Piecewise flat Ricci flow of compact without boundary three-manifolds

Abstract

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homogeneity, and include Nil and Gowdy manifolds, a three-torus initially embedded in Euclidean four-space, and a perturbation of a flat three-torus. The piecewise flat Ricci flow of the first two are shown to converge to known smooth Ricci flow solutions, with the remaining two flowing asymptotically to flat metrics.

Figures (12)

• Figure 1: The deficit angle at an edge, and a cross section of the region around an edge showing the vertex and edge volumes.
• Figure 2: The three different block types, with the six tetrahedra of the cubic block on the far left, and a slight separation of the three diamond shapes forming the diamond block.
• Figure 3: Relations between the $yz$-faces at $x=0$ and $x=1$ for $\lambda = 1$, and single and three-block triangulations with these face relations. The orthogonal frames for $\{\theta^i\}$ are also indicated for different values of $x$ along the three-block triangulation.
• Figure 4: Graphs of the metric functions in time for both the normalized and non-normalized Ricci flows, all showing convergence to the analytic solutions as the resolution is increased.
• Figure 5: A grid of six cubic blocks along the $\theta$-direction for the lowest resolution triangulation, with the vertices, edges and triangles on opposite sides identified to give a $T^3$ topology.

Theorems & Definitions (1)

• Remark 1