Manifolds With Group Actions and Elliptic Operators

This work studies equivariant linear second order elliptic operators $P$ on a connected noncompact manifold $X$ with a given action of a group $G$. The action is assumed to be cocompact, meaning that $GV=X$ for some compact subset $V$ of $X$. The aim is to study the structure of the convex cone of all positive solutions of $Pu=0$. It turns out that the set of all normalized positive solutions which are also eigenfunctions of the given $G$-action can be realized as a real analytic submanifold $\Gamma _0$ of an appropriate topological vector space $\mathcal H$. When $G$ is finitely generated, $\mathcal H$ has finite dimension, and in nontrivial cases $\Gamma _0$ is the boundary of a strictly convex body in $\mathcal H$. When $G$ is nilpotent, any positive solution $u$ can be represented as an integral with respect to some uniquely defined positive Borel measure over $\Gamma _0$. Lin and Pinchover also discuss related results for parabolic equations on $X$ and for elliptic operators on noncompact manifolds with boundary.