vanishes at $ t_{max} = 0.20628 $. The curves $ \Gamma_0 $ and $ \Gamma_{max} $ were reconstructed from $ \kappa $, $ \tau $ and $ g $. The position and orientation was partially recovered using Principal component analysis. However, the rotation around the $ z $-axis was not preserved
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