Related papers: (1,1) forms with specified Lagrangian phase: A pri…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian…

Given a (possibly non-K\"ahler) Calabi--Yau threefold $(X,\Omega)$, we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of $(X,\omega,\Omega)$, where $\omega$ is a Hermitian metric on $X$. The equations defining…

In this paper, we establish a doubling argument to obtain Hessian estimates for the special Lagrangian equation under general phase constraints. In particular, our approach does not rely on the Michael-Simon mean value inequality. As an…

Let L be a Lagrangian submanifold of a pseudo- or para-K\"ahler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect…

Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost K\"ahler manifold.…

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…

The Lagrange-d'Alembert equations with constraints belonging to $H^{1,\infty}$ have been considered. A concept of weak solutions to these equations has been built. Global existence theorem for Cauchy problem has been obtained.

In this paper, we consider the homogeneous complex k-Hessian equation in $\Omega\backslash\{0\}$. We prove the existence and uniqueness of the $C^{1,\alpha}$ solution by constructing approximating solutions. The key point for us is to…

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{5}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}\delta(2,2)\leq…

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic…

We give sharp $C^{2,\alpha}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous…

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary…

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric…

In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact K\"ahler manifolds to allow for the twisting function to be non-constant and slightly negative in all…

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to…

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 exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current…

We introduce a (variation of quadrics) ansatz for constructing explicit, real-valued solutions to broad classes of complex Hessian equations on domains in $\mathbb{C}^{n+1}$ and real Hessian equations on domains in $\mathbb{R}^{n+1}$. In…

Let $(M,I, \Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi:\; M \mapsto X$, and $\eta$ a closed form of Hodge type (1,1)+(2,0) on $X$. We prove that $\Omega':=\Omega+\pi^* \eta$ is…

A Hilbert manifold structure is described for the ADM phase space of asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev regularity. Solutions of the constraint equations form a Hilbert submanifold. A regularized…