Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
Abstract
We prove a Pólya--Szegö inequality for functions defined on an n-dimensional submanifold Σ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on ℝn, and the constant depends on the Ln-norm of the mean curvature of Σ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.
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BibTeX
@misc{do2026pólyaszegö,
title = {{Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications}},
author = {Anh Xuan Do and Nguyen Lam and Guozhen Lu and Raoní Ponciano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22676},
eprint = {2609.22676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22676},
}
RIS
TY - UNPB
TI - Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
AU - Anh Xuan Do
AU - Nguyen Lam
AU - Guozhen Lu
AU - Raoní Ponciano
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22676
UR - https://arxiv.org/abs/2609.22676
AB - We prove a Pólya--Szegö inequality for functions defined on an n-dimensional submanifold Σ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on ℝ^n, and the constant depends on the L^n-norm of the mean curvature of Σ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.
ER -