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

What surface maximizes entanglement entropy?