Ricci-DeTurck Flow Research Articles

Short-time existence of Ricci flow in standard static spacetimes

We prove the short-time existence of Ricci flow in the class of standard static spacetimes with a closed and compact Riemannian spatial component. The proof constitutes establishing parabolicity of the Ricci-DeTurck flow system which therefore, by existence and uniqueness theorem, admits a short-time solution. Finally pulling this solution back via an appropriate family of diffeomorphisms yields a solution to the original Ricci flow.

ADM mass for C 0 C^{0} metrics and distortion under Ricci–DeTurck flow

Abstract We show that there exists a quantity, depending only on C 0 C^{0} data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the C 0 C^{0} sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the C 0 C^{0} mass at infinity is independent of choice of C 0 C^{0} -asymptotically flat coordinate chart, and the C 0 C^{0} local mass has controlled distortion under Ricci–DeTurck flow when coupled with a suitably evolving test function.

Convergence stability for Ricci flow on manifolds with bounded geometry

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on recent work of Bahuaud, Guenther, Isenberg and Mazzeo where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric.

A statistical field theory underlying the thermodynamics of Ricci flow and gravity

This paper proposes a statistical field theory of quantum reference frame underlying Perelman’s analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a [Formula: see text] quantum nonlinear sigma model (NLSM), interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation quantization. The induced Ricci flow of the frame fields and the Ricci–DeTurck flow of the frame fields associated with the density matrix are deduced. In this framework, the diffeomorphism anomaly of the theory has a deep thermodynamic interpretation. The trace anomaly is related to a Shannon entropy in terms of the density matrix, which monotonically flows and achieves its maximal value at the flow limit, called the Gradient Shrinking Ricci Soliton (GSRS), corresponding to a thermal equilibrium state of spacetime. A relative Shannon entropy with respect to the maximal entropy gives a statistical interpretation to Perelman’s partition function, which is also monotonic and gives an analogous H-theorem to the statistical frame fields system. A temporal static three-space of a GSRS four-spacetime is also a GSRS in lower three-dimension, we find that it is in a thermal equilibrium state, and Perelman’s analogies between his formalism and the thermodynamics of the frame fields in equilibrium can be explicitly given in the framework. By extending the validity of the Equivalence Principle to the quantum level, the quantum reference frame fields theory at low energy gives an effective theory of gravity, a scale-dependent Einstein–Hilbert action plus a cosmological constant is recovered. As a possible underlying microscopic theory of the gravitational system, the theory is also applied to understand the thermodynamics of the Schwarzschild black hole.

Evolution of Lifshitz metric anisotropies in Einstein–Proca theory under the Ricci–DeTurck flow

By starting from a Perelman entropy functional and considering the Ricci–DeTurck flow equations, we analyze the behavior of Einstein–Hilbert and Einstein–Proca theories with Lifshitz geometry as functions of a flow parameter. In the former case, we found one consistent fixed point that represents flat space-time as the flow parameter tends to infinity. Massive vector fields in the latter theory enrich the system under study and have the same fixed point achieved at the same rate as in the former case. The geometric flow is parametrized by the metric coefficients and represents a change in anisotropy of the geometry toward an isotropic flat space-time as the flow parameter evolves. Indeed, the flow of the Proca fields depends on certain coefficients that vanish when the flow parameter increases, rendering these fields constant. We have been able to write down the evolving Lifshitz metric solution with positive, but otherwise arbitrary, critical exponents relevant to geometries with spatially anisotropic holographic duals. We show that both the scalar curvature and matter contributions to the Ricci–DeTurck flow vanish under the flow at a fixed point consistent with flat space-time geometry. Thus, the behavior of the scalar curvature always increases, homogenizing the geometry along the flow. Moreover, the theory under study keeps positive-definite, but decreasing, the entropy functional along the Ricci–DeTurck flow.

Ricci DeTurck flow on incomplete manifolds

In this paper we construct a Ricci DeTurck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by W.-X. Shi for complete manifolds [J. Differ. Geom. 30, No. 1, 223–301 (1989; Zbl 0676.53044)], this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci DeTurck flow for a short time.

Ricci de Turck Flow on Singular Manifolds

In this paper, we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long-time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.

Stability of Ricci de Turck flow on singular spaces

In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Scalar curvature rigidity and Ricci DeTurck flow on perturbations of Euclidean space

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $$\mathbb {R}^n$$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known $$L^p$$ bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.

Ricci flow on Finsler surfaces

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.

Parabolic equations with rough data

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local oscillation. The approach is sufficiently flexible to apply to boundary value problems, quasilinear and fully nonlinear equations.

A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows

It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.

Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua

The elliptic Einstein–DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. The Ricci–DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein–DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza–Klein, locally AdS or have extremal horizons. Using a maximum principle, we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that the Ricci–DeTurck flow preserves these classes of manifolds. As an example, we simulate the Ricci–DeTurck flow for a manifold with asymptotics relevant for AdS5/CFT4. Our maximum principle dictates that there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N2c) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.

An energy approach to the problem of uniqueness for the Ricci flow

We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu’s theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this technique we prove a further uniqueness theorem for subsolutions to a general class of mixed differential inequalities and obtain an extension of Chen-Zhu’s result to solutions (and initial data) of potentially unbounded curvature. Let M = M be a smooth manifold and g0 a Riemannian metric on M . We are interested in the question of uniqueness of solutions to the initial value problem (1) ∂ ∂t g(t) = −2 Rc(g(t)), g(0) = g0, associated to the Ricci flow on M . The broadest category in which uniqueness is currently known to hold in every dimension is that of complete solutions of uniformly bounded curvature. Theorem 1 (Hamilton [H1]; Chen-Zhu [CZ]). Suppose g0 is a complete metric and g(t) and g(t) are solutions to the initial value problem (1) satisfying sup M×[0,T ] |Rm |g(t), sup M×[0,T ] |Rm|g(t) ≤ K0. Then g(t) = g(t) for all t ∈ [0, T ]. The uniqueness of solutions to the Ricci flow is not an automatic consequence of the theory of parabolic equations, as the system (1) is only weakly-parabolic. For compact M , there are two basic arguments, both due to Hamilton. The first is a byproduct of the proof of the short-time existence of solutions in Hamilton’s orginal paper [H1] and is based on a Nash-Moser-type inverse function theorem. The second, given in [H2], effectively reduces the question of uniqueness to that for the strictly parabolic Ricci-DeTurck flow. The basis of this argument is the observation that the DeTurck diffeomorphisms, which are generally obtained as solutions to a system of ODE depending on a given solution to the Ricci-DeTurck flow, can also be represented as the solutions to a certain parabolic PDE – specifically, a harmonic map heat flow – which depends on the associated solution to the Ricci flow. As DeTurck’s method is applicable to many other geometric evolution equations with gauge-based degeneracies, this second argument of Hamilton’s gives rise to an elegant and flexible general prescription in which one exchanges the problem of uniqueness for one weakly parabolic system for the (separate) problems of existence and uniqueness for one or more auxiliary strictly parabolic systems. Date: May 2012. The author was supported in part by NSF grant DMS-0805834/DMS-1160613.

Geometric flows with rough initial data

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.