Sharp Sobolev inequalities on minimal immersions via optimal transport

We prove that, for $2\le p<n$, the optimal $L^p$-Sobolev constant on every complete $n$-dimensional Euclidean minimal immersion without boundary is the Euclidean Aubin–Talenti constant. The proof combines optimal transport to a suitably chosen target with a two-kernel comparison linking the tangential transport moment to deficits in the normal-fibre and Jacobian estimates.