A combinatorial Yamabe flow in three dimensions
By David Glickenstein. 2005
79 citations recorded by the source corpus.
A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the maximum principle need not hold. It is then shown that if the flow is nonsingular, the flow converges to a constant curvature metric.
Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/9658075ed586001a117fe086d8a31467ebc366d3
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In Applied Mathematics
In Computational Mechanics
In Computational Theory and Mathematics
Citations in the corpus
- Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension (2013, methodology): “Glickenstein proved ([6]) ∂αijkl ∂rj rjri = Aijkl lij .”
- Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, methodology): “, [23] [17]), but the proofs were explicit computations, which made them difficult to extend to more general cases.”
- Combinatorial Calabi flows on surfaces (2012, background): “Discrete Laplacian defined here is exactly the famous discrete dual-Laplacian ([13][15]), which will be explained carefully in Appendix A.”
- Geometric triangulations and discrete Laplacians on manifolds (2005, background): “It also appears in [20] [21] in three dimensions, where Thurston triangulations are considered such that dij depend only on i.”