Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature

2023/03/16 by Hui Ma, Jing Wu, Ma, Hui +1
Mathematics · #53C21 #53C40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2303.09285

openalex publication_date 2023/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Related

• Sobolev inequalities in manifolds with nonnegative curvature
• The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature
• Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature
• Geometric inequalities and the Alexandrov-Bakelman-Pucci technique
• Sobolev Inequalities In Manifolds With Asymptotically Nonnegative Curvature