The Spherical Boundary and Volume Growth. (26th March 2012) - Buckley, Stephen M.; Kokkendorff, Simon L.
Abstract : We consider the spherical boundary, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is "small" (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies to l 2 -products of length spaces, since we prove that a natural metric, generalizing such "norm-like" product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.