arXiv research

Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.

Study on a deformed Hermitian-Yang-Mills equation on compact Kähler manifolds.

Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.

Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.

Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.

Paper solves a complex equation for smooth domains.

Study disproves conjecture about Hermitian-Yang-Mills solutions.

The paper proves conditions for solutions of $J$-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.

New examples of deformed Hermitian-Yang-Mills connections found.

Paper proves solvability condition for complex equation on special submanifolds.

Solves supercritical dHYM on projective manifolds with specific conditions.

Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.

Study on a nonlinear elliptic equation on compact Hermitian manifolds.

Proves existence and uniqueness of weak solutions for specific equations.

The paper confirms the solvability of a complex equation for a 4D manifold.

Proves solvability of general inverse σ_k equations with constant coefficients.

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\…

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hatθ \in (\fracπ{2},\frac{3π}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path …

We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…

Paper confirms conjecture for projective manifolds in supercritical phase.

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 the equation, and some recent progress towards its solution. In dimension 3 we …

Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.

Paper establishes estimates for nonlinear equations on compact manifolds.

Paper generalizes sub-slope definition and solves complex equations on compact manifolds.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

Introduces a new PDE involving differential forms for Kähler geometry.

The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.

The note proves a metric equivalence for stable bundles on surfaces.

The paper finds explicit instantons on a specific 6-manifold.

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angl…

The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.

We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…

The paper examines inequalities for Chern numbers on specific 4D Kähler manifolds.

The paper studies conditions for graphs connecting level sets of harmonic polynomials.

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM eq…

The paper studies deformations of Hermitian Yang-Mills and Donaldson-Thomas connections on $G_2$-manifolds.

The paper studies singularities in a complex flow related to mean curvature.

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that th…

In this paper, we prove that for any Kähler metrics $ω_0$ and $χ$ on $M$, there exists $ω_\varphi=ω_0+\sqrt{-1}\partial\bar\partial\varphi>0$ satisfying the J-equation $\mathrm{tr}_{ω_\varphi}χ=c$ if and only if $(M,[ω_0],[χ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler met…

Proves stability of certain vector bundles on Kähler surfaces.

We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solom…

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In…

Let $L$ be a holomorphic line bundle over a compact Kähler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that $X$ is Calabi-Yau. We show that this equation is the Euler-Lagrange equ…

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vect…

Let $(X,ω)$ be a compact Kähler manifold of complex dimension $n$ and $(L,h)$ be a holomorphic line bundle over $X$. The line bundle mean curvature flow was introduced in \cite{JY} in order to find deformed Hermitian-Yang-Mills metrics on $L$. In this paper, we consider the stability of the line bundle mean curvature f…