Related papers: A special Lagrangian type equation for holomorphic…
In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a K\"{a}hler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable K\"{a}hler current with singularities in higher…
We derive a generalization of the flat space Yang's and Newman's equations for self-dual Yang-Mills fields to (locally) conformally Kahler Riemannian 4-manifolds. The results also apply to Einstein metrics (whose full curvature is not…
In this paper, we study the hypercritical deformed Hermitian-Yang-Mills equation on compact K\"ahler manifolds and resolve two conjectures of Collins-Yau.
We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to…
The K\"ahler-Yang-Mills equations are coupled equations for a K\"ahler metric on a compact complex manifold and a connection on a complex vector bundle over it. After briefly reviewing the main aspects of the geometry of the…
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of…
We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with…
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K\"ahler manifolds and we present…
In this work, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show…
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost K\"ahler manifold. The short time existence of the corresponding parabolic flow was…
For a given closed two-form, we introduce the cone Yang-Mills functional which is a Yang-Mills-type functional for a pair $(A,B)$, a connection one-form $A$ and a scalar $B$ taking value in the adjoint representation of a Lie group. The…
We prove that any complete non-compact K\"ahler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all…
Let $(X,\alpha)$ be a K\"ahler manifold of dimension n, and let $[\omega] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic…
By using the global deformation of almost complex structures which are compatible with a symplectic form off a Lebesgue measure zero subset, we construct a (measurable) Lipschitz Kahler metric such that the one-form type Calabi-Yau equation…
Consider a vector bundle over a K\"ahler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the K\"ahler manifold at a collection of points also admits a Hermitian Yang-Mills…
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K\"ahler manifolds and we present…
In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds. Let $ (E,\theta) $ be a Higgs bundle over a compact Hermitian manifold $(M,\omega_g) $. Suppose that there exists a…
This paper investigates the variational problems associated with the $L^2$-norms of the normal and tangent curvature tensors for submanifolds immersed in a unit sphere. We define the critical points of these functionals under normal…
The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in…
In this note we briefly present the results of our computation of special K\"ahler geometry for polynomial deformations of Berglund-H\"ubsch type Calabi-Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that…