The eigencurve, parametrizing p-adic overconvergent modular eigenforms, was constructed by Coleman and Mazur in 1998. Generalizing this, eigenvarieties parametrizing p-adic overconvergent cuspidal Hilbert modular eigenforms were constructed by Andreatta, Iovita, and Pilloni. This requires defining overconvergent Hilbert eigenforms as sections of certain invertible weight sheaves on certain open subsets of Hilbert modular varieties. We will discuss how to extend the domains on which these sections converge, and possible applications to the geometry of the corresponding eigenvarieties.