Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature

Abstract

We prove Michael-Simon type Sobolev inequalities for $n$-dimensional submanifolds in $(n+m)$-dimensional Riemannian manifolds with nonnegative $(n-1)$-th intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. 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 $(n-1)$-Ricci curvature setting. In particular, a simple application of these inequalities gives rise to some isoperimetric inequalities for minimal submanifolds in Riemannian manifolds.Comment: 11 pages. All comments are welcome