Showing papers on "Ricci flow published in 2012"

On the Riemann tensor in double field theory

Abstract: Double field theory provides T-duality covariant generalized tensors that are natural extensions of the scalar and Ricci curvatures of Riemannian geometry. We search for a similar extension of the Riemann curvature tensor by developing a geometry based on the generalized metric and the dilaton. We find a duality covariant Riemann tensor whose contractions give the Ricci and scalar curvatures, but that is not fully determined in terms of the physical fields. This suggests that α′ corrections to the effective action require α′ corrections to T-duality transformations and/or generalized diffeomorphisms. Further evidence to this effect is found by an additional computation that shows that there is no T-duality invariant four-derivative object built from the generalized metric and the dilaton that reduces to the square of the Riemann tensor.

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

Abstract: In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n − 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.

Rotational symmetry of self-similar solutions to the Ricci flow

Abstract: Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

Brain Surface Conformal Parameterization With the Ricci Flow

Abstract: In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T2 test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.

Three-dimensional lorentzian homogeneous ricci solitons

Abstract: We study three-dimensional Lorentzian homogeneous Ricci solitons, proving the existence of shrinking, expanding and steady Ricci solitons. For all the non-trivial examples, the Ricci operator is not diagonalizable and has three equal eigenvalues.

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

Abstract: In this paper, we study Perelman’s \({{\mathcal W}}\) -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the \({\mathcal{W}}\) -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the \({{\mathcal W}}\) -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the \({\mathcal{W}}\) -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.

New monotonicity formulas for Ricci curvature and applications. I

Abstract: We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten.

Ricci flow coupled with harmonic map flow

Abstract: We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.

Weakly z symmetric manifolds

Abstract: We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n .I f theZ tensor is singular we give condi- tions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally har- monic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.

Classification of ancient compact solutions to the Ricci flow on surfaces

Abstract: We consider an ancient solution $g(•, t)$ of the Ricci flow on a compact surface that exists for $t\in (−\infty, T)$ and becomes spherical at time $t = T$. We prove that the metric $g(•, t)$ is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor

Abstract: We introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2) from the Gauss equation. We then derive a new formula for the Ricci tensor of M in G2(ℂm+2). Finally, we prove that there does not exist any Hopf real hypersurface in complex two-plane Grassmannians G2(ℂm+2) with parallel Ricci tensor.

On the Positive Mass Theorem for Manifolds with Corners

Abstract: We study the positive mass theorem for certain non-smooth metrics following P. Miao’s work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

Bounds on volume growth of geodesic balls under Ricci flow

Abstract: We prove a so called $\kappa$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $\kappa$ non-collapsing property for Ricci flow. These two results together imply volume doubling property for Ricci flow without assuming Ricci curvature lower bound.

A numerical approach to finding general stationary vacuum black holes

Abstract: The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon, this equation has previously been shown to be elliptic, and Ricci flow and Newton’s method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergo-regions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes which is general enough to include many interesting higher dimensional solutions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson’s boundary conditions. We demonstrate both Newton’s method and Ricci flow to find these Lorentzian solutions.

The Chern-Ricci flow on complex surfaces

Abstract: The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci flow. This provides evidence that the Chern-Ricci flow carries out blow-downs of exceptional curves on non-minimal surfaces. We also describe explicit solutions to the Chern-Ricci flow for various non-Kahler surfaces. On Hopf surfaces and Inoue surfaces these solutions, appropriately normalized, collapse to a circle in the sense of Gromov-Hausdorff. For non-Kahler properly elliptic surfaces, our explicit solutions collapse to a Riemann surface. Finally, we define a Mabuchi energy functional for complex surfaces with vanishing first Bott-Chern class and show that it decreases along the Chern-Ricci flow.

On homogeneous Ricci solitons

Abstract: We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its omega-limit set consists of a single point. We also geometrically characterize algebraic solitons among homogeneous Ricci solitons as those for which the Ricci flow solution is simultaneously diagonalizable.

Fractional nonholonomic Ricci flows

Abstract: We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman’s functionals and derived the corresponding fractional evolution (Hamilton’s) equations. We apply in fractional calculus the nonlinear connection formalism originally elaborated in Finsler geometry and generalizations and recently applied to classical and quantum gravity theories. There are also analyzed the fractional operators for the entropy and fundamental thermodynamic values.

On the global structure of conformal gradient solitons with nonnegative ricci tensor

Abstract: In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product ℝ × Nn-1, or globally conformally equivalent to the Euclidean space ℝn or to the round sphere 𝕊n. 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 sphere theorems for manifolds with positive scalar curvature

Abstract: Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0 \gt \sigma_n K_{\rm max}$, where $\sigma_n \in (\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if $M^n(n \ge 3)$ is a compact manifold whose $(n − 2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n−2)}_{\rm min} \gt \tau_n(n −2)R_0$, where $\tau_n \in(\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

Perelman’s lambda-functional and the stability of Ricci-flat metrics

Abstract: In this article, we introduce a new method (based on Perelman’s λ-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (a) a Ricci-flat metric is a local maximizer of λ in a C 2,α -sense if and only if its Lichnerowicz Laplacian is nonpositive, (b) λ satisfies a Łojasiewicz-Simon gradient inequality, (c) the Ricci flow does not move excessively in gauge directions. As consequences, we obtain a rigidity result, a new proof of Sesum’s dynamical stability theorem, and a dynamical instability theorem.

The Bakry-Emery Ricci tensor and its applications to some compactness theorems

Abstract: Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry–Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.

Mean Curvature Flow in a Ricci Flow Background

Abstract: Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.

Smooth metric measure spaces and quasi-einstein metrics

Abstract: Smooth metric measure spaces have been studied from the two different perspectives of Bakry–Emery and Chang–Gursky–Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.

New logarithmic Sobolev inequalities and an \epsilon-regularity theorem for the Ricci flow

Abstract: In this note we prove a new \epsilon-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed entropy, with W_{x}(s) 0, depending only on T and on lower scalar curvature and \mu-entropy bounds for (M,g(-T)), such that W_{x_0}(s) > -\epsilon implies |Rm|< r^{-2} on P_{\epsilon r}(x,0), where r^2 = |s| and P_r(x,t) \equiv B_r(x,t)\times (t-r^2,t] is the parabolic ball. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s-average of W_x(s). To accomplish this, we require a new log-Sobolev inequality. It is well known by Perelman that the metric measure spaces (M,g(t),dv_{g(t)}) satisfy a log-Sobolev; however we prove that this is also true for the conjugate heat kernel weighted spaces (M,g(t),H_{x}(-,t)\,dv_{g(t)}). Our log-Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log-Sobolev has other consequences as well, including an average Gaussian upper bound on the conjugate heat kernel that only depends on a two-sided scalar curvature bound.

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Abstract: We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.