Uniform Perfectness, Geodesic Richness, and Rigidity for Sublinearly Morse Boundaries

Han and Liu gave a geometric characterization of uniform perfectness for the Morse boundary of a proper geodesic metric space: the Morse boundary is uniformly perfect if and only if the space is Morse geodesically rich, equivalently center--exhaustive. In this paper we prove the analogous statement for the sublinearly Morse boundary $\partial_{\kappa}X$. Here $\kappa$ is a fixed concave increasing sublinear function and $\partial_{\kappa}X$ is the boundary introduced by Qing--Rafi for CAT(0) spaces and extended by Qing--Rafi--Tiozzo to proper geodesic spaces. Assuming that $\partial_{\kappa}X$ has at least three points, we show that uniform perfectness of $\partial_{\kappa}X$ (for any $\kappa$--visual metric based at a fixed basepoint) is equivalent to $\kappa$--Morse geodesic richness and to $\kappa$--center--exhaustiveness. The geometric input is a sublinear thin--triangle statement for $\kappa$--Morse geodesics, together with the renormalization map $\rho_{\kappa}(t)=\int_0^t \frac{ds}{\kappa(s)}$, which converts $\kappa$--scale errors at radius $R$ into bounded errors in the $\rho_\kappa$--scale. As applications we obtain quantitative lower bounds on the lower Assouad dimension (and, under doubling hypotheses, on the Hausdorff dimension) of $\kappa$--visual metrics on $\partial_{\kappa}X$ in terms of the uniform perfectness constant. Finally, for $\kappa$--center--exhaustive spaces $X$ and $Y$ satisfying a mild additional growth condition on $\kappa$, we prove a rigidity statement in the sublinear category: every quasi-symmetric homeomorphism $\partial_{\kappa}X\to\partial_{\kappa}Y$ is induced by a sublinear bilipschitz equivalence $X\to Y$.