We study the space of triply periodic minimal surfaces in , 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 containing where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from . We also apply these results to several known examples.
Nous étudions l’espace des surfaces minimales triplement périodiques dans , 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 , 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 contenant , dont l’indice de Morse saute d’un entier impair, ceci démontrera l’existence d’une branche bifurquant depuis . Nous appliquons aussi ces résultats à plusieurs exemples connus.
Revised:
Accepted:
Published online:
Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory
Mot clés : Surfaces minimales triplement périodiques, famille H, famille rPD, famille tP, famille tD, théorie de bifurcation.
CC-BY-ND 4.0
@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},
pages = {2743--2778},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {68},
number = {6},
year = {2018},
doi = {10.5802/aif.3222},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3222/}
}
TY - JOUR
AU - Koiso, Miyuki
AU - Piccione, Paolo
AU - Shoda, Toshihiro
TI - On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$
JO - Annales de l'Institut Fourier
PY - 2018
SP - 2743
EP - 2778
VL - 68
IS - 6
PB - Association des Annales de l’institut Fourier
UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3222/
DO - 10.5802/aif.3222
LA - en
ID - AIF_2018__68_6_2743_0
ER -
%0 Journal Article
%A Koiso, Miyuki
%A Piccione, Paolo
%A Shoda, Toshihiro
%T On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$
%J Annales de l'Institut Fourier
%D 2018
%P 2743-2778
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3222/
%R 10.5802/aif.3222
%G en
%F AIF_2018__68_6_2743_0
Koiso, Miyuki; Piccione, Paolo; Shoda, Toshihiro. 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/articles/10.5802/aif.3222/
[1] On the manifold structure of the set of unparameterized embeddings with low regularity, Bull. Braz. Math. Soc. (N.S.), Volume 42 (2011) no. 2, pp. 171-183 | DOI | MR | Zbl
[2] Minimal surfaces and structures: from inorganic and metal crystals to cell membranes and biopolymers, Chem. Rev., Volume 88 (1988) no. 1, pp. 221-242 | DOI
[3] Bifurcation of periodic solutions to the singular Yamabe problem on spheres, J. Differ. Geom., Volume 103 (2016) no. 2, pp. 191-205 http://projecteuclid.org/euclid.jdg/1463404117 | MR | Zbl
[4] Deforming solutions of geometric variational problems with varying symmetry groups, Transform. Groups, Volume 19 (2014) no. 4, pp. 941-968 | DOI | MR | Zbl
[5] A generating function of a complex Lagrangian cone in (2013) (preprint)
[6] On a moduli theory of minimal surfaces, Prospects of differential geometry and its related fields, World Scientific, 2014, pp. 155-172 | MR | Zbl
[7] The Morse index of a triply periodic minimal surface, Differ. Geom. Appl., Volume 58 (2018), pp. 177-201 | Zbl
[8] On -periodic minimal surfaces, Z. Kristallogr., Volume 179 (1987) no. 1-4, pp. 31-52 | DOI | MR | Zbl
[9] Continuous transformations of cubic minimal surfaces, Eur. Phys. J. B, Volume 7 (1999) no. 1, pp. 91-104 | DOI
[10] 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, Volume 54 (2006) no. 4, pp. 509-524 | DOI
[11] Constant mean curvature surfaces in Euclidean three-space, Bull. Am. Math. Soc., Volume 17 (1987) no. 2, pp. 318-320 | DOI | MR | Zbl
[12] Complete constant mean curvature surfaces in Euclidean three-space, Ann. Math., Volume 131 (1990) no. 2, pp. 239-330 | DOI | MR | Zbl
[13] The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscr. Math., Volume 64 (1989) no. 3, pp. 291-357 | DOI | MR | Zbl
[14] Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995, xxii+619 pages (Reprint of the 1980 edition) | MR | Zbl
[15] Bifurcation theory. An introduction with applications to partial differential equations, Applied Mathematical Sciences, 156, Springer, 2012, viii+398 pages | DOI | MR | Zbl
[16] Bifurcation and symmetry breaking of nodoids with fixed boundary, Adv. Calc. Var., Volume 8 (2015) no. 4, pp. 337-370 | DOI | MR | Zbl
[17] The structure of complete embedded surfaces with constant mean curvature, J. Differ. Geom., Volume 30 (1989) no. 2, pp. 465-503 http://projecteuclid.org/euclid.jdg/1214443598 | MR | Zbl
[18] Constant mean curvature surfaces with Delaunay ends, Commun. Anal. Geom., Volume 9 (2001) no. 1, pp. 169-237 | DOI | MR | Zbl
[19] Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math., Volume 536 (2001), pp. 115-165 | DOI | MR | Zbl
[20] The theory of triply periodic minimal surfaces, Indiana Univ. Math. J., Volume 39 (1990) no. 3, pp. 877-936 | DOI | MR | Zbl
[21] Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin, 1990) (Lecture Notes in Mathematics), Volume 1481, Springer, 1991, pp. 147-174 | DOI | MR | Zbl
[22] Minimal varieties and harmonic maps in tori, Comment. Math. Helv., Volume 50 (1975), pp. 249-265 | DOI | MR | Zbl
[23] The space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J., Volume 45 (1996) no. 1, pp. 177-204 | DOI | MR | Zbl
[24] 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] One-sided complete stable minimal surfaces, J. Differ. Geom., Volume 74 (2006) no. 1, pp. 69-92 http://projecteuclid.org/euclid.jdg/1175266182 | MR | Zbl
[26] Schwarz’ and surfaces are stable, Differ. Geom. Appl., Volume 2 (1992) no. 2, pp. 179-195 | DOI | MR | Zbl
[27] Verification methods: rigorous results using floating-point arithmetic, Acta Numer., Volume 19 (2010), pp. 287-449 | DOI | MR | Zbl
[28] Nodal surfaces of Fourier series: Fundamental invariants of structured matter, Zeitschrift für Physik B Condensed Matter, Volume 83 (1991), pp. 407-412 | DOI
[29] Infinite periodic minimal surfaces without self-intersections, NASA Technical Note, D-5541, National Aeronautics and Space Administration, 1970, vii+92 pages | Zbl
[30] On the genus of triply periodic minimal surfaces, J. Differ. Geom., Volume 79 (2008) no. 2, pp. 243-275 http://projecteuclid.org/euclid.jdg/1211512641 | MR | Zbl
[31] The space of -dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J., Volume 36 (1987) no. 3, pp. 567-602 | DOI | MR | Zbl
Cited by Sources:
