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A method used to treat the elastic distortion of a uniaxial nematic liquid crystal induced by homogeneous anchoring on the surface grooves is generalized to biaxial nematic liquid crystals under the homeotropic anchoring condition.Employing some approximations for the elastic constants,we obtain an additional term in the elastic energy per unit area which depends on the angle between the minor director at infinity and the direction of the grooves,with a period of π/2.This leads to instability on the surface grooves so that two states with crossed minor directors are energetically indistinguishable.Our theoretical study explains why the homeotropic alignment method developed for uniaxial liquid crystals loses efficacy for biaxial nematics.In most liquid crystal devices,the liquid crystals are sandwiched between two substrates coated with alignment layers.In the absence of exteally applied fields,the orientation of the liquid crystal in the cell is determined by the anchoring condition of the alignment layer.[1-3] One usually distinguishes three main types of liquid crystalline director alignment near solid walls:homeotropic,homogeneous (or planar) and tilted orientations.Here we study the first of these and consider the biaxial nematic phase,which was observed in lyotropic systems as early as 1980[4] and has been confirmed by deuterium NMR spectroscopy.