New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

Manifolds with Group Actions and Elliptic Operators
 SEARCH THIS BOOK:
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

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.

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

• Appendix: analyticity of $$\Lambda (\xi , \scr L)$$