On bifurcation and local rigidity of triply periodic minimal surfaces in 3
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, p. 2743-2778
We study the space of triply periodic minimal surfaces in 3 , giving a result on the local rigidity and a result on the existence of bifurcation. We prove that, near a triply periodic minimal surface with nullity three, the space of triply periodic minimal surfaces consists of a smooth five-parameter family of pairwise non-homothetic surfaces. On the other hand, if there is a smooth one-parameter family of triply periodic minimal surfaces {X t } t containing X 0 where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from X 0 . We also apply these results to several known examples.
Nous étudions l’espace des surfaces minimales triplement périodiques dans 3 , obtenant un résultat sur la rigidité locale ainsi que sur l’existence de bifurcation. Nous démontrons que, près d’une surface minimale triplement périodique de nullité 3, l’espace des surfaces minimales triplement périodiques est une famille lisse à cinq paramètres de surfaces deux à deux non homothétiques. D’autre part, s’il y a une famille lisse à un paramètre de surfaces minimales triplement périodiques {X t } t contenant X 0 , dont l’indice de Morse saute d’un entier impair, ceci démontrera l’existence d’une branche bifurquant depuis X 0 . Nous appliquons aussi ces résultats à plusieurs exemples connus.
Received : 2015-07-22
Revised : 2016-03-16
Accepted : 2018-01-31
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3222
Classification:  53A10,  58J55,  58E12,  35J62
Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory
@article{AIF_2018__68_6_2743_0,
     author = {Koiso, Miyuki and Piccione, Paolo and Shoda, Toshihiro},
     title = {On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     pages = {2743-2778},
     doi = {10.5802/aif.3222},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_6_2743_0}
}
On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2743-2778. doi : 10.5802/aif.3222. https://aif.centre-mersenne.org/item/AIF_2018__68_6_2743_0/

[1] Alías, Luis J.; Piccione, Paolo On the manifold structure of the set of unparameterized embeddings with low regularity, Bull. Braz. Math. Soc. (N.S.), Tome 42 (2011) no. 2, pp. 171-183 | Article | MR 2833797 | Zbl 1245.58007

[2] Andersson, Sten; Hyde, Stephen T.; Larsson, Kåre; Lidin, Sven Minimal surfaces and structures: from inorganic and metal crystals to cell membranes and biopolymers, Chem. Rev., Tome 88 (1988) no. 1, pp. 221-242 | Article

[3] Bettiol, Renato G.; Piccione, Paolo; Santoro, Bianca Bifurcation of periodic solutions to the singular Yamabe problem on spheres, J. Differ. Geom., Tome 103 (2016) no. 2, pp. 191-205 http://projecteuclid.org/euclid.jdg/1463404117 | MR 3504948 | Zbl 1348.53044

[4] Bettiol, Renato G.; Piccione, Paolo; Siciliano, Gaetano Deforming solutions of geometric variational problems with varying symmetry groups, Transform. Groups, Tome 19 (2014) no. 4, pp. 941-968 | Article | MR 3278856 | Zbl 1320.58004

[5] Ejiri, Norio A generating function of a complex Lagrangian cone in H n (2013) (preprint)

[6] Ejiri, Norio; Shoda, Toshihiro On a moduli theory of minimal surfaces, Prospects of differential geometry and its related fields, World Scientific (2014), pp. 155-172 | MR 3203974 | Zbl 1297.53041

[7] Ejiri, Norio; Shoda, Toshihiro The Morse index of a triply periodic minimal surface, Differ. Geom. Appl., Tome 58 (2018), pp. 177-201 | Zbl 1388.53012

[8] Fischer, Werner; Koch, Elke On 3-periodic minimal surfaces, Z. Kristallogr., Tome 179 (1987) no. 1-4, pp. 31-52 | Article | MR 978627 | Zbl 0684.53010

[9] Fogden, Andrew S.; Hyde, Stephen T. Continuous transformations of cubic minimal surfaces, Eur. Phys. J. B, Tome 7 (1999) no. 1, pp. 91-104 | Article

[10] Fogden, Andrew S.; Schröder-Turk, G. E.; Hyde, Stephen T. Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces, Eur. Phys. J. B, Tome 54 (2006) no. 4, pp. 509-524 | Article

[11] Kapouleas, Nicolaos Constant mean curvature surfaces in Euclidean three-space, Bull. Am. Math. Soc., Tome 17 (1987) no. 2, pp. 318-320 | Article | MR 903742 | Zbl 0636.53010

[12] Kapouleas, Nicolaos Complete constant mean curvature surfaces in Euclidean three-space, Ann. Math., Tome 131 (1990) no. 2, pp. 239-330 | Article | MR 1043269 | Zbl 0699.53007

[13] Karcher, Hermann The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscr. Math., Tome 64 (1989) no. 3, pp. 291-357 | Article | MR 1003093 | Zbl 0687.53010

[14] Kato, Tosio Perturbation theory for linear operators, Springer, Classics in Mathematics (1995), xxii+619 pages (Reprint of the 1980 edition) | MR 1335452 | Zbl 0836.47009

[15] Kielhöfer, Hansjörg Bifurcation theory. An introduction with applications to partial differential equations, Springer, Applied Mathematical Sciences, Tome 156 (2012), viii+398 pages | Article | MR 2859263 | Zbl 1230.35002

[16] Koiso, Miyuki; Palmer, Bennett; Piccione, Paolo Bifurcation and symmetry breaking of nodoids with fixed boundary, Adv. Calc. Var., Tome 8 (2015) no. 4, pp. 337-370 | Article | MR 3403431 | Zbl 1381.58005

[17] Korevaar, Nicholas J.; Kusner, Rob; Solomon, Bruce The structure of complete embedded surfaces with constant mean curvature, J. Differ. Geom., Tome 30 (1989) no. 2, pp. 465-503 http://projecteuclid.org/euclid.jdg/1214443598 | MR 1010168 | Zbl 0726.53007

[18] Mazzeo, Rafe; Pacard, Frank Constant mean curvature surfaces with Delaunay ends, Commun. Anal. Geom., Tome 9 (2001) no. 1, pp. 169-237 | Article | MR 1807955 | Zbl 1005.53006

[19] Mazzeo, Rafe; Pacard, Frank; Pollack, Daniel Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math., Tome 536 (2001), pp. 115-165 | Article | MR 1837428 | Zbl 0972.53010

[20] Meeks, William H. Iii The theory of triply periodic minimal surfaces, Indiana Univ. Math. J., Tome 39 (1990) no. 3, pp. 877-936 | Article | MR 1078743 | Zbl 0721.53057

[21] Montiel, Sebastián; Ros, Antonio Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin, 1990), Springer (Lecture Notes in Mathematics) Tome 1481 (1991), pp. 147-174 | Article | MR 1178529 | Zbl 0744.58007

[22] Nagano, Tadashi; Smyth, Brian Minimal varieties and harmonic maps in tori, Comment. Math. Helv., Tome 50 (1975), pp. 249-265 | Article | MR 0390974 | Zbl 0326.53055

[23] Pérez, Joaquí N; Ros, Antonio The space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J., Tome 45 (1996) no. 1, pp. 177-204 | Article | MR 1406689 | Zbl 0864.53008

[24] Plateau, J. Experimental and theoretical statics of liquids subject to molecular forces only (facstaff.susqu.edu/brakke/aux/downloads/plateau-eng.pdf, translated by Kenneth A. Brakke)

[25] Ros, Antonio One-sided complete stable minimal surfaces, J. Differ. Geom., Tome 74 (2006) no. 1, pp. 69-92 http://projecteuclid.org/euclid.jdg/1175266182 | MR 2260928 | Zbl 1110.53009

[26] Ross, Marty Schwarz’ P and D surfaces are stable, Differ. Geom. Appl., Tome 2 (1992) no. 2, pp. 179-195 | Article | MR 1245555 | Zbl 0747.53010

[27] Rump, Siegfried M. Verification methods: rigorous results using floating-point arithmetic, Acta Numer., Tome 19 (2010), pp. 287-449 | Article | MR 2652784 | Zbl 1323.65046

[28] Von Schnering, Hans Georg; Nesper, R. Nodal surfaces of Fourier series: Fundamental invariants of structured matter, Zeitschrift für Physik B Condensed Matter, Tome 83 (1991), pp. 407-412 | Article

[29] Schoen, Alan Hugh Infinite periodic minimal surfaces without self-intersections, National Aeronautics and Space Administration, NASA Technical Note, Tome D-5541 (1970), vii+92 pages | Zbl 1071.53507

[30] Traizet, Martin On the genus of triply periodic minimal surfaces, J. Differ. Geom., Tome 79 (2008) no. 2, pp. 243-275 http://projecteuclid.org/euclid.jdg/1211512641 | MR 2420019 | Zbl 1167.53013

[31] White, Brian The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J., Tome 36 (1987) no. 3, pp. 567-602 | Article | MR 905611 | Zbl 0770.58005