{"paper":{"title":"Geodesic Rays and K\\\"ahler-Ricci Trajectories on Fano Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Tam\\'as Darvas, Weiyong He","submitted_at":"2014-11-04T03:28:13Z","abstract_excerpt":"Suppose $(X,J,\\omega)$ is a Fano manifold and $t \\to r_t$ is a diverging K\\\"ahler-Ricci trajectory. We construct a bounded geodesic ray $t \\to u_t$ weakly asymptotic to $t \\to r_t$, along which Ding's $\\mathcal F$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fields this proves the equivalence between geodesic stability of the $\\mathcal F$-functional and existence of K\\\"ahler-Einstein metrics. We also explore applications of our construction to Tian's $\\alpha$-invariant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0774","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}