The heat equation for Riemannian foliations

Abstract:

Let $\mathcal {F}$ be a Riemannian foliation on a closed oriented manifold $M$, with the transversal Laplacian ${\Delta _B}$ acting on the basic forms $\Omega _B^r(\mathcal {F})$ of degree $r \geqslant 0$. We construct the fundamental solution $e_B^r(x,y,t)$ for the basic heat operator $\partial /\partial t + {\Delta _B}$, and prove existence and uniqueness for the solution of the heat equation on $\Omega _B^r(\mathcal {F})$. As an application we give a new proof for the deRham-Hodge decomposition theorem for ${\Delta _B}$ in $\Omega _B^r(\mathcal {F})$, generalizing the approach to the classical deRham-Hodge theorem pioneered by Milgram and Rosenbloom.