Vol. 87, No. 1, 1980

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Fredholm theory of partial differential equations on complete Riemannian manifolds

Robert Colman McOwen

Vol. 87 (1980), No. 1, 169–185

This paper studies necessary and sufficient conditions for differential operators to be Fredholm on the Sobolev spaces of a complete (not necessarily compact) Riemannian manifold Ω. The conditions are formulated algebraically in terms of the nonvanishing of the operator’s principal symbol on Ω (ellipticity) and its “total symbol” at infinity of Ω. The operators considered arise by taking sums of products of vector fields, all of whose covariant derivatives vanish at infinity; and the study involves C-algebra techniques. The required technical restrictions on the curvature and topology of Ω near infinity are much weaker than those in earlier joint work with H. O. Cordes.

Mathematical Subject Classification 2000
Primary: 58G15
Secondary: 35B99
Received: 11 April 1979
Published: 1 March 1980
Robert Colman McOwen