Related papers: Hessian equations on closed Hermitian manifolds

In this paper, we derive the second order estimate to the $2$-nd Hessian type equation on a compact almost Hermitian manifold.

Let $(M,\omega)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $\omega$. $\ac$ has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit…

We derive second order estimates for $\chi$-plurisubharmonic solutions of complex Hessian equations with right hand sides depending on gradients on compact Hermitian manifolds.

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily K\"ahler, we solve the Hermitian-Einstein equation on analytically stable…

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$-sh function can be approximated from above by smooth $(\omega,m)$-sh functions. A…

We give a partial account of some problems concerning cohomological invariants and metric properties of complex non-K\"ahler manifolds.

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an…

Our recent work about fully non-linear elliptic equations on compact manifolds with a flat hyperk\"ahler metric is hereby extended to the parabolic setting. This approach will help us to study some problems arising from hyperhermitian…

In this paper, we shall study the existence of weak solutions to Hessian type equations on compact Riemannian manifolds without boundary.

In this paper, we prove new rigidity results related to some generalised Ricci-Hessian equation on Riemannian manifolds.

We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an…

In this paper, we investigate the $\partial$-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in $\mathbb{C}^n$,…

We study generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.

A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated K\"ahler manifolds. The solutions to these systems arise from sets of geodesic second order non-linear differential…

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the…

In this paper, we prove the existence of a solution for the exterior Dirichlet problem for Hessian equations on a non-convex ring. Moreover, the solution we obtained is smooth. This extends the result of [Bao-Li-Li, ``On the exterior…

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type…

This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by C. LeBrun in…

We consider a class of fully nonlinear second order elliptic equations on Hermitian manifolds closely related to the general notion of $\bfG$-plurisubharmonicity of Harvey-Lawson and an equation treated by Sz\'ekelyhidi-Tosatti-Weinkove in…

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such…