A maximum principle for combinatorial Yamabe flow

Glickenstein, David

doi:10.1016/j.top.2005.02.002

Cited by 49 publications

(61 citation statements)

References 16 publications

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“…The proof is essentially the same as the corresponding theorem for Delaunay triangulations, which goes back at least to Rivin [23]. Details may be found in [12] or simply proved by looking at when the length of the dual edge is equal to zero. The formula for h ij,k makes it easy to see that the condition for being weighted Delaunay is unchanged by a weight scaling of the type {w i → w i + c} i∈T 0 , where c is some constant independent of i, since h ij,k is unaffected by such a deformation.…”

Section: Definition 4 a Weighted Triangulation (T W) Is A Triangulamentioning

confidence: 83%

“…When the centers of triangles lie inside the triangles, it is possible to show that solutions to the discrete Dirichlet problem associated to these Laplacians will actually approximate solutions of the Dirichlet problem [6]. Special cases of Laplacians with this form have also been studied, for instance, in [2,3,11,12,17,18,20]. These Laplacians are all Laplacians on the graph determined by the one-skeleton of the triangulation although the coefficients may be negative.…”

Section: Definition 4 a Weighted Triangulation (T W) Is A Triangulamentioning

confidence: 99%

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A Monotonicity Property for Weighted Delaunay Triangulations

Glickenstein

2007

Discrete Comput Geom

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Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the "roughness") is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa's result.

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Section: Definition 4 a Weighted Triangulation (T W) Is A Triangulamentioning

confidence: 83%

Section: Definition 4 a Weighted Triangulation (T W) Is A Triangulamentioning

confidence: 99%

A Monotonicity Property for Weighted Delaunay Triangulations

Glickenstein

2007

Discrete Comput Geom

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“…Thus, it is a combinatorial analogy of the scalar curvature in the infinitesimal sense. The work [8] follows the approach of [5] and defines a combinatorial Yamabe flow for ball packing metrics. The PL scalar curvature defined in this paper is more closely related to Regge's calculus.…”

Section: 4mentioning

confidence: 99%

Combinatorial Yamabe Flow on Surfaces

Luo

2004

Commun. Contemp. Math.

201

267

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In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation. We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.

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“…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%

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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A maximum principle for combinatorial Yamabe flow

Cited by 49 publications

References 16 publications

A Monotonicity Property for Weighted Delaunay Triangulations

A Monotonicity Property for Weighted Delaunay Triangulations

Combinatorial Yamabe Flow on Surfaces

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

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