Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds

We study critical metrics for the squared L²-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature by making use of a curvature identity on 4-dimensional Riemannian manifolds.     

Weyl curvature and the Euler characteristic in dimension four

We give lower bounds, in terms of the Euler characteristic, for the L²-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.     

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.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four

We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L²-norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C1,α-close to the original metric on a smaller ball but still of definite size.     

Integral pinching theorems

Using Hamilton's Ricci flow we shall prove several pinching results for integral curvature. In particular, we show that if p>n/2$ and the L ^p norm of the curvature tensor is small and the diameter is bounded, then the manifold is an infra-nilmanifold. We also obtain a result on deforming metrics to positive sectional curvature. Received: 17 February 1999     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Almost-Schur lemma

Schur’s lemma states that every Einstein manifold of dimension n?≥ 3 has constant scalar curvature. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be small rather than identically zero. In particular, we provide an optimal L ² estimate under suitable assumptions and show that these assumptions cannot be removed.     

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π₁(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L¹- sense.Received: 19 July 2004     

Maskit combinations of Poincaré-Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S³ and S²×S¹ bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

Bundle stabilisation and positive Ricci curvature

If E is the total space of a vector bundle over a compact Ricci non-negative manifold, it is known that E×Rp admits a complete metric of positive Ricci curvature for all sufficiently large p. In this paper we establish a small, explicit lower bound for the dimension p.     

Rigidity of Riemannian manifolds with positive scalar curvature

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the \(L^{\frac{n}{2}}\)-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.     

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).     

On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric

In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kähler-Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kähler-Finsler metric F is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type (2,0) of the complex Berwald connection vanish; (iii) the complex Wrona metric in Cn is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S2n−1⊂Cn is explicitly obtained.     

Ricci Yang-Mills flow on surfaces

We study the behavior of the Ricci Yang-Mills flow for U(1) bundles on surfaces. By exploiting a coupling of the Liouville and Yang-Mills energies we show that existence for the flow reduces to a bound on the isoperimetric constant or the L⁴ norm of the bundle curvature. We furthermore completely describe the behavior of long time solutions of this flow on surfaces. Finally, in Appendix A we classify all gradient solitons of this flow on surfaces.     

Bakry–Emery Curvature Operator and Ricci Flow

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L ²– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University.     

The Webster scalar curvature flow on CR sphere. Part I

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p$L^p$ convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper.