The Cahn-Hoffman capillarity vector thermodynamics for curved anisotropic interfaces is adapted to soft liquid crystalline interfaces. The formalism is used to derive Herring's capillary pressure equation for anisotropic surfaces, where the role of anchoring energy of liquid crystals is made explicity. It is shown in detail that liquid crystal interfaces have three distinct contributions to capillary pressure: (i) area reduction, (ii) area rotation, and (iii) orientation curvature. General expressions representing these three mechanisms in terms of isotropic and anisotropic surface tensions are derived and used to analyze the Rayleigh capillary instability in thin fibers. It is shown that liquid crystal fibers and filaments are unstable to peristaltic and chiral surface ripple modes. The peristaltic mode leads to droplet formation, while chiral modes produce ripples in the curvature of the fiber. The role of liquid crystal orientation and anchoring energy on mode selection is elucidated and quantified.