We construct a discrete form of Hamilton's Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einstein's 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 Hamilton's 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.
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 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 t…
We define the Ricci curvature on simplicial complexes by modifying the definition of the Ricci curvature on graphs, and we prove the upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial complexes using the Ricci curvature.
Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $π:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $føπ$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below.
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical ph…
In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. We remark that in this paper we will mostly be dealing with non-singular Ricci flows. The existence of surgeries will not play an important role.
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 Hamilton's 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 corresp…
We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.
In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.
Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat R…
We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.
We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization of 3-manifolds, Ricci flow with surgery, and the simplicial volume approach to collapsing theorems. In the last section, Ricci flow with surgery on open 3-manifolds and obstructions to positive scalar curvature are discussed.
We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.
Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructed from a gradient shrinking Ricci soliton.
In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.
We study relation of the Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds. In particular, we construct Ricci-flat cohomogeneity one metrics with respect to 3-dimensional Lie groups.
In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of Chow-Lu-Ni (2006).
We simplify and improve the curvature estimates in the paper: On the conditions to extend Ricci flow(II). Furthermore, we develop some volume estimates for the Ricci flow with bounded scalar curvature. These estimates can be applied to study the singularities of the Ricci flow and convergence properties of the Kähler Ricci flow.
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped products), Ricci flow coupled with both harmonic map flow and Yang-Mills flow, and Ricci flow coupled with heat flow for the torsion of a metric-compatible connection. The methods used to prove stability follow a program outlined by Guenther, Isenberg, and Knopf, which uses maximal…
In this paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation of geometric curvature tensors under $*-$ Ricci flow. In the last two section of the paper, we have found the $\Im$-functional and $ω-$ functional for $*-$ Ricci flow respectively.
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finall…
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions on noncompact Riemannian manifolds with at most quadratic curvature growth, whose Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method imp…
We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.
In this note, a modified Kähler-Ricci flow is introduced and studied. The main point is to show the flexibility of Kähler-Ricci flow and summarize some useful techniques.
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds.
In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study the uniform Perelman's estimates of the twisted Kähler-Ricci flows. After that, we prove that the conical Kähler-Ricci flow must converge to a conical Kähler-Einstein metric if there exists one.
We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.
In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eigenvalues of Laplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of the shrinker and expanding entropies of the harmonic-Ricci flow.
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.
In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a quest…
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier work of Ishida and Sasahira are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov's simplicial volume is nontrivial, Perelman's lambda-bar invariant…
We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove $L^{p}$ bounds for the curvature tensor on time-slices for any $p<2$. In higher dimensions, we show that every singular point of …
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of R…
In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4π)^{\frac{n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.
In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.
We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.
Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some sense.
We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.
In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points are Ricci-flat metrics. Modifying the argument used by Shi and List, we prove the short time existence and higher order derivatives estimates.
We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $inj_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly…
Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.
We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.
A short proof of the convergence of the Kahler-Ricci flow on Fano manifolds admitting a Kahler-Einstein metric or a Kahler-Ricci soliton is given, using a variety of recent techniques
