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Tube structures of co-rank 1 with forms defined on compact surfaces. (English) Zbl 07328212
Summary: We study the global solvability of a locally integrable structure of tube type and co-rank 1 by considering a linear partial differential operator \(\mathbb{L}\) associated to a general complex smooth closed 1-form \(c\) defined on a smooth closed \(n\)-manifold. The main result characterizes the global solvability of \(\mathbb{L}\) when \(n=2\) in terms of geometric properties of a primitive of a convenient exact pullback of the form \(\mathfrak{Im}(c)\) as well as in terms of homological properties of \(\mathfrak{Re}(c)\) related to small divisors phenomena. Although the full characterization is restricted to orientable surfaces, some partial results hold true for compact manifolds of any dimension, in particular, the necessity of the conditions, and the equivalence when \(\mathfrak{Im}(c)\) is exact. We also obtain informations on the global hypoellipticity of \(\mathbb{L}\) and the global solvability of \(\mathbb{L}^{n-1}\) – the last non-trivial operator of the complex when \(M\) is orientable.
58J10 Differential complexes
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35N10 Overdetermined systems of PDEs with variable coefficients
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[1] Arnol’d, VI, Topological and ergodic properties of closed \(1\)-forms with incommensurable periods, Funkt. Anal. Prilozhen, 25, 2, 1-12, 96 (1991) · Zbl 0732.58001
[2] Bergamasco, AP; Kirilov, A., Global solvability for a class of overdetermined systems, J. Funct. Anal., 252, 2, 603-629 (2007) · Zbl 1158.58011
[3] Bergamasco, AP; Petronilho, G., Global solvability of a class of involutive systems, J. Math. Anal. Appl., 233, 1, 314-327 (1999) · Zbl 0942.35011
[4] Bergamasco, AP; Cordaro, PD; Malagutti, PA, Globally hypoelliptic systems of vector fields, J. Funct. Anal., 114, 2, 267-285 (1993) · Zbl 0777.58041
[5] Bergamasco, AP; Cordaro, PD; Petronilho, G., Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 2, 261-274 (1996) · Zbl 0863.58062
[6] Bergamasco, AP; Kirilov, A.; Nunes, WVL; Zani, SL, On the global solvability for overdetermined systems, Trans. Am. Math. Soc., 364, 9, 4533-4549 (2012) · Zbl 1275.35004
[7] Bergamasco, AP; Kirilov, A.; Nunes, WVL; Zani, SL, Global solutions to involutive systems, Proc. Am. Math. Soc., 143, 11, 4851-4862 (2015) · Zbl 1330.35078
[8] Bergamasco, AP; de Medeira, C.; Kirilov, A.; Zani, SL, On the global solvability of involutive systems, J. Math. Anal. Appl., 444, 1, 527-549 (2016) · Zbl 1359.37107
[9] Bergamasco, A.; Parmeggiani, A.; Zani, S.; Zugliani, G., Classes of globally solvable involutive systems, J. Pseudo Differ. Oper. Appl., 8, 4, 551-583 (2017) · Zbl 1382.58018
[10] Bergamasco, AP; Parmeggiani, A.; Zani, SL; Zugliani, GA, Geometrical proofs for the global solvability of systems, Math. Nachr., 291, 16, 2367-2380 (2018) · Zbl 1406.58015
[11] Berhanu, S., Cordaro, P.D., Hounie, J.: An Introduction to Involutive Structures. New Mathematical Monographs, vol. 6. Cambridge University Press, Cambridge (2008) · Zbl 1151.35011
[12] Cardoso, F.; Hounie, J., Global solvability of an abstract complex, Proc. Am. Math. Soc., 65, 1, 117-124 (1977) · Zbl 0335.58015
[13] Dattori da Silva, PL; Meziani, A., Cohomology relative to a system of closed forms on the torus, Math. Nachr., 289, 17-18, 2147-2158 (2016) · Zbl 1368.58011
[14] de Medeira, C.; Zani, SL, A class of globally non-solvable involutive systems on the torus, J. Pseudo Differ. Oper. Appl., 10, 2, 455-474 (2019) · Zbl 1417.58016
[15] Hatcher, A., Algebraic Topology (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1044.55001
[16] Hounie, J.; Zugliani, G., Global solvability of real analytic involutive systems on compact manifolds, Math. Ann., 369, 3-4, 1177-1209 (2017) · Zbl 1380.35129
[17] Hounie, J.; Zugliani, G., Global solvability of real analytic involutive systems on compact manifolds. Part 2, Trans. Am. Math. Soc., 371, 7, 5157-5178 (2019) · Zbl 1418.35002
[18] Munkres, J.R.: Elementary Differential Topology. Lectures Given at Massachusetts Institute of Technology, Fall, vol. 1961. Princeton University Press, Princeton (1966) · Zbl 0161.20201
[19] Treves, F., Study of a model in the theory of complexes of pseudodifferential operators, Ann. Math. (2), 104, 2, 269-324 (1976) · Zbl 0354.35067
[20] Trèves, F.: Hypo-analytic Structures. Princeton Mathematical Series: Local Theory, vol. 40. Princeton University Press, Princeton (1992) · Zbl 0787.35003
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