Uniform Perfectness and Centers in Sublinearly Morse Boundaries

The $κ$--Morse boundary was introduced for CAT(0) spaces by Qing and Rafi and extended to proper geodesic spaces by Qing, Rafi, and Tiozzo. Motivated by Han and Liu's work on uniformly perfect Morse boundaries, we ask when uniform perfectness of visual boundary data detects $κ$--center exhaustivity. Two locally finite trees show that fixed-basepoint uniform perfectness is insufficient and, when $κ$ is unbounded, a basepoint-independent absolute annular cutoff is not necessary. For locally finite trees, center exhaustivity is equivalent to fixed-basepoint uniform perfectness and $κ$--radial accessibility. Under explicit uniform visual-data hypotheses, analogous conditions imply center exhaustivity through a chosen boundary stratum in proper geodesic spaces; a normalized all-basepoint criterion is also obtained. Finally, we characterize the metric transforms $ϕ$ for which one distortion function, depending only on $ϕ$, works for every identity $(Z,d)\to(Z,ϕ\circ d)$, and analyze the resulting metrics on rooted $q$--ary trees.