Sobolev Inequalities in Manifolds With Nonnegative Intermediate Ricci Curvature

We prove Michael-Simon type Sobolev inequalities for n-dimensional submanifolds in (n+m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+m)$$\end{document}-dimensional Riemannian manifolds with nonnegative kth intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here k=min(n-1,m-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=\min (n-1,m-1)$$\end{document}. These inequalities extend Brendle’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with nonnegative sectional curvature Brendle (Commun. Pure Appl. Math. 76(9), 2192–2218 (2022)) and Dong-Lin-Lu’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with asymptotically nonnegative sectional curvature Dong et al. (Sobolev inequalities in manifolds with asymptotically nonnegative curvature, 2022) to the k-Ricci curvature setting. In particular, a simple application of these inequalities gives rise to some isoperimetric inequalities for minimal submanifolds in Riemannian manifolds.