arXiv subjects
Publications and source records attributed to Jiaogen Zhang.
In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator satisfies a structural Condition $T$, which encompasses all $O(n)$-invariant Gårding-Dirichlet operator, such prescribed curvature problems are always solvable.
In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-Λu \quad \textrm{in} \, Ω, \qquad u=0 \quad \textrm{on} \, \partial Ω. \] These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $Γ$-admissible eigenfunction on the smooth, strictly $Γ$-convex domain $Ω\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(Ω) \cap C^{1,1}(\overlineΩ)$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.
This article explores the relationship between Garding cones, demonstrating that the shift cone $\overlineΓ^{+}_{2}(α)$ is contained in $\overline{\mathcal{P}}_{m}$. By combining these results with the study of positivity properties of curvature operators, we establish several new connections between algebraic positivity conditions and the geometry of underlying Riemannian manifolds. Our main theorems reveal how shifted cone conditions on curvature operators-both standard and of the second kind-constrain topology, including vanishing theorems for Betti numbers and characterizations of spherical space forms.
The quaternionic Calabi conjecture, posed by Alesker and Verbitsky \cite{Alesker-Verbitsky (2010)}, predicts that the quaternionic Monge-Ampère equation can always be solved on any compact HKT manifold. Motivated by this conjecture, we will introduce a quaternionic version of the Gauduchon conjecture on any compact $SL(n,\mathbb{H})$-manifold, specifically addressing the existence of quaternionic Gauduchon metrics with prescribed volume form. We reframe this question as a special case of fully nonlinear elliptic equations of second order and subsequently establish a uniform estimate for the potential function.
Our recent work about fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metric is hereby extended to the parabolic setting. This approach will help us to study some problems arising from hyperhermitian geometry.
The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $Ω\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Ampère equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Ampère equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercritical phase. At last, we prove the existence of deformed Hermitian-Yang-Mills equation and complex Hessian quotient equations under supersolutions.
Let $Ω\subseteq M$ be a bounded domain with a smooth boundary $\partialΩ$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on $Ω$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.
Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to the hypercomplex setting, we prove some a priori estimates for solutions to such equations under the assumption of existence of $\mathcal{C}$-subsolutions. In the estimate of the quaternionic Laplacian, we need to further assume the existence of a flat hyperkähler metric. As an application of our results we prove that the quaternionic analogue of the Hessian equation and Monge-Ampère equation for $(n-1)$-plurisubharmonic functions can always be solved on compact flat hyperkähler manifolds.
In this paper, we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.
Let $(M,g)$ be a closed Riemannian $4$-manifold and let $E$ be a vector bundle over $M$ with structure group $G$, where $G$ is a compact Lie group. In this paper, we consider a new higher order Yang--Mills--Higgs functional, in which the Higgs field is a section of $Ω^0(\textmd{ad}E)$. We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that $E$ is a line bundle, we are able to use a different blow up procedure and obtain an improvement of the long time result in \cite{Z1}. The proof is rather relevant to the properties of the Green function, which is very different from the previous techniques in \cite{Ke,Sa,Z1}.
In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge--Ampère equation for $(n-1)$-plurisubharmonic equations in the almost Hermitian setting.
In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.