Twistor Bundle of a Neutral Kähler Surface

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.

On a topology property for the moduli space of Kapustin–Witten equations

In this article, we study the Kapustin–Witten equations on a closed, simply connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use Taubes’ compactness theorem [C. H. Taubes, Compactness theorems for {\mathrm{SL}(2;\mathbb{C})} generalizations of the 4-dimensional anti-self dual equations, preprint 2014, https://arxiv.org/abs/1307.6447v4] to prove that if {(A,\phi)} is a smooth solution to the Kapustin–Witten equations and the connection A is closed to a generic ASD connection {A_{\infty}}, then {(A,\phi)} must be a trivial solution. We also prove that the moduli space of the solutions to the Kapustin–Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all generic. At last, we extend the results for the Kapustin–Witten equations to other equations on gauge theory such as the Hitchin–Simpson equations and the Vafa–Witten on a compact Kähler surface.

A mean-curvature flow along a Kähler–Ricci flow

Let [Formula: see text] be a Kähler surface, and [Formula: see text] an immersed surface in [Formula: see text]. The Kähler angle of [Formula: see text] in [Formula: see text] is introduced by Chern and Wolfson [Am. J. Math. 105 (1983) 59–83]. Let [Formula: see text] evolve along the Kähler–Ricci flow, and [Formula: see text] in [Formula: see text] evolve along the mean-curvature flow. We show that the Kähler angle [Formula: see text] satisfies the evolution equation [Formula: see text] where [Formula: see text] is the scalar curvature of [Formula: see text]. The equation implies that if the initial surface is symplectic (Lagrangian), then, along the flow, [Formula: see text] is always symplectic (Lagrangian) at each time [Formula: see text], which we call a symplectic (Lagrangian) Kähler–Ricci mean-curvature flow. In this paper, we mainly study the symplectic Kähler–Ricci mean-curvature flow.