Sobolev Inequalities Relating Rough Riemannian Metrics to their Distance Functions
Abstract
In previous papers, the first named author and E. Bryden explored Sobolev like inequalities relating the norm of a Riemmanian metric g1, in terms of a background Riemannian metric g0, in order to estimate the distance function dg1. Although |g1|g0 is not the gradient of the distance function dg1, it was shown that a Morrey and Sobolev inequalities hold in their appropriate domains. In this paper we see that these results can be extended to the case where g1 is a Rough Riemannian metric, something roughly analogous to an L∞ metric. We show that the proof of the Morrey and Sobolev inequality go through, with appropriate adjustments, as well as convergence results. We also adapt previous examples from the literature to the rough Riemannian metric case which show the hypotheses from the main theorems are necessary.
Related
- Sobolev--Poincaré inequalities for piecewise W2,p functions over general polytopic meshes
- Sobolev inequalities for sections and projections of log-concave functions
- Infinitesimal Hilbertianity for manifolds with Sobolev regular Riemannian metrics through uniform lower capacity bounds
- An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations
- Riemannian Hessian analysis of Sobolev gradient flows for the Gross--Pitaevskii ground state problem
BibTeX
@misc{allen2026sobolev,
title = {{Sobolev Inequalities Relating Rough Riemannian Metrics to their Distance Functions}},
author = {Brian Allen and Lisa Naples},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37884},
eprint = {2610.37884},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37884},
}
RIS
TY - UNPB
TI - Sobolev Inequalities Relating Rough Riemannian Metrics to their Distance Functions
AU - Brian Allen
AU - Lisa Naples
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37884
UR - https://arxiv.org/abs/2610.37884
AB - In previous papers, the first named author and E. Bryden explored Sobolev like inequalities relating the norm of a Riemmanian metric g_1, in terms of a background Riemannian metric g_0, in order to estimate the distance function d_g_1. Although |g_1|_g_0 is not the gradient of the distance function d_g_1, it was shown that a Morrey and Sobolev inequalities hold in their appropriate domains. In this paper we see that these results can be extended to the case where g_1 is a Rough Riemannian metric, something roughly analogous to an L^∞ metric. We show that the proof of the Morrey and Sobolev inequality go through, with appropriate adjustments, as well as convergence results. We also adapt previous examples from the literature to the rough Riemannian metric case which show the hypotheses from the main theorems are necessary.
ER -