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 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.