On bifurcation and local rigidity of triply periodic minimal surfaces in 3
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 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.

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Accepted:
Published online:
DOI: 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
Koiso, Miyuki 1; Piccione, Paolo 2; Shoda, Toshihiro 3

1 Institute of Mathematics for Industry Kyushu University 744, Motooka Nishi-ku Fukuoka 819-0395 (Japan)
2 Departamento de Matemática Universidade de São Paulo Rua do Matão 1010 CEP 05508-900, São Paulo, SP (Brazil)
3 Faculty of Education Saga University Saga, SAGA 840-8502 (Japan)
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     title = {On bifurcation and local rigidity of triply periodic minimal surfaces in $\protect \mathbb{R}^3$},
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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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