Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds

Abstract

Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is convex and close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.