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A duality theorem for a four dimensional Willmore energy
We prove an analogue of the energy identities underlying Bryant's duality theorem for a four-dimensional Willmore energy $\mathcal{E}_{\rm GR}$ obtained by Graham--Reichert and Zhang in codimension one. We show that, for an immersion $Φ$ of a compact four-dimensional manifold without boundary $Σ$ in
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