We explore the question of which shape a manifold is compelled to take when
immersed in another one, provided it must be the extremum of some functional.
We consider a family of functionals which depend quadratically on the extrinsic
curvatures and on projections of the ambient curvatures. These functionals
capture a number of physical setups ranging from holography to the study of
membranes and elastica. We present a detailed derivation of the equations of
motion, known as the shape equations, placing particular emphasis on the issue
of gauge freedom in the choice of normal frame. We apply these equations to the
particular case of holographic entanglement entropy for higher curvature three
dimensional gravity and find new classes of entangling curves. In particular,
we discuss the case of New Massive Gravity where we show that non-geodesic
entangling curves have always a smaller on-shell value of the entropy
functional. Then we apply this formalism to the computation of the entanglement
entropy for dual logarithmic CFTs. Nevertheless, the correct value for the
entanglement entropy is provided by geodesics. Then, we discuss the importance
of these equations in the context of classical elastica and comment on terms
that break gauge invariance.Comment: 54 pages, 8 figures. Significantly improved version, accepted for
publication in Annals of Physics. New section on logarithmic CFTs. Detailed
derivation of the shape equations added in appendix B. Typos corrected,
clarifications adde
