Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature

We prove Michael-Simon type Sobolev inequalities for $n$-dimensional submanifolds in $(n+m)$-dimensional Riemannian manifolds with nonnegative $k$-th intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here $k=\min(n-1,m-1)$. These inequalities extends Brendle's Michael-Simon type Sobolev inequalities on Riemannian manifolds with nonnegative sectional curvature (arXiv:2009.13717) and Dong-Lin-Lu's Michael-Simon type Sobolev inequalities on Riemannian manifolds with asymptotically nonnegative sectional curvature (arXiv:2203.14624) 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.