https://neckpinch.com/authors/yuan-liao/New arXiv papers by Yuan Liao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.34665arxiv:2610.34665Mon, 26 Oct 2026 00:00:00 GMTYuan Liao — We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)²; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.Ricci flowGeneralized & G₂ flowsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2610.13897arxiv:2610.13897Fri, 09 Oct 2026 00:00:00 GMTYuan Liao, Ziyang Qin, Jui-Hui Chung, Ayush Khaitan, Zhengyang Zhang, Hongzhou Lin, Zijian Zhang, Ligeng Zhu, Peng Lu, Bennett Chow — We formalize the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture. The public Lean declaration is reproduced in this paper. Its geometric interpretation follows, and verification is recorded in 4.Ricci flowGeometrization https://arxiv.org/abs/2609.33842arxiv:2609.33842Sun, 27 Sep 2026 00:00:00 GMTZiyang Qin, Yuan Liao, Ayush Khaitan, Bennett Chow — We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.Ricci flowSingularitiesSurgeryPoincaré conjecture3-manifolds https://arxiv.org/abs/2608.21502arxiv:2608.21502Fri, 21 Aug 2026 00:00:00 GMTBennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.Ricci flowSingularities(Non)collapsingCurvature pinching3-manifoldsLimit spaces & RCDHarmonic maps & DeTurck trick