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We construct a discrete form of Hamiltons Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einsteins theory of general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation is naturally associated to each edge, L, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice, S, and its circumcentric dual lattice, S*. In particular, the RRF equation associated to L is naturally defined on a d-dimensional hybrid block connecting $ell$ with its (d-1)-dimensional circumcentric dual cell, L*. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume weighted average of the fractional rate of change of the edges, lambda in L*, of the circumcentric dual lattice, S*, that are in the dual of L. The inherent orthogonality between elements of S and their duals in S* provide a simple geometric representation of Hamiltons RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.
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Hamiltons Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are 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.
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamiltons Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The 2-sphere cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice well-centered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model.
Hamiltons Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurstons geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure -- a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Formans expression of Ricci curvature on graphs.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In addition, for Ricci harmonic flow, we use the monotonicity of functional $ u_alpha$ to show the connection between finite-time singularity and shrinking Ricci harmonic soliton. At last, we explore the property of ancient solutions for Ricci harmonic flow.
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kahler-Ricci flow. This leads to global convergence results, and applications to complex geometry. A purely mathematical introduction to the physical idea of T-duality is given, and a discussion of its relationship to generalized Ricci flow.
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