Some remarks on basic \(L^2\)-cohomology (Q2771407)

The authors discuss the de Rham-Hodge theory in a situation where the ambient manifold \(M\) with a Riemannian foliation \(F\) is complete. First they prepare some results on the basic subcomplex \((\Omega_{B,c},d_B)\) of basic forms with compact support following a similar argument as in \textit{J.A. Alvarez López} [Ann. Global Anal. Geom. 10, No. 2, 179-194 (1992; Zbl 0712.57012)]. The next task is to define basic \(L^2\)-cohomology associated to \((\Omega_{B,c},d_B)\). \(d_B\) has closed extensions to \(L^2_B(F)\) the completion of \(\Omega_{B,c}\) with respect to the pre-Hilbert metric \(\langle \cdot,\cdot\rangle\), which all lie between the closure \(d_{B,min}\) and the maximal extension \(d_{B,max}\). Using these operators, several basic \(L^2\)-cohomology for \(F\) are defined. Furthermore, in addition to the assumption that \(F\) is minimal, i.e., the mean curvature form \(\kappa=0\), the authors deduce the basic Hodge isomorphism, which gives a relation between the basic \(L^2\)-cohomology and the basic \(L^2\)-harmonic spaces. NEWLINENEWLINENEWLINEFinally they apply their results to obtain some transversal vanishing results for the basic Dirac operator, which generalize the results of \textit{J. F. Glazebrook} and \textit{F. W. Kamber} [Commun. Math. Phys. 140, No. 2, 217-240 (1991; Zbl 0744.53014)].NEWLINENEWLINEFor the entire collection see [Zbl 0953.00036].