Related papers: The Fu-Yau equation on compact astheno-K\"ahler ma…

In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed K\"ahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which…

Transversally K\"ahler foliations are a generalisation of K\"ahler manifolds, appearing naturally in the complex non-K\"ahler setting. We give a self-contained proof of how the classical methods used in the proof of the Aubin-Yau theorem…

The aim of this paper is to further develop the theory of the degenerate complex Hessian equations on compact Hermitian manifolds. Building upon the generalization of the Bedford-Taylor pluripotential theory to complex Hessian equations by…

We establish a C^0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version…

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact K\"ahler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved…

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

Let $X$ be a normal compact K\"ahler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the…

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…

This survey explores a range of classical findings and recent developments related to our understanding of astheno-K\"ahler manifolds. Furthermore, we provide various examples of astheno-K\"ahler manifolds and analyze the challenges…

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact complex three-folds conditioned on a…

We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.

On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a K\"ahler metric with semi-negative holomorphic sectional curvature.…

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture…

Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of elliptic equations on compact hyperhermitian manifolds. By adapting the approach of Sz\'ekelyhidi to the hypercomplex setting, we prove some a priori estimates…

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with…

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian…

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar…

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…

We consider a complex Plateau problem for strongly pseudoconvex contours in non K\"ahler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a…

In this note, We proved that a compact Sasakian manifold $( M, {\xi}, {\eta}, {\Phi} , g )$ with negative transverse holomorphic sectional curvature must have has a Sasakian structure $( {\xi}, {\eta} , {\Phi} , g )$ with negative…