Neckpinch — Discrete & graph Ricci

Two-step nilpotent Lie algebras obtained by quivers and geometric structures

Mon, 19 Oct 2026 00:00:00 GMT

Fumika Mizoguchi — In the study of geometric structures on nilmanifolds, two-step nilpotent Lie algebras obtained by graphs play an important role. Recently, additional examples of nilpotent Lie algebras have been constructed from finite quivers without cycles. These latter examples can have arbitrarily high degrees of nilpotency, and admit Riemannian Ricci soliton metrics. In this paper, we study nilpotent Lie algebras obtained by finite quivers without cycles that are two-step nilpotent, and we prove that they can also be obtained by graphs. Using this relationship, we demonstrate that every two-step nilpotent Lie algebra obtained by a finite quiver without cycles admits a pseudo-Riemannian Ricci-flat metric. Additionally, we also classify these nilpotent Lie algebras that admit symplectic structures.

Combinatorial Calabi flows on infinitely triangulated surfaces

Mon, 19 Oct 2026 00:00:00 GMT

Yuerong Bian, Xu Xu — We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the L^∞ norm.

Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

Mon, 28 Sep 2026 00:00:00 GMT

Soichiro Uemura — We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.

Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

Thu, 17 Sep 2026 00:00:00 GMT

Ping-Hung Lee — We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.

Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees

Fri, 04 Sep 2026 00:00:00 GMT

Shuliang Bai, Bobo Hua — We study the inverse-weight Lin–Lu–Yau Ricci flow on finite trees, and prove the convergence of the curvature on every edge via its diffusion equation. Moreover, the limiting curvature is the unique minimum-norm point of a polyhedron determined by the tree. We characterize possible limits of the normalized edge weights, give criteria for their convergence, and show that different positive initial metrics have equivalent omega-limit sets.

Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry

Sat, 08 Aug 2026 00:00:00 GMT

Yeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang — We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.

Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs

Thu, 06 Aug 2026 00:00:00 GMT

Shiping Liu, Yunyan Yang — In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the p-Ollivier curvature up to scaling whenever the idleness parameter p≥ 1/2. Moreover, the threshold 1/2 is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

Tue, 28 Jul 2026 00:00:00 GMT

Yuerong Bian, Xu Xu — We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.

Infinite Combinatorial Yamabe Flows in Three Dimensions

Sun, 26 Jul 2026 00:00:00 GMT

Bohao Ji — In this paper, we study three-dimensional combinatorial Yamabe flows on locally finite infinite triangulations in Euclidean and hyperbolic background geometries. Under suitable non-degeneracy and bounded-degree assumptions, we establish the short-time existence and uniqueness of the original flows. We further introduce the extended flows by using the continuous extension of solid angles, and prove long-time existence for both extended flows under suitable initial assumptions.

Solvable Lie algebras obtained by quivers and Einstein metrics

Wed, 08 Jul 2026 00:00:00 GMT

Fumika Mizoguchi — In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers. Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons. Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.

Evolving edge weights via local entropy flow and cohesion flow on graphs

Thu, 25 Jun 2026 00:00:00 GMT

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao — In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

Nilpotent Lie algebras obtained by ordered sets and Ricci solitons

Tue, 23 Jun 2026 00:00:00 GMT

Yihao Zheng, Shenglin Zhu — Nilpotent Lie groups with left-invariant metrics provide nontrivial examples of Ricci solitons. Some typical examples are given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs and the class of nilpotent Lie algebras obtained from finite acyclic quivers. In this paper, we generalize the construction of nilpotent Lie algebras that are algebraic Ricci solitons obtained from finite acyclic quivers. We use some special ordered sets to construct nilpotent Lie algebras, which can also be obtained from some special quivers with relations. A transitively and antisymmetrically ordered set (or TAOS, for short) is a set together with a binary relation that is transitive and antisymmetric. Utilizing the concept of incidence algebras of TAOSs, we construct nilpotent Lie algebras. We modify the method introduced by Mizoguchi and Tamaru and use it to show that the nilpotent Lie algebras with arbitrarily high degrees of nilpotency obtained from some special finite transitively and antisymmetrically ordered sets, called array TAOSs, are algebraic Ricci solitons. We also give some generalizations of this result, which yield more nilpotent Lie algebras that are algebraic Ricci solitons. Moreover, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons.

The Ollivier Ricci flow with prescribed curvature on infinite graphs

Mon, 08 Jun 2026 00:00:00 GMT

Bobo Hua, Yong Lin, Shuang Liu — In this paper, we consider the Ricci flow with prescribed curvature on infinite graphs, which reads as \beginequation* \fracddtω(t)=-(κ(t)-κ^*)ω(t), t>0, \endequation* where ω is the edge weight, κ and κ^* are Lin-Lu-Yau Ricci curvature and the prescribed curvature on the set of edges, respectively. First, we establish the existence and uniqueness of the solution to the Ricci flow. Furthermore, we prove the convergence of the Ricci flow for graphs with girth at least 6 under two different conditions. Our convergence result aligns with the conclusion of Rodin and Sullivan (J Differ Geom, 26(2) 1987) that a circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

The Calabi flow with prescribed curvature on finite graphs

Fri, 03 Apr 2026 00:00:00 GMT

Yi Li, Jie Wang, Pingsan Yuan, Chao Zheng — In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.

The Ricci flow with prescribed curvature on graphs

Wed, 11 Mar 2026 00:00:00 GMT

Yong Lin, Shuang Liu — In this paper, we consider the Ricci flow with prescribed curvature on the finite graph G=(V,E). For any e in E, (dω(t,e))/dt = −(κ(t,e)−κ^*(e))ω(t,e), t > 0, where ω is the weight function, κ is Lin-Lu-Yau Ricci curvature, and κ^* is the prescribed curvature. By imposing invariance of the graph distance with respect to time t, the Ricci flow introduced above characterizes the weight evolution governed by the Lin-Lu-Yau curvature. We first establish the existence and uniqueness of the solution to this equation on general graphs. Furthermore, for graphs with girth of at least 6, we prove that the Ricci flow converges exponentially to weights of κ^* if and only if κ^* is attainable (namely, there exist weights realizing κ^*). In particular, we prove that the weights for constant curvature exist if and only if max_(∅ ≠ Ωsubsetneq V) (|E(Ω)|)/(|Ω|) < (|E|)/(|V|), where E(Ω) denotes the set of edges within the induced subgraph of Ω, and |A| is the cardinality of the set A. Viewing edge weights as metrics on surface tilings with girth of at least 5 or the duals of triangulations with vertex degrees exceeding 5, we demonstrate that our constant Lin-Lu-Yau curvature flow serves as an analog to the 2D combinatorial Ricci flow for piecewise constant curvature metrics, thereby providing an affirmative answer to Question 2 posed by Chow and Luo (J Differ Geom, 63(1) 2002).

The character of ideal circle patterns

Mon, 23 Feb 2026 00:00:00 GMT

Chang 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.

Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary

Wed, 21 Jan 2026 00:00:00 GMT

Xinrong Zhao — In this paper, we study a combinatorial Ricci flow on closed pseudo 3-manifolds (M,𝓣). We prove that if every edge in the triangulation 𝓣 has valence at least 9, then the combinatorial Ricci flow converges exponentially fast to the unique zero-curvature hyper-ideal metric. As a consequence, for any compact 3-manifold N with boundary admitting an ideal triangulation 𝓣_N whose edges all have valence at least 9, there exists a unique complete hyperbolic metric with totally geodesic boundary on N such that 𝓣_N is isotopic to a geometric decomposition of N. This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer to Thurston's geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.

The weighted Forman and Lin-Lu-Yau Ricci flow on graphs

Tue, 06 Jan 2026 00:00:00 GMT

Shuliang Bai, Shuang Liu, Xin Lai — In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results.

Neural Feature Geometry Evolves as Discrete Ricci Flow

Fri, 26 Sep 2025 00:00:00 GMT

Moritz Hehl, Max von Renesse, Melanie Weber — Deep neural networks learn feature representations via complex geometric transformations of the input data manifold. Despite the models' empirical success across domains, our understanding of neural feature representations is still incomplete. In this work we investigate neural feature geometry through the lens of discrete geometry. Since the input data manifold is typically unobserved, we approximate it using geometric graphs that encode local similarity structure. We provide theoretical results on the evolution of these graphs during training, showing that nonlinear activations play a crucial role in shaping feature geometry in feedforward neural networks. Moreover, we discover that the geometric transformations resemble a discrete Ricci flow on these graphs, suggesting that neural feature geometry evolves analogous to Ricci flow. This connection is supported by experiments on over 20,000 feedforward neural networks trained on binary classification tasks across both synthetic and real-world datasets. We observe that the emergence of class separability corresponds to the emergence of community structure in the associated graph representations, which is known to relate to discrete Ricci flow dynamics. Building on these insights, we introduce a novel framework for locally evaluating geometric transformations through comparison with discrete Ricci flow dynamics. Our results suggest practical design principles, including a geometry-informed early-stopping heuristic and a criterion for selecting network depth.

On the Ricci flow on Trees

Fri, 26 Sep 2025 00:00:00 GMT

Shuliang Bai, Bobo Hua, Yong Lin, Shuang Liu — In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

Ricci Flow on Weighted Digraphs with Balancing Factor

Wed, 24 Sep 2025 00:00:00 GMT

Shuliang Bai, Rui Li, Shuang Liu, Xin Lai — Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph representation learning. However, their development on directed graphs remains limited, with Ricci flow being especially underexplored. In this work, we introduce a rigorous formulation of Ricci flow on directed weighted graphs, which evolves edge weights while preserving distances, and establish both the existence and uniqueness of its solutions. To capture the essence of asymmetry in directed networks and to enhance the capability of modeling more flexible structures, we incorporate a node-wise balancing factor that regulates between outflow and inflow. Building on the continuous Ricci flow evolution framework, we propose a discrete Ricci flow algorithm that is applicable to numerical computing. Numerical studies on various directed graph examples demonstrate the capacity of the proposed flow to reveal structural asymmetry and dynamic evolutions.

Core detection via Ricci curvature flows on weighted graphs

Sat, 02 Aug 2025 00:00:00 GMT

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao — Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

Discrete conformal structures on surfaces with boundary (III) – Deformation

Thu, 24 Jul 2025 00:00:00 GMT

Xu Xu, Chao Zheng — The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

The existence and uniqueness of infinite combinatorial Yamabe flows

Wed, 16 Jul 2025 00:00:00 GMT

Bohao Ji — In this paper, we study the combinatorial Yamabe flow on infinite triangulated surfaces in Euclidean background geometry, aiming for solving discrete Yamabe problem on noncompact surfaces. Under suitable conditions, we establish the short-time existence and uniqueness of the flow. We further introduce an extended version of the flow and prove its long-time existence. As an application, we prove the convergence result of the Yamabe flow in the case of hexagonal triangulations of the plane.

Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts

Sun, 06 Jul 2025 00:00:00 GMT

Mauricio Angel — We present a comprehensive analysis of Bredon's trick, a powerful local-to-global extension principle with broad applications across differential geometry and computational topology. Our main contributions include: (1) novel applications to stratified pseudomanifolds via Verona cohomology with explicit verification of axiomatic conditions; (2) new frameworks for Ricci flow singularity analysis using local curvature concentration; (3) stability theorems for persistent homology in distributed computational settings; and (4) rigorous applications to medical imaging and neural network topology. By systematically developing the theoretical foundations and providing concrete implementations, this work establishes Bredon's trick as a unifying framework for modern local-to-global arguments in geometric analysis and applied topology.

Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

Thu, 05 Jun 2025 00:00:00 GMT

Huabin Ge, Bobo Hua, Hao Yu, Puchun Zhou — In his seminal work, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.

Branched α-combinatorial Ricci flows on closed surfaces with Euler characteristic χ≤ 0

Fri, 30 May 2025 00:00:00 GMT

Wenjun 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.

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

Mon, 26 May 2025 00:00:00 GMT

Xinrong Zhao, Puchun Zhou — In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

Infinite combinatorial Ricci flow in spherical background geometry

Fri, 09 May 2025 00:00:00 GMT

Chang Li, Yangxiang Lu, Hao Yu — Since the fundamental work of Chow-Luo, Ge et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci flow with prescribed total geodesic curvatures in spherical background geometry. For infinite cellular decompositions, we establish the existence of a solution to the flow equation for all time. Furthermore, under an additional condition, we prove that the solution converges as time tends to infinity. To the best of our knowledge, this is the first study of an infinite combinatorial curvature flow in spherical background geometry.

Generalized circle patterns on surfaces with cusps

Sat, 12 Apr 2025 00:00:00 GMT

Zhiwen Xiong, Xu Xu — Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.