| Conformally flat spaces of bounded curvature are generalizations of Riemannian spaces. In two dimensions these are the two dimensional manifolds of bounded curvature. Reshetnyak has shown that the logarithms of the conformal factors of such spaces are δ-subharmonic functions. For higher dimensional spaces, Slavskii has shown that Riemannian spaces with curvature bounded below by zero have a conformal factor that has subharmonic logarithm on all two dimensional slices, under all conformal transformations.; Here we show that if the logarithm of the conformal factor is subharmonic then the space has curvature bounded above by zero, and, subject to a growth constraint, if the logarithm of the conformal factor is subharmonic under all conformal transformations then the curvature is bounded below by zero. If the space has Lipschitz conformal factor and curvature bounded below by zero then the two dimensional subspaces have curvature bounded below by K, depending only on the Lipschitz constant and the size of the function. |
