Sharp Sobolev inequalities on minimal immersions via optimal transport

Abstract: We prove that, for $2\leq p&lt;n$, the optimal $L<sup>p$-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.