# Manifolds with Group Actions and Elliptic Operators

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*Vladimir Ya. Lin; Yehuda Pinchover*

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.

#### Table of Contents

# Table of Contents

## Manifolds with Group Actions and Elliptic Operators

- Table of Contents v6 free
- 1. Introduction 18 free
- 2. Some notions connected with group actions 613 free
- 3. Some notions and results connected with elliptic operators 1320
- 4. Elliptic operators and group actions 1926
- 5. Positive multiplicative solutions 2633
- 6. Nilpotent groups: extreme points and multiplicative solutions 4552
- 7. Nonnegative solutions of parabolic equations 5562
- 8. Invariant operators on a manifold with boundary 6572
- 9. Examples and open problems 7077
- 10. Appendix: analyticity of A(ξL) 7481
- References 7683