  For a given quantum field theory, provided the area of the entangling surface
is fixed, what surface maximizes entanglement entropy? We analyze the answer to
this question in four and higher dimensions. Surprisingly, in four dimensions
the answer is related to a mathematical problem of finding surfaces which
minimize the Willmore (bending) energy and eventually to the Willmore
conjecture. We propose a generalization of the Willmore energy in higher
dimensions and analyze its minimizers in a general class of topologies
$S^m\times S^n$ and make certain observations and conjectures which may have
some mathematical significance.
