A sphere theorem for 2-dimensional CAT(1)-spaces. (Q1858332)

Then main result of the paper is the following Theorem: Let \(X\) be a compact, geodesically complete CAT(1)-space whose 2-dimensional Hausdorff measure is at most 3/2 of that of the unit sphere \(S^2\), and the Hausdorff dimension of every sufficiently small neighborhood of every point in \(X\) equals 2. Then either \(X\) is homeomorphic to \(S^2\), or \(X\) is isometric to the 2-polyhedron which is the union of \(S^2\) and the half-sphere \(HS^2\) along the equator. The proof is based on previous results of the author on the volume convergence for Alexandrov spaces of curvature bounded above.