Related papers: Pinched self-dual Weyl curvature on Einstein four-…

We consider compact oriented four-manifolds with harmonic self-dual Weyl curvature in addition to a pinching condition.

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article,…

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) K\"ahler-Einstein, provided that $\lambda_2 \geq -\frac{S}{12}$, where $\lambda_2$ is the middle…

We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally K\"ahler if and only if the determinant of the self-dual Weyl curvature is positive.

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…

We show that a compact almost-K\"ahler four manifold $(M, g, \omega)$ with harmonic self-dual Weyl curvature and constant scalar curvature is K\"ahler if $c_{1}\cdot\omega\geq 0$. We also prove an integral curvature inequality for compact…

We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including…

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of…

It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need…

In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of…

The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A…

We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new…

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the self-dual part of Weyl tensor, then it will be shown that there is no metric $g$ on $S^2 \times…

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and…

The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature.…

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact…

Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The…

We show the existence of strictly almost-Kahler anti-self-dual metrics on certain 4-manifolds by deforming scalar-flat Kahler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of…

We show that a compact Riemannian manifold with weakly 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form.