Related papers: A Gap Theorem for Half-Conformally Flat Manifolds

We generalize to nearly K\"ahler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly K\"ahler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly K\"ahler manifold of any…

We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic $2$-form which tames the underlying almost complex structure.

Let $(M,\omega)$ be a geometrically bounded symplectic manifold, $N\subseteq M$ a closed, regular (i.e. "fibering") coisotropic submanifold, and $\phi:M\to M$ a Hamiltonian diffeomorphism. The main result of this article is that the number…

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,…

Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-K\"ahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the K\"ahler case. The key idea is…

We study rigidity on certain K\"ahler manifolds with nonnegative Ricci curvature. Among others things, we show that a complete noncompact K\"ahler surface with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature…

Let $(M,J,g,\omega)$ be a complete Hermitian manifold of complex dimension $n\ge2$. Let $1\le p\le n-1$ and assume that $\omega^{n-p}$ is $(\partial+\overline{\partial})$-bounded. We prove that, if $\psi$ is an $L^2$ and $d$-closed…

For a compact K\"ahler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic…

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with singularities conjugate to ADE-type is proved. In 1988, Claude Lebrun gave examples of scalar-flat K\"ahler ALE metrics with negative mass, on the…

Let $M$ be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number $b_2$ in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in…

Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat K\"ahler metric. Then if M is blown up at…

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally…

In complex dimensions $\geq 3$, we provide a geometric existence for generalized ALG complete non-compact Ricci flat K\"ahler manifolds with Schwartz decay i.e. metric decay in any polynomial rate to an ALG model $\mathbb{C}\times Y$ modulo…

It is known that the overlap of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an…

We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally…

The underlying complex structure of an ALE K\"ahler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE K\"ahler…

We relate the dimensions of $L^p$ reduced cohomology spaces in degree k of an ALE manifold to the dimension of some spaces of decaying harmonic forms, depending both on p and on k. In this class of manifolds, this provides an extension to…

Let $(M,g)$ be a compact Ricci-flat 4-manifold. For $p \in M$ let $K_{max}(p)$ (respectively $K_{min}(p)$) denote the maximum (respectively the minimum) of sectional curvatures at $p$. We prove that if $$K_{max} (p) \le \ -c K_{min}(p)$$…

Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass…

In this note we prove the following result: There is a positive constant $\epsilon(n,\Lambda)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $\Lambda$, diameter…