We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.
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 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.
In this paper, we study conformal Ricci solitons and conformal gradient Ricci solitons on generalized ($κ,μ$)-space forms. The conditions for the solitons to be shrinking, steady, and expanding are derived in terms of conformal pressure p. We show under what conditions a Ricci semi-symmetric generalized ($κ,μ$)-space form equipped with a conformal Ricci soliton forms an Einstein manifold.
The goal of our present paper is to deliberate $*$-conformal $η$-Ricci soliton within the framework of Kenmotsu manifolds. Here we have shown that a Kenmotsu metric as a $*$-conformal $η$-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have evolved the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $*$-conformal $η$-Ricci soliton. Next, we have contrived $*$-conformal $η$-Ricci soliton admitting $(κ,μ)^\prime$-almost Kenmotsu manifold and proved that the manifold is Ricci fla…
We analyze an energy functional associated to Conformal Ricci Flow along closed manifolds with constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of both the metric and the pressure function along Conformal Ricci Flow.
In this paper we study *-Conformal η-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting *-Conformal η-Ricci soliton. We obtain some significant results on *-Conformal η-Ricci soliton in Sasakian manifolds satisfying R(ξ,X).S = 0, S(ξ,X).R = 0, {\overline}P(ξ,X).S = 0, where {\overline}P is Pseudo-projective curvature tensor.The conditions for *-Conformal η-Ricci soliton on Φ-conharmonically flat and Φ-projectively flat Sasakian manifolds have been obtained in this article. Lastly we give an example of 5-dimensional Sasakian manifolds …
In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.
The present paper is to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field ξ in connection with conformal Ricci-Yamabe metric and conformal η-Ricci-Yamabe metric. Here we have delineated the conditions for conformal Ricci-Yamabe soliton to be expanding, steady, or shrinking. Later, we have acquired Laplace equation from conformal η-Ricci-Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type. Lastly, we have designated perfect fluid with Robertson-Walker spacetime and some applications of physics and gravity.
In this paper, we initiate the study of conformal $η$-Ricci soliton and almost conformal $η$-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $η$-Ricci soliton, then the manifold is $η$-Einstein and either the soliton vector field $V$ is Killing or it leaves $ϕ$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $η$-Ricci soliton and vector field is pointwise collinear with the characteristic vector field $ξ$. Next, we show that a para-S…
In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the manifold to be conformal to a sphere.
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
Walker manifolds of signature (2,2) have been used to provide examples of Osserman and of conformal Osserman manifolds of signature (2,2). We study questions of geodesic completeness and Ricci blowup in this context.
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. In this paper, we introduce combinatorial $α$-curvature for discrete conformal structures on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. Then we prove the local and global rigidity of combinatorial $α$-curvature with respect to discrete conformal structures on polyhedral surfaces, which confirms parameterized Glickenstein rigidity conjecture. To s…
The aim of this paper is characterize a class of contact metric manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensional $N(k)$-contact metric manifold $M$ admits $\ast$-conformal Ricci soliton or $\ast$-conformal gradient Ricci soliton, then the manifold M is $\ast$-Ricci at and locally isometric to the Riemannian of a flat $(n + 1)$-dimensional manifold and an $n$-dimensional manifold of constant curvature 4 for $n > 1$ and flat for $n = 1$. Further, for the first case, the soliton vector field is conformal and for the $\ast$-gradient case, the potenti…
We show that a complete gradient Ricci soliton $(M^n,\,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar c…
Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. This paper deals with the study of conformal Ricci solitons within the framework of warped product manifolds which extends the notion of usual Riemannian product manifolds. First, we prove that if a warped product manifold admits conformal Ricci soliton then the base and the fiber also share the same property. In the next section the characterization of conformal Ricci solitons on warped product manifolds in terms of Killing and conformal vector fields has been studied. Next, we prove that a warped produc…
Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in some sense solvable, without assuming any closeness near a special metric.
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional $(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant principal Ricci curvatures is a Riemannian locally symmetric space.
In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product $\mathbb{R}\times N^{n-1}$, or globally conformally equivalent to the Euclidean space $\mathbb{R}^{n}$ or to the round sphere $\mathbb{S}^{n}$. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.
The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $η$-Ricci soliton. Here, we have investigated the nature of the conformal $η$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $η$-Einstein Kenmotsu manifold admitting conformal $η$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $η$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $η$-Ri…
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preser…
We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters $k=2$ and $k=3$ naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric $(2,3,5)$-distribution (described locally by a certain function $φ(x,q)=\frac{q^2}{H''(x)}$) to a Ricci-flat one.
Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of positive Ricci curvature, through a smooth, radial, positive, integrable function of R^m, for m > 1. These results are motivated by some recent questions on Yamabe constants.
We study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of Cotton-York tensor and the L1-norm of it under the Ricci flow. In particular, we investigate the behavior of the L1-norm of the Cotton-York tensor under the Ricci flow on three-dimensional simply-connected homogeneous manifolds which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the L1-norm of the Cotton-York tensor for the product metric of the Rosenau solution for the Ricci flow on a two-sphere and the standard metric of a cir…
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a $C$-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane wav…
We study singularity formation of Kähler-Ricci flow on a Kähler manifold that admits a horizontally homothetic conformal submersion into another Kähler manifold. We will derive necessary and sufficient conditions for the preservation of horizontally homothetic conformal submersion along the flow and establish the formation of type I singularity together with a standard splitting of the Cheeger-Gromov limit. This generalizes the setup of Calabi symmetry that was discussed in \cite{1} and \cite{2}, thus gives new proofs for the results listed there.
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works by Q.-M. Cheng, M.H. Noronha, B.-L. Chen and X.-P. Zhu, and S. Zhu.
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in \cite{GL} \cite{Lee} \cite{Bi} of conformally compact Einstein metrics with conform…
In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial $(0,3)$-tensors related to the geome…
We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the the…
A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green's function and…
In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metrics to a euclidean space and invariant by rotation are gradient Ricci almost soliton.
We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the $σ_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the $σ_k$-Ricci problem. By adopting results of Mazzeo-Pacard, we show an interesting relationship between the complete metrics we construct and the existence of Poincaré-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case…
Let $\overline{M}^{n+1}$ be a semi-Riemannian manifold of constant sectional curvature, and endowed with a conformal vector field . Consider a Riemannian manifold $M^n$, isometrically immersed into $\overline{M}^{n+1}$. With these hypotheses in mind, the ultimate goal of this paper is to investigate the intimate relationship between conformal vector fields on $\overline{M}^{n+1}$ and the Ricci almost soliton structure on $M^n$. Assuming that the latter manifold is connected and totally umbilic, we prove that it is naturally given a Ricci almost soliton structure by means of the tangential part…
In this paper, we characterize the potential function $f$ of the almost conformal gradient Ricci soliton on a Sasakian manifold in terms of the non-dynamical scalar field $p$ and deduce the necessary condition for the potential function $f$ to be constant. Furthermore, a relation between $λ$ and the potential function $f$ has been established. Finally, we prove a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton and also a characterization of the soliton in terms of shrinking, steady or expanding has been done.
We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.
We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $β(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0 \leq β(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq β(M^4,[g]) < 4(1 + ε)$ for $ε> 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to…
The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss-Codazzi constraint equations under deformations are also discussed.
In this paper, we generalize Huber's finite point conformal compactification theorem to a higher dimensional manifold, which is conformally compact with $L^\frac{n}{2}$ integrable Ricci curvatures.
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.
The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of critical Kähler Finsler metrics is obtained. A Yamabe type problem for mean Ricci curvature is considered.
We descrive examples of metrics in the conformal class $[g]$ on complete conformally flat Riemannian manifolds $(M,g].$ These metrics have a constant scalar curvature and an harmonic curvature with non parallel Ricci tensor.
In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci curvature.
We consider the normalized Ricci flow $\del_t g = (ρ- R)g$ with initial condition a complete metric $g_0$ on an open surface $M$ where $M$ is conformal to a punctured compact Riemann surface and $g_0$ has ends which are asymptotic to hyperbolic cusps. We prove that when $χ(M) < 0$ and $ρ< 0$, the flow $g(t)$ converges exponentially to the unique complete metric of constant Gauss curvature $ρ$ in the conformal class.
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.
