We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound.
Topological regularity of spaces with an upper curvature bound
Geodesically complete spaces with an upper curvature bound
A lower bound for the curvature integral under an upper curvature bound
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