Semicontinuity theorems for holomorphic and CR automorphism groups

We review the semicontinuity property for automorphism groups of CR manifolds proved in Joo (Pac J Math 285:225–241, 2016 [19]) which belongs to the line of research on the semicontinuity for Riemmanian isometry groups and holomorphic automorphism groups of domains Ebin (Proceedings of symposia in pure mathematics, pp 11–40, 1968 [6]), Greene and Kim (Math Z 277:909–916, 2014 [10]), Greene et al. (Pac J Math 262:365–395, 2013 [12]), Greene and Krantz (Math Ann 261:425–446, 1982 [13]), Kim (Arch Math (Basel) 49:450–455, 1987 [21]), Krantz (Real Anal Exch 36, 421–433, 2010/11 [22]). A brief explanation of the proof and some related open problems as well as the motivation of the study are provided.