Long time behavior of leafwise heat flow for Riemannian foliations (Q2711521)

Let \(M\) be a closed, Riemannian manifold, and let \(\mathcal{F}\) be a smooth foliation on \(M\). Let \(d\) denote a first order leafwise differential operator on sections of a bundle \(E\) over \(M\) that generates a leafwise elliptic differential complex. Let \(\delta\) denote the formal adjoint of \(d\) on \(M\). The associated Laplacian \(\Delta= (d+\delta)^{2}\) is an essentially self-adjoint operator on \(L^{2}(E)\), and the leafwise heat operator \(e^{-t\Delta}\) on \(L^{2}(E)\) is defined for each \(t>0\). NEWLINENEWLINENEWLINEAccording to the work of \textit{J. Ro}e [Math. Proc. Camb. Philos. Soc. 102, 459-466 (1987; Zbl 0646.58024)], the leafwise heat operator is a continuous operator on smooth sections of \(E\) which depends continuously on \(t\geq 0\). As \(t \to \infty\), the operator \(e^{-t\Delta}\) converges strongly to \(\Pi\), the orthogonal projection onto the kernel of \(\Delta\). The operator \(\Pi\) need not preserve the space of smooth sections, \(C^\infty(E)\), because the operator \(\Delta\) is typically not elliptic on \(M\). The main result of this paper is that if \(M\) admits a first order, transversally elliptic differential operator \(A\) on sections of E such that \(Ad\pm dA=Gd +dH\) and \(A\delta\pm \delta A=K\delta + \delta L\) for some zeroth order operators \(G\), \(H\), \(K\), and \(L\), then \(\Pi\) actually maps \(C^\infty(E)\) to itself. Furthermore, the leafwise Hodge decomposition results: \(C^\infty(E)=\ker(\Delta)\oplus \overline{\text{im}(\Delta)}\). NEWLINENEWLINENEWLINEThe authors apply this result to the setting of a Riemannian foliation, where the normal bundle of the foliation is endowed with a holonomy-invariant metric. It is always possible to choose a metric on \(M\) that restricts to this normal bundle metric; such a metric is called bundle-like. In this case, the leafwise differential complex of the foliation admits a Hodge decomposition, and the leafwise heat flow takes smooth forms to smooth, leafwise harmonic forms. This result is extended to the case of forms with values in a vector bundle that is compatible with the foliation structure. As a corollary, the space of bundle-like metrics on \(M\) is shown to be a deformation retract of the space of all metrics on \(M\). NEWLINENEWLINENEWLINEThe results of this paper may be used to construct examples of dense Riemannian foliations on a closed, Riemannian manifold such that the space of smooth, leafwise harmonic forms is infinite dimensional. The authors also use these techniques to shed new light on the second term in the differential spectral sequence of the foliation.