On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ3

Miyuki Koiso; Paolo Piccione; Toshihiro Shoda

doi:10.5802/aif.3222

On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ³

Miyuki Koiso, Paolo Piccione, Toshihiro Shoda

Division of Fundamental mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We study the space of triply periodic minimal surfaces in ℝ³, 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 oneparameter family of triply periodic minimal surfaces {X_t}_t containing X₀ where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from X₀. We also apply these results to several known examples.

Original language	English
Pages (from-to)	2743-2778
Number of pages	36
Journal	Annales de l'Institut Fourier
Volume	68
Issue number	6
DOIs	https://doi.org/10.5802/aif.3222
Publication status	Published - 2018

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Geometry and Topology

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title = "On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ3",

abstract = "We study the space of triply periodic minimal surfaces in ℝ3, 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 oneparameter family of triply periodic minimal surfaces {Xt}t containing X0 where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from X0. We also apply these results to several known examples.",

author = "Miyuki Koiso and Paolo Piccione and Toshihiro Shoda",

note = "Funding Information: Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory. 2010 Mathematics Subject Classification: 53A10, 58J55, 58E12, 35J62. (*) The first author is supported in part by JSPS KAKENHI Grant Numbers JP25287012, JP26520205, JP26610016, and the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development. The second author is partially supported by Fapesp and CNPq, Brazil. The third author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05134. Publisher Copyright: {\textcopyright} Association des Annales de l'institut Fourier, 2018.",

year = "2018",

doi = "10.5802/aif.3222",

language = "English",

volume = "68",

pages = "2743--2778",

journal = "Annales de l'Institut Fourier",

issn = "0373-0956",

publisher = "Association des Annales de l'Institut Fourier",

number = "6",

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TY - JOUR

T1 - On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ3

AU - Koiso, Miyuki

AU - Piccione, Paolo

AU - Shoda, Toshihiro

N1 - Funding Information: Keywords: triply periodic minimal surfaces, H-family, rPD-family, tP-family, tD-family, tCLP-family, bifurcation theory. 2010 Mathematics Subject Classification: 53A10, 58J55, 58E12, 35J62. (*) The first author is supported in part by JSPS KAKENHI Grant Numbers JP25287012, JP26520205, JP26610016, and the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development. The second author is partially supported by Fapesp and CNPq, Brazil. The third author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05134. Publisher Copyright: © Association des Annales de l'institut Fourier, 2018.

PY - 2018

Y1 - 2018

N2 - We study the space of triply periodic minimal surfaces in ℝ3, 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 oneparameter family of triply periodic minimal surfaces {Xt}t containing X0 where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from X0. We also apply these results to several known examples.

AB - We study the space of triply periodic minimal surfaces in ℝ3, 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 oneparameter family of triply periodic minimal surfaces {Xt}t containing X0 where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from X0. We also apply these results to several known examples.

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On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ³

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