Pólya-Szegő inequalities on submanifolds with small total mean curvature

TL;DR

This work develops Pólya-Szegő-type inequalities for Sobolev functions on n-dimensional submanifolds of Euclidean space under a small total mean curvature bound, introducing the main gradient comparison $\|\nabla u^*\|_{L^p(ℝ^n)} \leq PS(n,K) \|\nabla_\Sigma u\|_{L^p(Σ)}$ with $PS(n,K)=\dfrac{\text{IC}(n)}{1-\text{IC}(n)K}\,n\omega_n^{1/n}$. The analysis relies on Schwartz rearrangement adapted to submanifolds and the Michael–Simon isoperimetric inequality, establishing a bridge from Euclidean level-set geometry to Σ under the curvature bound. The authors derive Euclidean-type first-order inequalities (Sobolev, Gagliardo–Nirenberg, spectral gap) as corollaries, and obtain sharp $p$-Log-Sobolev inequalities for minimal submanifolds in codimensions 1–2; a monotonicity principle ties these to classical constants via $PS(n,K)$, with explicit constants when codimension is small. They also prove asymptotic sharpness of the curvature bound as the ambient dimension grows and present a model-space reformulation in an appendix. Overall, the results extend Euclidean functional inequalities to a broad class of submanifolds without Ricci curvature assumptions, governed by the total mean-curvature bound.

Abstract

We establish Pólya-Szegő-type inequalities (PSIs) for Sobolev-functions defined on a regular $n$-dimensional submanifold $Σ$ (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 $W^{1,p}_0(Σ)$ 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 $K$.

Theorems & Definitions (27)

• Theorem 1: Pólya-Szegő inequality
• Corollary 2: Sharp $p$-Log-Sobolev inequality for minimal submanifolds
• Definition : Schwartz rearrangement
• Proposition 3
• Lemma 4
• Lemma 5
• proof
• Proposition 6: Isoperimetric/Sobolev inequality for $\Sigma$ in $\mathcal{STC}({K})$
• proof
• proof : Proof of Theorem \ref{['th:PS']}