Abstract
We prove Michael-Simon type Sobolev inequalities for n-dimensional submanifolds in \((n+m)\)-dimensional Riemannian manifolds with nonnegative kth 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.
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Acknowledgements
The authors were partially supported by the National Natural Science Foundation of China under grants No. 11831005 and Nos. 12061131014. They would like to thank Professor Chao Qian for his helpful suggestions. They also thank Kai-Hsiang Wang for bringing their attention to [23, 24] and for helpful comments on the earlier version of this paper.
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Ma, H., Wu, J. Sobolev Inequalities in Manifolds With Nonnegative Intermediate Ricci Curvature. J Geom Anal 34, 93 (2024). https://doi.org/10.1007/s12220-023-01486-5
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DOI: https://doi.org/10.1007/s12220-023-01486-5