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We study nematic configurations within three-dimensional (3D) cuboids, withplanar degenerate boundary conditions on the cuboid faces, in the Landau-deGennes framework. There are two geometry-dependent variables: the edge lengthof the square cross-section, $\lambda$, and the parameter $h$, which is ameasure of the cuboid height. Theoretically, we prove the existence anduniqueness of the global minimiser with a small enough cuboid size. We developa new numerical scheme for the high-index saddle dynamics to deal with thesurface energies. We report on a plethora of (meta)stable states, and theirdependence on $h$ and $\lambda$, and in particular, how the 3D states areconnected with their two-dimensional counterparts on squares and rectangles.Notably, we find families of almost uniaxial stable states constructed from thetopological classification of tangent unit-vector fields and study transitionpathways between them. We also provide a phase diagram of competing(meta)stable states, as a function of $\lambda$ and $h$.

Multistability for Nematic Liquid Crystals in Cuboids with Degenerate Planar Boundary Conditions