Convex decomposition theory (Q2778027)

The present paper continues the work started by the same authors in [Geom. Topol. 4, 219-242 (2000; Zbl 0961.57009)], with the purpose of studying relationships between taut foliations and tight contact structures [\textit{Y. M. Eliashberg} and \textit{W. P. Thurston}, Confooliations, Univ. Lect. Ser. 13 (1998; Zbl 0893.53001)] by means of the notion of convex decomposition. The paper makes essential use of Gabai's sutured manifold decomposition theory [\textit{D. Gabai}, J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013)], as well as of \textit{V. Colin}'s key ideas [Bull. Soc. Math. Fr. 127, No. 1, 43-69; addendum ibid. 127, 623 (1999; Zbl 0930.53053); Sur la torsion de structures de contact tendues, Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, No. 2, 267-286 (2001; Zbl 1035.53119)] in the context of convex decomposition theory. NEWLINENEWLINENEWLINEIn particular, if \((M,\gamma)\) is an oriented, compact, connected, irreducible, sutured 3-manifold which has boundary, is taut and has annular sutures, the authors provide an alternative proof of the existence of a universally tight contact structure (originally due to Gabai-Eliashberg-Thurston); this proof is performed by direct application of a suitable gluing theorem, and resorts neither 4-dimensional symplectic filling techniques nor perturbation of taut foliations into tight contact structure (as it happens in the previous paper by the same authors). NEWLINENEWLINENEWLINEMoreover, similar ideas are applied to prove that, if \(M\) is an oriented, closed, connected, irreducile 3-manifold which contains an incompressible torus, then \(M\) carries infinitely many isomorphism classes of universally tight contact structures. Note that this result is deeply related to a conjecture studied (and partially proved) in [\textit{E. Giroux}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, No. 6, 697-705 (1994; Zbl 0819.53018), Invent. Math. 135, No. 3, 789-802 (1999; Zbl 0969.53044); \textit{K. Yutaka}, Commun. Anal. Geom. 5, No. 3, 413-438 (1997; Zbl 0899.53028); \textit{V. Colin}, Sur la torsion des structures \dots, loc. cit.].