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Manifolds With Group Actions and Elliptic Operators (Memoirs of the American Mathematical Society, 540)
Manifolds With Group Actions and Elliptic Operators - Memoirs of the American Mathematical Society, 540 Author:Vladimir Iakovlevich Lin, Yehuda Pinchover, Vladimir Ya. Lin 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 ... more »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.« less