Ricci Flow IS a Renormalization Group Flow: Where Two Lineages Shake Hands over Taming the Infinite, a Census of a Royal-Road Bridge.
Two lineages run through this corpus: the royal road of GEOMETRY (Riemann -> … -> Ricci flow / Perelman, 91/103) and the royal road of ANALYSIS / FIELD THEORY (the 1/16pi^2 of loop integrals, the renormalization group, 109). This paper records, as a LAYER-2 CENSUS of a BRIDGE, that the two shake hands not by metaphor but by ONE PROVEN EQUATION. Keystone = FRIEDAN’s theorem (1980): the renormalization-group (RG) beta function of a 2D nonlinear sigma model’s target metric g_ij equals, at one loop, the Ricci tensor itself: beta^g_ij = alpha’ R_ij + O(alpha’^2). Hence the RG flow of the metric IS Ricci flow (dg/dt = -alpha’ R_ij vs Hamilton-Perelman dg/dtau = -2 R_ij, up to a constant and direction). The fixed point beta=0 is Ricci-flat = the vacuum Einstein equation (the route by which string theory “derives” gravity; CFMP 1985). Shared backbone = TAMING THE INFINITE (geometry: curvature blow-up / surgery; field theory: UV divergences / renormalization). The paper sorts findings into THREE TIERS: GOLD (proven) = Ricci flow = RG flow (Friedan); SILVER (a real resonance, UNSETTLED) = Perelman’s monotone entropy <-> Zamolodchikov’s c-theorem (both give the flow an arrow); GOLD-PLATED (a RHYME) = surgery ~ renormalization (same theme, DIFFERENT machinery: topological excision vs absorbing divergences — not used as proof). HONEST LINE: the identity is exact only at one loop (alpha’ leading order); higher orders add higher-curvature corrections. IMPORTANT (scope): a Layer-2 structural interpretation / census of established results (Friedan, Perelman, Zamolodchikov, the string derivation of GR); no new theorem and no prediction; NOT a GUT — one bridge (102-type); resonance is a signature not a proof (100); no post-hoc adjustment (101). The Universe OS (“the universe renormalizing itself”) is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Ricci Flow IS a Renormalization Group Flow [Paper 110]. Zenodo.
DOI: 10.5281/zenodo.21837476
ORCID: 0009-0006-8587-3450