Integral Pinching Conditions Research Articles

A note on rigidity of Riemannian manifolds with positive scalar curvature

In this short note, we obtain an integral inequality for closed Riemannian manifolds with positive scalar curvature and give some rigidity characterization of the equality case, which generalizes the recent results of Catino which deal with the conformally flat case, and of Huang and Ma which deal with the harmonic curvature case. Moreover, we obtain an integral pinching condition with non-negative constant $$\sigma _2(A^{\tau })$$ , which can be seen as a complement to Bo and Sheng who considered conformally flat manifolds with constant quotient curvature of $$\sigma _k(A^{\tau })$$ .

Rigidity Theorems of Complete Kähler-Einstein Manifolds and Complex Space Forms

We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kahler manifolds. We derive some elliptic differential inequalities from Weitzenb¨ock formulas for the traceless Ricci tensor of Kahler manifolds with constant scalar curvature and the Bochner tensor of Kahler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L∞ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature and complex space forms among Kahler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kahler manifolds and noncompact Kahler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.

Some remarks on Bach-flat manifolds with positive constant scalar curvature

We give some rigidity results for Bach-flat manifolds with positive constant scalar curvature under pointwise or integral pinching conditions which have the additional property of being sharp.

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

Let $$M^n(n\ge 3)$$ be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $$M^n$$ satisfies some integral pinching conditions. We give some rigidity theorems. In particular, Theorems 1.4 and 1.10 are sharp for our conditions have the additional properties of being sharp. By this, we mean that we can precisely characterize the case of equality.

Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space

Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L2 harmonic 1-forms, we can further prove some one-end theorems.

A remark on compact hypersurfaces with constant mean curvature in space forms

In this note we characterize compact hypersurfaces of dimension n≥2 with constant mean curvature H immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when n≥3 and H≠0, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf–Chern result.

On conformally flat manifolds with constant positive scalar curvature

We classify compact conformally flat n-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either S with the round metric, S ×Sn−1 with the product metric or S × Sn−1 with a rotationally symmetric Derdzinski metric.

On the structure of conformally flat Riemannian manifolds

In this paper we will prove vanishing and finiteness theorems for L2-harmonic 1-forms on a locally conformally flat Riemannian manifold which satisfies an integral pinching condition on the traceless Ricci tensor, and for which the scalar curvature is non-positive or satisfies some integral pinching conditions. Based on these vanishing and finiteness theorems, combining with the work of Li–Tam, we can obtain some one-end and finite ends theorems.

Integral pinching results for manifolds with boundary

We prove that some Riemannian manifolds with boundary satisfying an explicit integral pinching condition are spherical space forms. More precisely, we show that three-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L-norm of the scalar curvature and the L-norm of the Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that four-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of the (Q,T )-curvature and the L-norm of the Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence, we show that a certain conformally invariant operator, which plays an important role in Conformal Geometry, is non-negative and has trivial kernel if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the (Q,T )-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant Q-curvature, constant T -curvature, and zero mean curvature under the latter assumptions.

The volume-preserving mean curvature flow in Euclidean space

We study the convergence of the volume-preserving mean curvature flow of hypersurfaces in Euclidean space under some initial integral pinching conditions. We prove that if the traceless second fundamental form is sufficiently small, the flow will exist for all time and converge exponentially fast to a round sphere.