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.
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Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory
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@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/
UR - https://doi.org/10.5802/aif.3222
DO - 10.5802/aif.3222
LA - en
ID - AIF_2018__68_6_2743_0
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%J Annales de l'Institut Fourier
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%P 2743-2778
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%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3222
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%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/
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