P\'olya-Szeg\H{o} inequalities on submanifolds with small total mean curvature

Abstract

We establish P\'olya-Szeg\H{o}-type inequalities (PSIs) for Sobolev-functions defined on a regular $n$-dimensional submanifold $\Sigma$ (possibly with boundary) of a $(n+m)$-dimensional Euclidean space, under explicit upper bounds of the total mean curvature. The $p$-Sobolev and Gagliardo-Nirenberg inequalities, as well as the spectral gap in $W^{1,p}_0(\Sigma)$ are derived as corollaries. Using these PSIs, we prove a sharp $p$-Log-Sobolev inequality for minimal submanifolds in codimension one and two. The asymptotic sharpness of both the multiplicative constant appearing in PSIs and the assumption on the total mean curvature bound as $n\to \infty$ is provided. A second equivalent version of our PSIs is presented in the appendix of this paper, introducing the notion of model space $(\mathbb{R}^+,\mathfrak{m}_{n,K})$ of dimension $n$ and total mean curvature bounded by $K$.