The Willmore Conjecture in the Real Projective Space

We prove that for any torus $M$ immersed in the real projective space \hbox{\scriptsize $\mathbb R \mathrm P^3(1)$} with mean curvature $H$, we have that $\int_M (1 + H^2)dA \geq \pi^2$ and that the equality holds only for the minimal Clifford torus. In terms of the three sphere, this result says that the Willmore conjecture is true for immersed tori in $S^3(1)$ invariant under the antipodal map.%}

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Published 1 January 1999