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Manifolds with Group Actions and Elliptic Operators
Vladimir Ya. Lin and Yehuda Pinchover
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Memoirs of the American Mathematical Society
1994; 78 pp; softcover
Volume: 112
ISBN-10: 0-8218-2604-2
ISBN-13: 978-0-8218-2604-1
List Price: US$37
Individual Members: US$22.20
Institutional Members: US$29.60
Order Code: MEMO/112/540
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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.

Readership

Analysts, specialists in partial differential equations and mathematical physics, and graduate students in analysis.

Table of Contents

  • Introduction
  • Some notions connected with group actions
  • Some notions and results connected with elliptic operators
  • Elliptic operators and group actions
  • Positive multiplicative solutions
  • Nilpotent groups: extreme points and multiplicative solutions
  • Nonnegative solutions of parabolic equations
  • Invariant operators on a manifold with boundary
  • Examples and open problems
  • Appendix: analyticity of \(\Lambda (\xi , \scr L)\)
  • References
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