Nonnegative Isotropic Curvature Research Articles

Aronson–Bénilan estimates for the fast diffusion equation under the Ricci flow

We study the fast diffusion equation with a linear forcing term, (0.1)∂u∂t=div(|u|p−1∇u)+Ru,under the Ricci flow on a complete manifold M such that M×R2 has bounded curvature and nonnegative isotropic curvature, where 0<p<1 and R=R(x,t) is the evolving scalar curvature of M at time t. We prove Aronson–Bénilan and Li–Yau–Hamilton type differential Harnack estimates for positive solutions of (0.1). In addition, we use similar method to prove certain Li–Yau–Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in Cao and Hamilton (2009), Cao (2008), and Kuang and Zhang (2008) to the noncompact setting.

Four-Dimensional Gradient Shrinking Solitons with Positive Isotropic Curvature

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of |$\mathbb{S}^4$| or a quotient of |$\mathbb{S}^3 \times \mathbb R $|⁠. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with non-negative isotropic curvature.

Lower bound of Ricci flow's existence time

Let ( M n , g ) be a compact n-dimensional ( n ⩾ 2 ) manifold with nonnegative Ricci curvature, and if n ⩾ 3 , then we assume that ( M n , g ) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on ( M n , g ) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for n = 3 under Rc ⩾ 0 assumption among others. Combining these results, we proved the short-time existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

On the Structure of Submanifolds in Euclidean Space with Flat Normal Bundle

In this paper we study the structure of an immersed submanifold Mn in a Riemannian manifold with flat normal bundle in two ways. Firstly, we prove that if Mn is compact and satisfies some pointwise pinching condition, and assume further that the ambient space has pure curvature tensor and non-negative isotropic curvature, then the Betti numbers βp(M) = 0 for 2 ≤ p ≤ n−2. Secondly, suppose that Mn is a complete non-compact submanifold in the Euclidean space with finite total curvature in the sense that its traceless second fundament form has finite Ln-norm, then we show that the spaces of L2 harmonic p-forms on Mn have finite dimensions for all 2 ≤ p ≤ n−2.

Mean value inequalities and conditions to extend Ricci flow

This paper concerns conditions related to the rst nite singularity time of a Ricci ow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality method, suggested by N. Le (13) and F. He (11). We also display a close connection between this method and time slice analysis as in (23). As an application, we prove several inequalities for a Ricci ow solution on a manifold with nonnegative isotropic curvature.

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators \(C(S)\), which are nonnegative in a suitable sense, to every \(Ad_{SO(n,\mathbb{C })}\) invariant subset \(S \subset \mathbf{so}(n,\mathbb{C })\). In this article we show that if \(S\) is an \(Ad_{SO(n,\mathbb{C })}\) invariant subset of \(\mathbf{so}(n,\mathbb{C })\) such that \(S\cup \{0\}\) is closed and \(C_+(S)\subset C(S)\) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in \(C_+(S)\) also admits a metric with curvature operator in \(C_+(S)\) (b) The normalized Ricci flow on any compact Riemannian manifold \(M\) with curvature operator in \(C_+(S)\) converges to a metric of constant positive sectional curvature. We also point out that if \(S\) is an arbitrary \(Ad_{SO(n,\mathbb{C })}\) subset, then \(C(S)\) is contained in the cone of curvature operators with nonnegative isotropic curvature.

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.

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.

Einstein manifolds with nonnegative isotropic curvature are locally symmetric

Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.

On stable minimal disks in manifolds with nonnegative isotropic curvature

On stable minimal disks in manifolds with nonnegative isotropic curvature

Almost-Einstein manifolds with nonnegative isotropic curvature

Almost-Einstein manifolds with nonnegative isotropic curvature

Isotropic Curvature and the Ricci Flow

In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.

A generalization of Hamilton's differential Harnack inequality for the Ricci flow

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R2 has non-negative isotropic curvature.

Compact Hermitian surfaces and isotropic curvature

It is shown that on a Kahler surface the non-negativity (resp. non-positivity) of the isotropic curvature is implied by the non-negativity (resp. non-positivity) of the holomorphic bisectional curvature. The compact Hermitian surfaces of non-negative isotropic curvature are described. The full list of compact half conformally flat Hermitian surfaces of non-positive isotropic curvature is also given.

Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.

Metrics with nonnegative isotropic curvature

Metrics with nonnegative isotropic curvature

Manifolds with nonnegative isotropic curvature

We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature,then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.