an:06798242
Zbl 1380.35129
Hounie, Jorge; Zugliani, Giuliano
Global solvability of real analytic involutive systems on compact manifolds
EN
Math. Ann. 369, No. 3-4, 1177-1209 (2017).
00371446
2017
j
35N10 58J10
global solvability; linear operators; compact manifolds
Let \(b\) be a real analytic closed non-exact 1-form defined on a compact and without boundary connected \(n\)-dimensional manifold \(M\), \((n > 1)\). The focus of this work is the smooth global solvability of the differential operator \({\mathbb L}:C^\infty(M\times {\mathbb S}^1)\to \Lambda^1 C^\infty(M\times {\mathbb S}^1)\) given by \({\mathbb L}u =d_{t_j}u + ib(t) \wedge \partial_x u\), where \(x\in {\mathbb S}^1\), and \(d_t\) is the exterior derivative on \(M\).
The approach relies on defining an appropriate covering projection \(\tilde M \to M\) such that the pullback of \(b\) has a primitive \(\tilde B\) and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of \(\tilde B\) are connected. In case of orientable manifolds \(M\), further charactirizations are made.
Marius Ghergu (Dublin)