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:
Classification: 53A10, 58J55, 58E12, 35J62
Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory
Author's affiliations:
@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 TI - On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$ JO - Annales de l'Institut Fourier PY - 2018 DA - 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 ER -
%0 Journal Article %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://doi.org/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/
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