Abstract. We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.Introduction.
We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.
Academic Press
Unique continuation results are proved for metrics with prescribed Ricci
curvature in the setting of bounded metrics on compact manifolds with boundary,
and in the setting of complete, conformally compact metrics. Related to this
issue, an isometry extension property is proved: continuous groups of
isometries at conformal infinity extend into the bulk of any complete
conformally compact Einstein metric. Relations of this property with the
invariance of the Gauss-Codazzi constraint equations under deformations are
also discussed.Comment: 32 pages, supercedes math.DG/0501067; final published versio
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