Article Contents
Article Contents

# Plateau-rayleigh instability of singular minimal surfaces

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.

Mathematics Subject Classification: Primary: 53C42, 53A10; Secondary: 58E30, 35J60, 35P30, 65L15.

 Citation:

• 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

Figure 2.  Solutions of (2.3) for values $\alpha = -2$ (right), $\alpha = 0.8$ (middle) and $\alpha = 2$ (right)

Figure 3.  Case $\alpha = 1$. Left: the function $I_1(a)$. Right: the function $L\mapsto I_2(a,L)$ (here $a = 1$)

Figure 4.  The function $L_0 = L_0(a)$ given in (4.6)

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

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

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

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
•  [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".

Figures(4)

Tables(4)

• on this site

/