The Willmore conjecture in the real projective space

Abstract

We prove that for any torus M immersed in the real projective space ℝP3(1) with mean curvature H, we have that ∫M(1 + H2)dA ≥ π2 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 S3(1) invariant under the antipodal map.