Simplicial Ricci Flow

Miller, Warner A.; McDonald, Jonathan R.; Alsing, Paul M.; Gu, Daqing; Yau, Shing–Tung

doi:10.1007/s00220-014-1911-6

Cited by 25 publications

(36 citation statements)

References 33 publications

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“…Curvature flow on hyperbolic 3-manifolds with totally geodesic boundary was also studied in [42]. An alternative approach of simplicial Ricci flow was proposed in [48] and successive papers.…”

Section: Three Dimensionsmentioning

confidence: 99%

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Chow¹,

Glickenstein²,

Luo³

2016

Bull. Amer. Math. Soc.

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One of the goals in the field of modern geometric analysis is to study canonical structures on manifolds and vector bundles-existence, uniqueness, and moduli. One of the pioneers of analytical approaches in this field is Yau, who, with Schoen and other coauthors solved a number of outstanding geometric problems, many of which are related to elliptic PDEs. A personal account of these developments is in the two-volume set [65]. Around the same time, there were remarkable developments in other areas of geometry. A more geometric theory, including the fundamental theories of compactness and collapse, was developed by Cheeger, Gromov, and others. Thurston revolutionized low-dimensional topology by developing a geometric theory aimed at understanding 3-manifolds via the geometrization conjecture and related ideas. Soon after, inspired by the earlier work of Eells and Sampson on harmonic maps, Hamilton invented the Ricci flow, a parabolic PDE, and through a number of very original works nearly single-handedly developed it into a compelling program to approach the Poincaré and geometrization conjectures. Two decades later, Perelman solved the Poincaré and geometrization conjectures by introducing a plethora of deep and powerful ideas and methods into Ricci flow to complete Hamilton's program. At the present time, more than another decade later, geometric analysis is a thriving field with many active subfields, including Kähler geometry, influenced by the work of Tian, Donaldson, Chen, and others, the study of Ricci curvature by Cheeger, Colding, Minicozzi, and others, geometric flows such as the mean curvature flow by Huisken, White, and others, and the work of Brendle to solve long-standing conjectures, to just name a few of the many directions this field has taken.The idea that a discrete set of values determines geometry is the main tenet of discrete differential geometry. These ideas come from a number of disparate directions. Whitney used numerical analysis to connect de Rham theory and simplicial homology [64], and these ideas form the central tenets of discrete exterior calculus [20]. In a separate intellectual stream, Regge introduced a notion of discrete geometry that he considered a way of developing computational and quantum gravity from first principles that were motivated by, but not dependent on, the theory of smooth manifolds (see [34,51]). Thurston introduced the idea of circle packings for approximating Riemann mappings, and this developed into a whole field of circle packing and related topics (see Stephenson's book [58], reviewed in [6]). A discrete form of integrable systems was developed by Bobenko and Suris [5] into a geometric theory, while other groups [1,23] have developed similar or competing theories arising from numerical analysis and mathematical physics. All of these directions may be characterized as structure-preserving discretizations.

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Section: Three Dimensionsmentioning

confidence: 99%

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Chow¹,

Glickenstein²,

Luo³

2016

Bull. Amer. Math. Soc.

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“…We refer to this discretization as simplicial Ricci flow (SRF) [20]. SRF is founded on Regge calculus and upon the mathematical foundations of Alexandrov [9,21,22,23].…”

Section: Exploring Simplicial Ricci Flow In 3dmentioning

confidence: 99%

“…The recent definition of the SRF equations and the dual-edge SRF equations in [20] made use of elements from both the simplicial lattice geometry (in this case the frustum lattice geometry with end caps, F), and from its circumcentric dual lattice F * . The SRF and dual-edge SRF equations for the S 3 frustum geometry, F are functions of the (N a − 1) axial edges,…”

Section: Theorem: the Srf Equations For The Frustum Geometry Are The mentioning

confidence: 99%

“…We use the results of the Appendix and the definitions in [20] to examine each term of this equation and to examine the Taylor series expansion to obtain the continuum limit expression. We do this in three steps by examining (1) the two circumcentric volume weighting factors on the right-hand side of the dual SRF equation, (2) the two Gaussian curvature expressions also on the right-hand side of Eq.…”

Section: The Dual α I -Edge Srf Equation At the Point O I And Its Conmentioning

confidence: 99%

“…In [20] we defined the Gaussian curvature of edge s i to be the deficit angle, si , and distributed uniformly over the circumcentric dual area s * i ,…”

Section: Gaussian Curvaturementioning

confidence: 99%

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Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Miller

Alsing

Corne

et al. 2014

Geometry, Imaging and Computing

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Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d ≥ 2. The structure of the simplicial Ricci flow (SRF) equations is dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that the SRF equations converge to the corresponding continuum RF equations as reported by Angenent and Knopf. Exploring simplicial Ricci flow in 3DHamilton's Ricci flow (RF) continues to yield new insights into problems in pure and applied mathematics and proves to be a useful tool across a broad spectrum of engineering fields [1,2,3,4]. Here the time evolution of the metric is proportional to the Ricci tensor,Hamilton showed that this yields a forced diffusion equation for the curvature, e.g. the scalar curvature evolves asThe bulk of the applications of this curvature flow have utilized the numerical evolution of piecewise-flat simplicial 2-surfaces [5,6]. It is a widely accepted verity in computational science that a geometry with complex topology is most naturally represented in a coordinate-free way by an unstructured mesh. This is apparent in the engineering applications utilizing arXiv: 1404.4055 333

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

2016

Calc. Var.

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In this paper, we introduce two discrete curvature flows, which are called α-flows on two and three dimensional triangulated manifolds. For triangulated surface M , we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo [3]), aiming at evolving α order discrete Gauss curvature to a constant. When αχ(M ) ≤ 0, we prove that the convergence of the flow is equivalent to the existence of constant α-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of Thurston's combinatorial-topological condition, for the existence of constant α-curvature metric. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in [7,9] to α-order. We also study the α-order flow carefully, aiming at evolving α order combinatorial scalar curvature, which is a generalization of Cooper and Rivin's combinatorial scalar curvature, to a constant.

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Simplicial Ricci Flow

Cited by 25 publications

References 33 publications

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

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

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