Nonnegative Curvature Research Articles

Complete noncompact manifolds with harmonic curvature

Let (Mn, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mn, g) is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mn, g) with positive scalar curvature.

Remarks on scalar curvature of Yamabe solitons

In this article, we consider the scalar curvature of Yamabe solitons. In particular, we show that, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that a complete non-compact Yamabe soliton with the quadratic decay at infinity of its Ricci curvature has non-negative scalar curvature. A new proof of Kazdan–Warner condition is also presented.

The curvature operator at the soul

We prove two splitting theorems, one topological, the other metric, for open manifolds with nonnegative sectional curvature.

Rigidity of gradient almost Ricci solitons

In this paper, we show that either, a Euclidean space $\mathbb{R}^{n}$, or a standard sphere $\mathbb{S}^{n}$, is the unique manifold with nonnegative scalar curvature which carries a structure of a gradient almost Ricci soliton, provided this gradient is a non trivial conformal vector field. Moreover, in the spherical case the field is given by the first eigenfunction of the Laplacian. Finally, we shall show that a compact locally conformally flat almost Ricci soliton is isometric to Euclidean sphere $\mathbb{S}^{n}$ provided an integral condition holds.

A mass-decreasing flow in dimension three

In this article, we introduce a mass-decreasing flow for asymptotically flat three-manifolds with nonnegative scalar curvature. This flow is defined by iterating a suitable Ricci flow with surgery and conformal rescalings and has a number of nice properties. In particular, wormholes pinch off and nontrivial spherical space forms bubble off in finite time. Moreover, a noncompact variant of the Perelman-energy is monotone along the flow. Assuming a certain inequality between the mass and this Perelman-energy a priori, we can prove that the flow squeezes out all the initial mass.

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 Mn is simply connected and the k-th Ricci curvature of Mn is bounded below by a quantity involving the mean curvature of Mn and the curvature of the ambient manifold, then Mn 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 Mn 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.

Estimate and monotonicity of the first eigenvalue under the Ricci flow

In this paper, we first derive a monotonicity formula for the first eigenvalue of \({-\Delta +aR (0 < a \leq 1/2)}\) on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces.

A Note on surfaces with parallel mean curvature

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space.

A new curvature condition preserved by the Ricci flow

In this paper, we establish a new curvature condition preserved by the Ricci flow, which is named as 2-parameters nonnegative curvature condition. It relies on the first, second and third eigenvalues of the Riemannian curvature operator. Based on this, we prove the strong maximum principle for the 2-parameters nonnegativity along Ricci flow.

An almost Schur theorem on 4-dimensional manifolds

In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.

A renormalized Perelman-functional and a lower bound for the ADM-mass

In the first part of this short article, we define a renormalized F -functional for perturbations of non-compact steady Ricci solitons. This functional motivates a stability inequality which plays an important role in questions concerning the regularity of Ricci-flat spaces and the non-uniqueness of the Ricci flow with conical initial data. In the second part, we define a geometric invariant λ A F for asymptotically flat manifolds with nonnegative scalar curvature. This invariant gives a quantitative lower bound for the ADM-mass from general relativity, motivates a Ricci flow proof of the rigidity statement in the positive mass theorem, and eventually leads to the discovery of a mass-decreasing flow in dimension three.

Normal scalar curvature and a pinching theorem in $\mathbb{S}^m $ × ℝ and ℍ m × ℝ

We obtain an inequality in $\mathbb{S}^m $ × ℝ and ℍ m × ℝ which is similar to DDVV conjecture. As an application, we show that a minimal submanifold in ℍ m × ℝ with nonnegative scalar curvature must be a surface of the type γ × ℝ, where γ is a geodesic in ℍ m . In addition, we get a pinching theorem in $\mathbb{S}^m $ × ℝ.

Viscosity Solutions to Second Order Parabolic PDEs on Riemannian Manifolds

In this work we consider viscosity solutions to second order parabolic PDEs u t +F(t,x,u,du,d 2 u)=0 defined on complete Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold M has nonnegative sectional curvature, we get the finest results. If one additionally requires F to depend on d 2 u in a uniformly continuous manner, the assumptions on curvature can be thrown away.

Splitting Theorems for Submanifolds of Nonnegative Isotropic Curvature

We study isometric immersions of complete manifolds of nonnegative isotropic curvature that have spaces of relative nullity. These manifolds decompose into a Riemannian product $${\bar{M}\times \mathbb R^k}$$ . In the case that k ≥ 2, we use recent results of Brendle-Schoen to completely classify these manifolds. We then apply this classification to study complete non-compact Kahler submanifolds with relatively low codimension.

A PL-manifold of nonnegative curvature homeomorphic to 𝑆²×𝑆² is a direct metric product

Let M 4 M^4 be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, S 2 × S 2 S^2 \times S^2 . We prove that M M is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, M M is a direct metric product of the surfaces of two convex polyhedra in R 3 \mathbb {R}^3 . The classical H. Hopf hypothesis states: for any Riemannian metric on S 2 × S 2 S^2 \times S^2 of nonnegative sectional curvature the curvature cannot be strictly positive at all points. The result of this paper can be viewed as a PL-version of Hopf’s hypothesis. It confirms the remark of M. Gromov that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to the condition of a nonnegative curvature operator.

Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

The authors give a short survey of previous results on generalized normal homogeneous ( δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp ( l ) / U ( 1 ) ⋅ Sp ( l − 1 ) = C P 2 l − 1 , l ⩾ 2 , supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval ( 1 / 16 , 1 / 4 ) . This implies very unusual geometric properties of the adjoint representation of Sp ( l ) , l ⩾ 2 . Some unsolved questions are suggested.

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L 2 harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L 2 harmonic two-form. We obtain that an almost Kahler manifold of non-negative isotropic curvature is Kahler and a symplectic manifold can not admit any almost Kahler structure of positive isotropic curvature.

Cohomogeneity one manifolds with a small family of invariant metrics

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.