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doi: 10.3934/cpaa.2022086
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Plateau-rayleigh instability of singular minimal surfaces

Rafael López

Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

Received  August 2021 Revised  April 2022 Early access May 2022

Fund Project: This work has been partially supported by the Projects I+D+i PID2020-117868GB-I00, A-FQM-139-UGR18 and P18-FR-4049

We prove a Plateau-Rayleigh criterion of instability for singular minimal surfaces, providing explicit bounds on the amplitude and length of the surface. More generally, we study the stability of $ \alpha $-singular minimal hypersurfaces considered as hypersurfaces in weighted manifolds. If $ \alpha<0 $ and the hypersurface is a graph, then we prove that the hypersurface is stable. If $ \alpha>0 $ and the surface is cylindrical, we give numerical evidences of the instability of long cylindrical $ \alpha $-singular minimal surfaces.

Keywords: Singular minimal surface, stability, cylindrical surface, Plateau-Rayleigh criterion.
Mathematics Subject Classification: Primary: 53C42, 53A10; Secondary: 58E30, 35J60, 35P30, 65L15.
Citation: Rafael López. Plateau-rayleigh instability of singular minimal surfaces. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022086
References:
[1]

V. Bayle, Propriétés de Concavité du Profil Isopérimétrique et Applications, Ph.D. Thesis, Institut Joseph Fourier, Grenoble, 2004.

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-74311-8.

[3]

R. Böhme, S. Hildebrandt and E. Taush, The two-dimensional analogue of the catenary, Pacific J. Math., 88 (1980), 247-278. 

[4]

K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys., 79 (2014), 14-28.  doi: 10.1016/j.geomphys.2014.01.013.

[5]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, Ann. Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.

[6]

L. Colter, Cylindrical liquid bridges, Involve, 8 (2015), 695-705.  doi: 10.2140/involve.2015.8.695.

[7]

U. Dierkes, A Bernstein result for energy minimizing hypersurfaces, Calc. Var. Partial Differ. Equ., 1 (1993), 37-54.  doi: 10.1007/BF02163263.

[8]

U. Dierkes, Singular minimal surfaces, in Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003.

[9]

U. Dierkes and G. Huisken, The $n$-dimensional analogue of the catenary: existence and nonexistence, Pacific J. Math., 141 (1990), 47-54. 

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998.

[11]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Commun. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.

[12]

M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215.  doi: 10.1007/s000390300004.

[13]

R. López, A criterion on instability of cylindrical rotating surfaces, Archiv Math., 94 (2010), 91-99.  doi: 10.1007/s00013-009-0085-5.

[14]

R. López, Bifurcation of cylinders for wetting and dewetting models with striped geometry, SIAM J. Math. Anal., 44 (2012), 946-965.  doi: 10.1137/11082484X.

[15]

R. López, Invariant singular minimal surfaces, Ann. Global Anal. Geom., 53 (2018), 521-541.  doi: 10.1007/s10455-017-9586-9.

[16]

J. McCuan, Extremities of stability for pendant drops, in Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/599/11944.

[17]

J. McCuan, The stability of cylindrical pendant drops, Mem. Amer. Math. Soc., 250 (2017), no. 1189. doi: 10.1090/memo/1189.

[18]

F. Otto, Zugbeanspruchte Konstruktionen, Berlin, Frankfurt, Wien: Ullstein, 1962.

[19]

B. Palmer and O. Perdomo, Equilibrium shapes of cylindrical rotating liquid drops, Bull. Braz. Math. Soc., 46 (2015), 515-561. 

[20]

J. A. F. Plateau, Statique Expérimentale et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires, vol. 2. Gauthier-Villars, 2018.

[21]

J. W. S. Rayleigh, On the instability of jets, Proc. London Math. Soc., 10 (1879), 4-13.  doi: 10.1112/plms/s1-10.1.4.

[22]

R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, in Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983.

[23]

L. Shahriyari, Translating graphs by mean curvature flow, Geom Dedicata, 175 (2015), 57-64.  doi: 10.1007/s10711-014-0028-6.

[24]

J. Sun, Lagrangian L-stability of Lagrangian translating solitons, Manuscripta Math., 161 (2020), 247-255.  doi: 10.1007/s00229-018-1089-x.

[25]

Wikipedia, Colegio Teresiano de Barcelona, "https://es.wikipedia.org/w/index.php?title=Colegio_Teresiano_de_Barcelona&oldid=134544852".

show all references

References:
[1]

V. Bayle, Propriétés de Concavité du Profil Isopérimétrique et Applications, Ph.D. Thesis, Institut Joseph Fourier, Grenoble, 2004.

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-74311-8.

[3]

R. Böhme, S. Hildebrandt and E. Taush, The two-dimensional analogue of the catenary, Pacific J. Math., 88 (1980), 247-278. 

[4]

K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys., 79 (2014), 14-28.  doi: 10.1016/j.geomphys.2014.01.013.

[5]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, Ann. Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.

[6]

L. Colter, Cylindrical liquid bridges, Involve, 8 (2015), 695-705.  doi: 10.2140/involve.2015.8.695.

[7]

U. Dierkes, A Bernstein result for energy minimizing hypersurfaces, Calc. Var. Partial Differ. Equ., 1 (1993), 37-54.  doi: 10.1007/BF02163263.

[8]

U. Dierkes, Singular minimal surfaces, in Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003.

[9]

U. Dierkes and G. Huisken, The $n$-dimensional analogue of the catenary: existence and nonexistence, Pacific J. Math., 141 (1990), 47-54. 

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998.

[11]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Commun. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.

[12]

M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215.  doi: 10.1007/s000390300004.

[13]

R. López, A criterion on instability of cylindrical rotating surfaces, Archiv Math., 94 (2010), 91-99.  doi: 10.1007/s00013-009-0085-5.

[14]

R. López, Bifurcation of cylinders for wetting and dewetting models with striped geometry, SIAM J. Math. Anal., 44 (2012), 946-965.  doi: 10.1137/11082484X.

[15]

R. López, Invariant singular minimal surfaces, Ann. Global Anal. Geom., 53 (2018), 521-541.  doi: 10.1007/s10455-017-9586-9.

[16]

J. McCuan, Extremities of stability for pendant drops, in Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/599/11944.

[17]

J. McCuan, The stability of cylindrical pendant drops, Mem. Amer. Math. Soc., 250 (2017), no. 1189. doi: 10.1090/memo/1189.

[18]

F. Otto, Zugbeanspruchte Konstruktionen, Berlin, Frankfurt, Wien: Ullstein, 1962.

[19]

B. Palmer and O. Perdomo, Equilibrium shapes of cylindrical rotating liquid drops, Bull. Braz. Math. Soc., 46 (2015), 515-561. 

[20]

J. A. F. Plateau, Statique Expérimentale et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires, vol. 2. Gauthier-Villars, 2018.

[21]

J. W. S. Rayleigh, On the instability of jets, Proc. London Math. Soc., 10 (1879), 4-13.  doi: 10.1112/plms/s1-10.1.4.

[22]

R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, in Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983.

[23]

L. Shahriyari, Translating graphs by mean curvature flow, Geom Dedicata, 175 (2015), 57-64.  doi: 10.1007/s10711-014-0028-6.

[24]

J. Sun, Lagrangian L-stability of Lagrangian translating solitons, Manuscripta Math., 161 (2020), 247-255.  doi: 10.1007/s00229-018-1089-x.

[25]

Wikipedia, Colegio Teresiano de Barcelona, "https://es.wikipedia.org/w/index.php?title=Colegio_Teresiano_de_Barcelona&oldid=134544852".

Figure 1.  Left: corridor in the Colegio de las Teresianas, Barcelona ([25]). Right: the singular minimal surface $ \{(s,t,\cosh(s)):s,t\in{\mathbb R}\} $ constructed by repeating a catenary (blue) in a horizontal direction
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Figure 2.  Solutions of (2.3) for values $ \alpha = -2 $ (right), $ \alpha = 0.8 $ (middle) and $ \alpha = 2 $ (right)
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Figure 3.  Case $ \alpha = 1 $. Left: the function $ I_1(a) $. Right: the function $ L\mapsto I_2(a,L) $ (here $ a = 1 $)
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Figure 4.  The function $ L_0 = L_0(a) $ given in (4.6)
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Table 1.  Values of $ I(a,L) $ for $ \alpha = 2 $
$ L $ $ 1 $ $ 10 $ $ 20 $ $ 30 $ $ 40 $ $ 50 $
$ a $
0.2 -0.02389 -0.02031 -0.02028 -0.02028 -0.02028 -0.02028
0.3 -0.09371 -0.06398 -0.06375 -0.06371 -0.06370 -0.06369
0.4 -0.27824 -0.13500 -0.13392 -0.13372 -0.13365 -0.13362
0.5 -0.74222 -0.21609 -0.21210 -0.21137 -0.21111 -0.21099
0.6 -1.93468 -0.253826 -0.24109 -0.23873 -0.23790 -0.237527
0.7 -5.20226 -0.12268 -0.08420 -0.07708 -0.07458 -0.07343
0.8 -15.15280 $\fbox{0.45027} $ 0.5684 0.5903 0.59802 0.60157
0.9 -50.9705 $ \fbox{2.0161}$ 2.41751 2.49185 2.51786 2.52991
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$ L $ $ 1 $ $ 10 $ $ 20 $ $ 30 $ $ 40 $ $ 50 $
$ a $
0.2 -0.02389 -0.02031 -0.02028 -0.02028 -0.02028 -0.02028
0.3 -0.09371 -0.06398 -0.06375 -0.06371 -0.06370 -0.06369
0.4 -0.27824 -0.13500 -0.13392 -0.13372 -0.13365 -0.13362
0.5 -0.74222 -0.21609 -0.21210 -0.21137 -0.21111 -0.21099
0.6 -1.93468 -0.253826 -0.24109 -0.23873 -0.23790 -0.237527
0.7 -5.20226 -0.12268 -0.08420 -0.07708 -0.07458 -0.07343
0.8 -15.15280 $\fbox{0.45027} $ 0.5684 0.5903 0.59802 0.60157
0.9 -50.9705 $ \fbox{2.0161}$ 2.41751 2.49185 2.51786 2.52991
Table 2.  Case $ \alpha = 2 $. Values of $ I(0.72,L) $
$ L $ $ 22 $ $ 24 $ $ 26 $ $ 28 $ $ 30 $ $ 32 $
-0.0053 -0.0032 -0.0015 -0.0002 $ \fbox{0.0007}$ 0.0016
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$ L $ $ 22 $ $ 24 $ $ 26 $ $ 28 $ $ 30 $ $ 32 $
-0.0053 -0.0032 -0.0015 -0.0002 $ \fbox{0.0007}$ 0.0016
Table 3.  Case $ \alpha = 2 $. Values of $ I(a,L) $ when $ a>0.72 $
$ L $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ 7 8 9
$ a $
0.75 -2.061 -0.820 -0.386 -0.185 -0.076 -0.010 $\fbox{0.032} $ 0.061
0.80 -3.332 -1.143 -0.377 -0.022 $\fbox{0.170} $ 0.286 0.361 0.413
0.90 -10.829 -3.395 -0.793 $ \fbox{0.410}$ 1.064 1.459 1.715 1.890
1.00 -49.743 -17.942 -6.812 -1.660 $\fbox{1.137} $ 2.824 3.920 4.670
1.05 -126.723 -49.655 -22.681 -10.196 -3.414 $ \fbox{0.674}$ 3.328 5.148
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$ L $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ 7 8 9
$ a $
0.75 -2.061 -0.820 -0.386 -0.185 -0.076 -0.010 $\fbox{0.032} $ 0.061
0.80 -3.332 -1.143 -0.377 -0.022 $\fbox{0.170} $ 0.286 0.361 0.413
0.90 -10.829 -3.395 -0.793 $ \fbox{0.410}$ 1.064 1.459 1.715 1.890
1.00 -49.743 -17.942 -6.812 -1.660 $\fbox{1.137} $ 2.824 3.920 4.670
1.05 -126.723 -49.655 -22.681 -10.196 -3.414 $ \fbox{0.674}$ 3.328 5.148
Table 4.  Case $ \alpha = 3 $. Values of $ I(a,L) $ when $ a>0.53 $
$ L $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ 10 11 12
$ a $
0.54 -0.394 -0.216 -0.119 -0.060 -0.022 $ \fbox{0.003}$ 0.021 0.035 0.046
0.56 -0.531 -0.247 -0.093 $\fbox{0.000} $ 0.059 0.100 0.130 0.152 0.168
0.58 -0.906 -0.433 -0.176 -0.021 $\fbox{0.079} $ 0.148 0.197 0.234 0.262
0.60 -1.928 -1.079 -0.618 -0.340 -0.160 -0.036 $ \fbox{0.051}$ 0.117 0.167
Table Options

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$ L $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ 10 11 12
$ a $
0.54 -0.394 -0.216 -0.119 -0.060 -0.022 $ \fbox{0.003}$ 0.021 0.035 0.046
0.56 -0.531 -0.247 -0.093 $\fbox{0.000} $ 0.059 0.100 0.130 0.152 0.168
0.58 -0.906 -0.433 -0.176 -0.021 $\fbox{0.079} $ 0.148 0.197 0.234 0.262
0.60 -1.928 -1.079 -0.618 -0.340 -0.160 -0.036 $ \fbox{0.051}$ 0.117 0.167
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