The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.     

The Ricci curvature of finite dimensional approximations to loop and path groups

Let W(G) and L(G) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [0,1]. We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric.     

Non-branching geodesics and optimal maps in strong CD(K,\infty )-spaces

We prove that in metric measure spaces where the entropy functional is \(K\) -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant.     

On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Summary The purpose of this paper is to prove that every ellipsoidal domain in Cⁿ admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cⁿ is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kähler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cⁿ, with the Bergman metric. Before the only known example of complete simply-connected Kähler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cⁿ.     

On the four-dimensional T<Superscript>2</Superscript>-manifolds of positive Ricci curvature

We prove that there is a T ²-invariant Riemannian metric of positive Ricci curvature on every four-dimensional simply connected T ²-manifold.     

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

Prescription de la courbure de Ricci au voisinage de la métrique hyperbolique

On the unit ball of, one considers the standard hyperbolic metric H₀ whose Ricci curvature equals R₀ and Riemann-Christoffel curvature is. We prove that, for any symmetric tensor R near R₀, there exists a unique metric H near H₀ whose Ricci curvature is R. We deduce in the C^∞ case that the image of the Riemann-Christoffel operator is a submanifold in a neighborhood of. Finally, we study more precisely the Ricci equation in dimension 2.     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Ricci Deformation of the Metric on Riemannian Orbifolds

In this note we generalize the Huisken’s (J Diff Geom 21:47–62, 1985) result to Riemannian orbifolds. We show that on any n-dimensional (n ≥ 4) orbifold of positive scalar curvature the metric can be deformed into a metric of constant positive curvature, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not large compared to the scalar curvature at each point, and therefore generalize 3-orbifolds result proved by Hamilton [Three- orbifolds with positive Ricci curvature. In: Cao HD, Chow B, Chu SC, Yau ST (eds) Collected Papers on Ricci Flow, Internat. Press, Somerville, 2003] to n-orbifolds (n ≥ 4).     

Yamabe Flow and Myers Type Theorem on Complete Manifolds

In this paper, we prove the following Myers type theorem: If (M ⁿ ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M ⁿ must be compact.     

Porous media equation and Sobolev inequalities under negative curvature

We study the link between some modified porous media equation and Sobolev inequalities on a Riemannian manifold M whose Ricci curvature tensor is bounded below by a negative constant −ρ. The method used deals with entropy-energy differentiation and follows the way the author got inequalities under nonnegative Ricci curvature assumptions. The key of the proof is the curvature-dimension criterion.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

Jost maps, ball-homogeneous and harmonic manifolds

Given a real number ε>0, small enough, an associated Jost map J_ε between two Riemannian manifolds is defined. Then we prove that connected Riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball (i.e. for which the identity is a J_ε map) are ball-homogeneous. In the analytic case we characterize such manifolds in terms of the Euclidean Laplacian and we show that they have constant scalar curvature. Under some restriction on the Ricci curvature we prove that Riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic.     

Etude locale d'opérateurs de courbure sur l'espace hyperbolique

We show in this article that, on the unit ball in, the operators of covariant and contravariant Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the hyperbolic metric h₀. We deduce in the C^∞ case that the image of the Riemann-Christoffel curvature operator is a submanifold in a neighborhood of h₀. We deal also with some obstructions related to the asymptotic behavior of metrics near h₀, and we treat more precisely the case of the Ricci equation in dimension 2.     

Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry

In this paper, we study complete Riemannian n-manifolds (n ≥ 3) with asymptotically nonnegative Ricci curvature and weak bounded geometry. We show among other things that the total Betti number of such a manifold has polynomial growth of degree n ² + n. Further more, such a manifold is of finite topological type if the volume growth rate of the metric ball around the base point is less than This work is partly supported by the National Natural Science Foundation (10371047) of China. Received: 13 June 2006     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.