What surface maximizes entanglement entropy?

Understand

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.