https://neckpinch.com/authors/aijin-lin/New arXiv papers by Aijin Lin, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2602.19748arxiv:2602.19748Mon, 23 Feb 2026 00:00:00 GMTChang Li, Aijin Lin, Liangming Shen — Let S be an oriented closed surface with a cellular decomposition 𝓓 and a weight Φ∈(0, π). It is crucial to determine when S supports an ideal 𝓓-type circle pattern 𝓟 with the exterior intersection angles given by Φ. Rivin, Bobenko-Springborn and Ge-Hua-Zhou provided perfect solutions and gave wonderful criteria for the existence and uniqueness of ideal circle patterns. However, all criteria established by Rivin, Bobenko-Springborn and Ge-Hua-Zhou are extremely difficult to verify for the given cellular decomposition 𝓓 and the weight Φ. In this paper, we introduce the character 𝓛(𝓓,Φ) depends only on the data of the weighted cellular decomposition (𝓓, Φ) on S, and give some quite simple criteria for the existence of ideal circle patterns realizing (𝓓,Φ). It seems that our character-type criteria are the first conditions totally different from criteria of Rivin, Bobenko-Springborn and Ge-Hua-Zhou, and provide more easily verifiable criteria. Our new character-type theorems may be of some independent interest. As an application, we give a new descriptions of the curvature image set K(ℝ^N_(>0)). To approach our results, we shall use the combinatorial Ricci flows with ideal circle patterns introduced by Ge-Hua-Zhou as a fundamental tool. The main difficulty in the proof of our results is to establish the compactness of the solution to the flows. To circumvent the difficulty, we borrow the techniques developed by Ge and his collaborators.Ricci flowLimit spaces & RCDStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2507.03901arxiv:2507.03901Sat, 05 Jul 2025 00:00:00 GMTAijin Lin, Longxiang Wu — Ge in his thesis introduced the combinatorial Calabi flows and established the long time existence and convergence of solutions to the flows in both hyperbolic and Euclidean background geometries. It is noteworthy that the existence of solutions to the combinatorial Calabi flows in hyperbolic background geometry proves to be more intricate and challenging compared to the Euclidean background geometry. The main difficulty is to establish the compactness, especially the lower boundeness along the flow equations. In this paper, we give two explicit estimates for the discrete Laplace operator based on the Glickenstein-Thomas formulation for discrete hyperbolic conformal structures. As applications, we give new proofs of the long time existence of solutions to the combinatorial Calabi flows established by Ge-Xu, Ge-Hua and the combinatorial p-th Calabi flows established by Lin-Zhang in hyperbolic background geometry.Other curvature flowsKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2505.24762arxiv:2505.24762Fri, 30 May 2025 00:00:00 GMTWenjun Li, Rongyuan Liu, Guohao Chen, Aijin Lin — In this paper we introduce the branched α-flows on closed surfaces with Euler characteristic χ≤ 0. Based on the strict convexity of the branched α-potentials, we establish the long time existence and convergence of the solutions to the branched α-flows, which generalizes Ge and Xu's main results on the α-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition χ(M)∈ ℤ via alternative α-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.Ricci flowSurfacesLong-time existence & convergenceDiscrete & graph Ricci