UC Riverside

We present two results. The first is a converse to a theorem first proved by Wongwhich says the ratio of intrinsic measures approaches 1 near the boundary of a strongly pseudoconvex domain; we show that for a particular type of domain the boundary is strongly pseudoconvex if the ratio of intrinsic measures approaches 1 near the boundary. The argument is primarily one from Zimmer using the scaling method. What we did is show that the ratio of intrinsic measures is a function which respects this scaling process. Our second contribution was done in an attempt to use one particular step of Huang and Xiao’s proof of the S.-Y. Cheng conjecture to settle the Ramadanov conjecture. While unsuccessful in this regard, we were able to make this step more direct and we ultimately show that if the Bergman metric is asymptotically Kähler-Einstein enough near the boundary of a C ∞ strongly pseudoconvex domain Ω then the boundary ∂Ω must be spherical. This result is of interest on its own but it also provides a more direct proof of the S.-Y. Cheng conjecture and may be used in further work on the Ramadanov conjecture.