2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/104376We study a model lipid bilayer composed of a mixture of two incompatible lipid types which have a natural tendency to segregate in the absence of membrane fluctuations. The membrane is mechanically characterized by a local bending rigidity $κ(ϕ)$ which varies with the average local lipid composition $ϕ$. We show, in the case where $κ$ varies weakly with $ϕ$, that the effective interaction between lipids of the same type can either be everywhere attractive or can have a repulsive component at intermediate distances greater than the typical lipid size. When this interaction has a repulsive component, it can prevent macro-phase separation and lead to separation in mesophases with a finite domain size. This effect could be relevant to certain experimental and numerical observations of mesoscopic domains in such systems.9 pages RevTex, 1 eps figureSoft Condensed Mattertext