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

Miyuki Koiso, Paolo Piccione, Toshihiro Shoda

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)2743-2778
Number of pages36
JournalAnnales de l'Institut Fourier
Volume68
Issue number6
DOIs
Publication statusPublished - Jan 1 2018

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Minimal surface
Rigidity
Bifurcation
Morse Index
Nullity
Pairwise
Jump
Branch
Odd
Integer
Family

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology

Cite this

On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ3. / Koiso, Miyuki; Piccione, Paolo; Shoda, Toshihiro.

In: Annales de l'Institut Fourier, Vol. 68, No. 6, 01.01.2018, p. 2743-2778.

Research output: Contribution to journalArticle

Koiso, Miyuki ; Piccione, Paolo ; Shoda, Toshihiro. / On bifurcation and local rigidity of triply periodic minimal surfaces in ℝ3. In: Annales de l'Institut Fourier. 2018 ; Vol. 68, No. 6. pp. 2743-2778.
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