Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

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.     

Some pinching theorems for minimal submanifolds in \mathbb{S}^m (1) \times \mathbb{R}

Let Mn be an n-dimensional compact minimal submanifolds in Sm(1)×R.We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively.In fact,we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.     

On the number of solutions to the discrete two-dimensional L0-Minkowski problem

Our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices, if they exist. Such polygons are solutions to the discrete L₀-Minkowski problem in the plane proposed, in a larger generality, by Lutwak in his extension of the Brunn-Minkowski theory. The method exploits the equivalence between the L₀-polygons and the homothetic solutions to appropriate anisotropic crystalline flows. Hence as a by-product we obtain a positive answer to Jean Taylor's conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature, without any symmetry assumptions.     

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.     

On closed minimal hypersurfaces with constant scalar curvature in \mathbb{S}^7

We consider minimal closed hypersurfaces ${M \subset \mathbb{S}^7(1)}$ with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.     

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     

Real minimal hypersurfaces in complex two-plane Grassmannians

We give a pinching condition for compact minimal hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) in terms of sectional curvature and the squared norm of the shape operator.     

INEQUALITIES FOR Lp-MIXED CURVATURE IMAGES

Lutwak, Yang, and Zhang posed the notion of Lp-curvature images and established several Lp analogs of the affne isoperimetric inequality. In this article, the notion of Lp-mixed curvature images is introduced, Lp-curvature images being a special case. The properties and Lp analogs of the affne isoperimetric inequality are established for Lp-mixed curvature images.     

A Concentration‐Collapse Decomposition for L2 Flow Singularities

We exhibit a concentration‐collapse decomposition of singularities of fourth‐order curvature flows, including the L² curvature flow and Calabi flow, in dimensions n ≤ 4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small energy and a volume growth lower bound, in the vein of Perelman's pseudo locality result. Next, we generalize our technique from prior work to exhibit local smoothing estimates for the L² flow in the presence of a curvature‐related bound. A final key ingredient is a new local ?‐regularity result for L² critical metrics with possibly nonconstant scalar curvature. Applications of these results include new compactness and diffeomorphism‐finiteness theorems for smooth compact 4‐manifolds satisfying the necessary and effectively minimal hypotheses of L² curvature pinching and a volume‐noncollapsing condition. © 2015 Wiley Periodicals, Inc.     

Rigidity of noncompact complete manifolds with harmonic curvature

Let (M, g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that the L ₂ norms of the traceless Ricci curvature are finite. We prove that (M, g) is Einstein if n ?? 5 and the L _n/2 norms of the Weyl curvature and traceless Ricci curvature are small enough.     

Lp-bounds on curvature,elliptic estimates and rectifiability of singular setsBornes Lp sur la courbure,estimées elliptiques et rectifiabilité d'ensembles singuliers

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov–Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for |Hess_h|_L2p, (|?Hess_h·|Hess_h|^p?2)_L2, where Δh=c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2. To cite this article: J. Cheeger, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 195–198.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces M of these spaces N. Indeed, we prove that if M is included in a ball of radius small enough then the Hausdorff-distance between M and a geodesic sphere S of N is small. Moreover M is diffeomorphic and quasi-isometric to S. As other application, we obtain rigidity results for almost umbilic hypersurfaces.     

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

Potential function estimates for quasi-Einstein metrics

In this paper, we derive the growth pinching estimates for potential functions of τ-quasi-Einstein metrics on complete noncompact connected manifolds, based on the estimates for the scalar curvature and the using of the weighted measure comparison theorem. Our results show that the estimates for potential functions rely on the sign of constants λ and μ.     

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.     

On the zero dynamics of linear nonstationary discrete systems with a single input and a scalar output

For a linear discrete nonstationary system with one-dimensional input and output, we find conditions for the existence of a feedback providing the existence of a subspace L _t of constant nonzero dimension such that this subspace is invariant under the closed system and the output variable is zero for all motions in this subspace. Under certain assumptions, we indicate equations that describe the solutions located in L _t . A criterion is established for the subspace L _t to be stationary (i.e., independent of t).     

Chern forms,chern numbers and curvature conditions on a Kähler-Einstein surface

In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL ²-norm of (3C ₂ ?C ₁ ² )/C ₁ ² is sufficiently small, whereC ₁ andC ₂ are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C ₂ ?C ₁ ² = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.