{"paper":{"title":"Moment map, convex function and extremal point","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.SG"],"primary_cat":"math.DG","authors_text":"Jacob Sturm, King leung Lee, Xiaowei Wang","submitted_at":"2022-08-07T13:54:47Z","abstract_excerpt":"The moment map $\\mu$ is a central concept in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this short note, we propose a theory of moment maps coupled with an $\\mathrm{Ad}_K$-invariant convex function $f$ on $\\mathfrak{k}^{\\ast}$, the dual of Lie algebra of $K$, and study the properties of the critical point of $f\\circ\\mu$. Our motivation comes from Donaldson \\cite{Donaldson2017} which is an example of infinite dimensional version of our setting. As an application, we interpret K\\\"ahler-Ricci solitons as a special case of the generalized extremal metric"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.03724","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.03724/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}