We will discuss questions about the geometry of Chenevier's eigenvarieties for automorphic forms on definite unitary groups. For example, we will give bounds on the eigenvalues of the $U_p$ Hecke operator that appear in these eigenvarieties. These bounds generalize ones of Liu-Wan-Xiao for rank 2, which they used to prove the Coleman-Mazur-Buzzard-Kilford conjecture in that setting, to all ranks. If time permits, we will discuss possible avenues for recovering additional information not obtainable from these bounds and coming closer to fully generalizing Liu-Wan-Xiao's results.