Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{ 1,1}$ geodesics between two smooth non-degenerate K\"ahler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi-definite.
Categories: math.DG