Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds

Glickenstein, David

doi:10.4310/jdg/1304514973

Cited by 86 publications

(112 citation statements)

References 40 publications

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“…Such a Laplacian also arises in certain variations of curvature formulas, such as in [14,47]. The work of He [36] and subsequent work [21,27,31] describe the connection between the two forms of the Laplacian. The main point is that a geometric structure is necessary on the Poincaré dual of the triangulation.…”

Section: Curvature and Laplacianmentioning

confidence: 99%

“…Hyperbolic embedding theorems from dihedral angle data had been treated by Hodgson and Rivin [39] and more recently [24,25]. Sphere packing metrics have been studied by a number of authors [15,[26][27][28], and there is some general theory on angle variations in three-dimensional piecewise Euclidean manifolds in [31] and further work on hyperbolic manifolds in [60]. Curvature flow on hyperbolic 3-manifolds with totally geodesic boundary was also studied in [42].…”

Section: Three Dimensionsmentioning

confidence: 99%

“…in the Euclidean background, and there is an extra linear reaction term in the other constant curvature backgrounds (see [31] for Euclidean background and [32] and [66] for other constant curvature backgrounds). Thus the stability of the zero curvature problem is related to the definiteness of the Laplacian.…”

Section: Application Of Curvature Flowmentioning

confidence: 99%

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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: Curvature and Laplacianmentioning

confidence: 99%

Section: Three Dimensionsmentioning

confidence: 99%

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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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“…More recent contributions are due to Alsing et al [18] Glickenstein [19] and Trout [20]. However, our approach is quite different and owns nothing to the mentioned works.…”

Section: Introductionmentioning

confidence: 99%

Metric Ricci Curvature forPLManifolds

Saucan

2013

Geometry

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“…While SRF is one approach to extending combinatorial Ricci flow from two to higher dimensions, there are other active and independent approaches that are being explored. In particular, Yin et al [15] study discrete curvature flow for hyperbolic 3-manifolds whose boundary consists of high genus surfaces, where Glickenstein [13,14] studies discrete conformal variations and scalar curvature on piecewise flat two and threedimensional manifolds and constructs discrete Laplacians on manifolds. In addition, Ge and Xu [16] define discrete quasi-Einstein metrics as critical points for discrete total curvature on simplicial 3-manifolds, where Forman [17] defined a new notion of curvature for cell complexes corresponding to the Ricci curvature for Riemannian manifolds.…”

Section: Exploring Simplicial Ricci Flow In 3dmentioning

confidence: 99%

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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Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds

Cited by 86 publications

References 40 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

Metric Ricci Curvature forPLManifolds

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

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