Explicit conformally constrained Willmore minimizers in arbitrary codimension

摘要

In our previous work (Ndiaye and Sch?tzle, 2014), we proved that the flat constant mean curvature tori T_r:= r S~1 × (1 ? r~2)/(1/2)S~1 ? S~3 for 0 < r ≤ 1/2/(1/2) minimize theWillmore energy in their conformal class in codimension one when r ≈ 1/2/(1/2), that is T_r is close to the Clifford torus T_(Cli f f) = T_(1/2/(1/2). In this article, we extend this to arbitrary codimension. Moreover we prove that the Clifford torus minimizes the Willmore energy in an open neighbourhood of its conformal class, again in arbitrary codimension, but the neighbourhood may depend on the codimension.