$$\alpha $$ α -curvatures and $$\alpha $$ α -flows on low dimensional triangulated manifolds

Ge, Huabin; Xu, Xu

doi:10.1007/s00526-016-0951-5

Cited by 22 publications

(14 citation statements)

References 21 publications

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“…The proof of this theorem is similar to those appeared in [3,5,6]. We show that g * is an asymptotically stable point of the normalized discrete Ricci flow (5.2).…”

Section: Proofsupporting

confidence: 57%

“…If λ inf (∆ * L ) > λ * , then by similar methods in [3,5,6], we can show that all eigenvalues of the matrix D g * Γ(g) are negative (up to scaling of metric l Similarly, for any α ∈ R − {0, −1}, we can also define the normalized α-order discrete Ricci flow as…”

Section: Proofmentioning

confidence: 96%

“…Similar to the works by Ge and Xu [4,5], almost all procedures about discrete metric g i and discrete curvature R i can be generalized to α order. We can define α-order discrete Einstein metric, i.e.…”

Section: Discrete Einstein Metric Of α-Ordermentioning

confidence: 99%

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A Note on Discrete Einstein Metrics

Ge¹

2019

JMS

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In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge's Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version in [3]. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.

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“…The proof of this theorem is similar to those appeared in [3,5,6]. We show that g * is an asymptotically stable point of the normalized discrete Ricci flow (5.2).…”

Section: Proofsupporting

confidence: 57%

Section: Proofmentioning

confidence: 96%

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A Note on Discrete Einstein Metrics

Ge¹

2019

JMS

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“…2 is reduced to the parameterized combinatorial curvature for circle packings studied in [9,11,[13][14][15][16]29]. If ε ≡ 0, the combinatorial curvature R α in Definition 1.2 is reduced to the parameterized combinatorial curvature for vertex scaling studied in [33,35].…”

Section: Methodsmentioning

confidence: 99%

“…In the special case of α = 0, Theorem 1.3 was proved by the first author in [34]. If ε i = 1 for all i ∈ V , Theorem 1.3 is reduced to the rigidity of circle packings on surfaces obtained in [2,9,13,15,16,23,27,29] and others. If ε i = 0 for all i ∈ V , Theorem 1.3 is reduced to the rigidity of vertex scaling on polyhedral surfaces obtained in [1,26,33,35] and others.…”

Section: Methodsmentioning

confidence: 99%

Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

Xu¹,

Chen²

2021

Preprint

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Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. It unifies and generalizes tangential circle packing, Thurston's circle packing, inversive distance circle packing and the vertex scaling on polyhedral surfaces. In this paper, we introduce combinatorial α-curvature for discrete conformal structures on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. Then we prove the local and global rigidity of combinatorial α-curvature with respect to discrete conformal structures on polyhedral surfaces, which confirms parameterized Glickenstein rigidity conjecture in [19]. To study the Yamabe problem for combinatorial α-curvature, we introduce combinatorial α-Ricci flow for discrete conformal structures on polyhedral surfaces, which is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings [2] and Luo's combinatorial Yamabe flow for vertex scaling [26] on polyhedral surfaces. To handle the potential singularities of the combinatorial α-Ricci flow, we extend the flow through the singularities by extending the inner angles in triangles by constants. Under the existence of a discrete conformal structure with prescribed combinatorial curvature, the solution of extended combinatorial α-Ricci flow is proved to exist for all time and converge exponentially fast for any initial value. This confirms a parameterized generalization of another conjecture of Glickenstein in [19] on the convergence of combinatorial Ricci flow, gives an almost equivalent characterization of the solvability of Yamabe problem for combinatorial α-curvature in terms of combinatorial α-Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial α-curvatures.

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Combinatorial Calabi flow with surgery on surfaces

Zhu

2019

Calc. Var.

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We study the combinatorial Calabi flow for Euclidean and hyperbolic polyhedral metrics on surfaces, which is an analogue of the smooth surface Calabi flow. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Then we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast to a constant combinatorial curvature metric.Mathematics Subject Classification (2010). 53C44, 52B70.

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$$\alpha $$ α -curvatures and $$\alpha $$ α -flows on low dimensional triangulated manifolds

Cited by 22 publications

References 21 publications

A Note on Discrete Einstein Metrics

A Note on Discrete Einstein Metrics

Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

Combinatorial Calabi flow with surgery on surfaces

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