Related papers: The harmonic $2$-forms on $K3$ surfaces converging…

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-K\"ahler metric. One motivation for this paper is the role of such…

Let $X$ be a closed, oriented four-manifold with $b_2^+ \leq 3$, and suppose $X$ contains a collection of pairwise disjoint embedded $(-2)$-spheres. We prove that there is a Riemannian metric on $X$ such that the Poincare dual of each of…

In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a 3-manifold $M$ and prove the orbits of the constrained Hamiltonian dynamical system correspond to $G_2$-manifolds foliated by…

The author has proved that a crepant resolution Y of a Ricci-flat K\"{a}hler cone X admits a complete Ricci-flat K\"{a}hler metric asymptotic to the cone metric in every K\"{a}hler class in H^2_c(Y,\R). These manifolds are generalizations…

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the…

Oeljeklaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $Q$ and a subgroup of units $U$. We characterize the existence of pluriclosed…

This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of…

We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally…

In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By…

We present a method in nonlinear elliptic systems to study curvature decays on asymptotically locally Euclidean (ALE) manifolds. In particular, we show that scalar flat Kahler and harmonic ALE metrics of real dimension n are of order n-2.

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the…

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on…

A new construction is presented of scalar-flat Kaehler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable…

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set,…

We find all hyper-K\"ahler 4-manifolds admitting conformal K\"ahler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual…

After a review of symmetries and classical solutions involved in the AdS3/CFT2 correspondence, we apply a similar analysis to asymptotically flat spacetimes at null infinity in 3 and 4 dimensions. In the spirit of two dimensional conformal…

We construct all axi-symmetric non-gradient $m$-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to $m=2$, as well as a family of new regular metrics…

We show that simply connected toric hyperK\"ahler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In…

Assume $(X, \omega)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $\mu$. We prove that a finite energy symplectic vortex exponentially…