Displacement convexity of Invariant Measures and curvature bounds of isometric actions

[2310.15332] Displacement convexity of Invariant Measures and curvature bounds of isometric actions Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2310.15332 (math) [Submitted on 23 Oct 2023 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: Displacement convexity of Invariant Measures and curvature bounds of isometric actions Authors: André Magalhães de Sá Gomes , Christian S. Rodrigues View PDF HTML (experimental) Abstract: Let $G\curvearrowright M$ be a proper isometric action of a compact Lie group on a complete, connected and orientable Riemannian manifold of dimension $N$. We characterize the local $K$-displacement convexity of the internal-energy functional $H$ on the space $\mathcal P^{ac}_G(M)$ of absolutely continuous $G$-invariant probability measures. Via disintegration along the principal orbits, $H$ reduces to the internal energy of a transversal density against the orbit-volume--weighted measure $\mathfrak m = V\operatorname{vol}_{M/G}$, and the convexity of $H$ is equivalent to the $N$-Bakry--Émery condition $\operatorname{Ric}^{\Psi}_N \ge K$ on the weighted quotient $(M/G,\mathfrak m)$. Written on $M$, this bound reads $\operatorname{Ric}^{\mathcal H}_M(v) + 3\|A_v\|^2 - \operatorname{Hess}(\log V)(v,v) - \langle\vec H,v\rangle^2/m \ge K|v|^2$ at every principal point and horizontal direction $v$, where $A$ is the O'Neill integrability tensor of $\pi\colon M\to M/G$, $\vec H$ the mean-curvature vector of the orbits, and $m=\dim(G\cdot x)$ the orbit dimension. The terms on the left encode, in this order, the horizontal curvature of $M$, the non-integrability of the horizontal distribution, and --- in the last two --- the variation of the orbit volume. When the orbits are points one recovers the theorem of von Renesse--Sturm. Comments: Revised and extended version, 47 pages Subjects: Differential Geometry (math.DG) ; Dynamical Systems (math.DS); Probability (math.PR) Cite as: arXiv:2310.15332 [math.DG] (or arXiv:2310.15332v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2310.15332 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Christian S. Rodrigues [ view email ] [v1] Mon, 23 Oct 2023 19:54:06 UTC (22 KB) [v2] Wed, 26 Aug 2026 21:02:08 UTC (48 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2023-10 Change to browse by: math math.DS math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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