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March 2021 S-stable foliations on flow-spines with transverse Reeb flow
Shin Handa, Masaharu Ishikawa
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Hiroshima Math. J. 51(1): 77-99 (March 2021). DOI: 10.32917/h2020026


The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $\beta$ defining a foliation on a branched simple polyhedron $P$ satisfies $d\beta > 0$, which means that the foliation is possibly a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $\beta$ on $P$ with $d\beta > 0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $\beta$ with $d\beta > 0$ on the abalone.

Funding Statement

This work is partially supported by JSPS KAKENHI Grant Number JP17H06128. The second author is supported by JSPS KAKENHI Grant Numbers JP19K03499 and Keio University Academic Development Funds for Individual Research.


We would like to thank Ippei Ishii for precious comments and especially for telling us the 3-manifold of the spine in Figure 13. We are also grateful to Yuya Koda and Hironobu Naoe for useful conversation. Finally, we thank the anonymous referee for insightful comments on improving the paper.


Download Citation

Shin Handa. Masaharu Ishikawa. "S-stable foliations on flow-spines with transverse Reeb flow." Hiroshima Math. J. 51 (1) 77 - 99, March 2021.


Received: 24 March 2020; Revised: 31 July 2020; Published: March 2021
First available in Project Euclid: 19 April 2021

Digital Object Identifier: 10.32917/h2020026

Primary: 57M50
Secondary: 557M25 , 57M20

Keywords: branched spine , contact structure , Flow-spine

Rights: Copyright © 2021 Hiroshima University, Mathematics Program


Vol.51 • No. 1 • 2021
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