Bibliographic Information: Assouline, R. (2024). Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds. arXiv preprint arXiv:2409.08001v2.
Research Objective: This paper investigates the relationship between magnetic Ricci curvature and the Brunn-Minkowski inequality on Riemannian manifolds endowed with a magnetic field. The author aims to establish a magnetic analogue of the known equivalence between nonnegative Ricci curvature and the Brunn-Minkowski inequality in the standard Riemannian setting.
Methodology: The author introduces the concept of magnetic Ricci curvature, denoted as mRic, which incorporates the geometry of the magnetic field. By employing a needle decomposition technique and analyzing the behavior of magnetic geodesics, the author derives a magnetic version of the Borell-Brascamp-Lieb inequality. This inequality is then used to prove the equivalence between nonnegative mRic and the magnetic Brunn-Minkowski inequality.
Key Findings: The paper's central result is the proof that non-negative magnetic Ricci curvature (mRic ≥ 0) is both necessary and sufficient for the magnetic Brunn-Minkowski inequality to hold on a Riemannian manifold equipped with a magnetic field. This result is established by first proving a more general weighted magnetic Borell-Brascamp-Lieb inequality.
Main Conclusions: This work provides a novel characterization of magnetic Ricci curvature through the lens of geometric inequalities. The equivalence between mRic ≥ 0 and the magnetic Brunn-Minkowski inequality offers a new perspective on the interplay between curvature, magnetic fields, and volume growth in the context of Riemannian manifolds.
Significance: This research significantly contributes to the fields of Riemannian geometry and geometric analysis by introducing new tools and concepts for studying manifolds with magnetic fields. The magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities, along with the notion of magnetic Ricci curvature, provide a framework for investigating geometric and analytical properties of these spaces.
Limitations and Future Research: The paper primarily focuses on smooth manifolds and smooth magnetic fields. Exploring similar relationships in the context of less regular spaces and magnetic fields could be a promising avenue for future research. Additionally, investigating potential applications of these results in other areas, such as optimal transport or geometric measure theory, could be of interest.
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by Rotem Assoul... at arxiv.org 10-10-2024
https://arxiv.org/pdf/2409.08001.pdfDeeper Inquiries